PHẦN V: BIẾN ĐỔI LƯỢNG GIÁC · tÀi liỆu hỌc tẬp toÁn 10 – chẤt lƯỢng bÀi giẢng: biẾn ĐỔi lƯỢng giÁc admin trẦn vĂn tÀi – 0977.413.341 -

TÀI LIỆU HỌC TẬP TOÁN 10 – CHẤT LƯỢNG BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC

ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC–TRUNG–NAM 1 | THBTN

sintan

cos

. coscot

sin

.

2 2sin cos 1 . tan .cot 1 .

2

2

11 tan

cos

. 2

2

11 cot

sin

.

BA I TÂP A P DUNG

Bài1. Chưng minh ca c đăng thư c sau

1/ 2 2 2cos x sin x 1 2 sin x . 2/ 2 22cos x 1 1 2 sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 2 23 4 sin x 4 cos x 1 . 4/ sin x cot x cos x tan x sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 4 4 2 2sin x cos x 1 2 sin x cos x . 6/ 4 4 2 2cos x sin x cos x sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 24 cos x 3 1 2 sin x 1 2 sin x . 8/ 2 2 21 cosx sin x cosx cos x sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

PHẦN V: BIẾN ĐỔI LƯỢNG GIÁC

A – HỆ THỨC LƯỢNG GIÁC CƠ BẢN

TÀI LIỆU HỌC TẬP TOÁN 10- CHẤT LƯỢNG BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC

2 | THBTN – CA FILE WORD LIÊN HỆ - THẦY TÀI: 0977.413.341

...................................................................................................................................................

...................................................................................................................................................

9/ 4 4 2 2sin x cos x 1 2cos x 2sin x 1 . 10/ 3 3sin x cos x sin x cos x sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 2 2 2 2tan x sin x tan x sin x . 12/ 2 2 2 2cot x cos x cot x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài2. Chưng minh cac đăng thư c sau

1/ 1tan x cotx

sin x cos x . 2/ 1 cos x sin x

sin x 1 cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 1 11

1 tan x 1 cotx

. 4/ 21 1

1 1 tan x 0cos x cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 2

2

2

1 sin x1 2 tan x

1 sin x

. 6/ tan x tan y

tan x tan ycotx cot y

.

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 4

2 4

2 11 cot x

sin x sin x . 8/ cos x 1

tan x1 sin x cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 2 11 cos x 1 cot x

1 cos x

. 10/ 1 cos x 1 cos x 4 cot x

1 cos x 1 cos x sin x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ sin x 1 cos x 21 cos x sin x sin x

. 12/ sin x cos x 1 cos x

sin x cos x 1 1 sin x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 2

2

sin x 2cos x 1 cos x1 cos x2 cos x cos x

. 14/ sin x cos x 1 2cos x

1 cos x sin x cos x 1

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................




1/ 6 6 2 2sin x cos x 1 3 sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 6 6 2 2 2 2sin x cos x sin x cos x 1 sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 28 8 2 2 4 4sin x cos x 1 2 sin x cos x 2 sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 8 8 2 2 2 2sin x cos x sin x cos x 1 2sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ 1 sin x cos x tan x 1 cos x 1 tan x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 1 tan x 1 cot x sin x cos x 1 2 sin x cos x .

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 22 21 tan x cos x 1 cotx sin x sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 2 2sin x tan x cos x cot x 2 sin x cos x tan x cot x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 2 2 2 2 2sin x tan x 4 sin x tan x 3 cos x 3 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ 2 2sin cos tan 1

1 2 sin cos tan 1

. 2/ 2 2

6

2 2

tan sintan

cot cos

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 2 2cos sin

1 sin cos1 tan 1 cot

. 4/

3

tan sin 1

sin cos 1 cos

.



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 2 2

2 2

1tan cot 2

sin cos

. 6/

22

2 2

1 3 tantan 1

cos cos

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 2 2 2 2

2 2 2 2

tan tan sin sin

tan tan sin sin

. 8/

22

1 cos1 cos1 2 cot

sin sin

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 2 11 cos 1 cot

1 cos

. 10/ 2 3

3

sin cos1 tan tan tan

cos

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 2

2

sin cos 1 cotsin cos cos sin 1 cot

. 12/

2 2 4

2 2 2 2

tan 1 cot 1 tan.

1 tan cot tan cot

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



13/ 2

21 sin 1 sin4 tan

1 sin 1 sin

. 14/

2

21 cos 1 cos4 cot

1 cos 1 cos

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài6. Ru t gon cac biêu thư c sau

1/ 4 2 21

P sin sin cos . 2/ 4 4 22

P sin cos cos .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 2 2 23

P sin sin cot . 4/ 2 2 24

P cos cos cot .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 2 2 25

P 1 sin cot 1 cot . 6/ 2 26

P sin tan cos cot 2sin cos .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 2

7

2cos 1P

sin cos

. 8/ 8 2

1 cos 1P

1 cossin

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 9

cot cotP

tan tan

. 10/ 10

cosP tan

1 sin

.

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 11

sin x tan xP sin x cotx

tan x

. 12/ 12 2

cos x tan xP cot x cos x

sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 2 2

13 2 2

cos x cot xP

sin x tan x

. 14/

2 22

14 2

1 sin x cos xP cos x

cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

15/ 3 2

15

sin x sin x cos x cos xP

1 2 sin x cos x

. 16/

216

sin x cos x 1P

tan x sin x cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

17/ 2 2 2

17 2 2 2

sin x sin x tan xP

cos x cos x tan x

. 18/

2 22

18 2

cot x cos xP 1 cos x

cot x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



19/ 2 2

2 219 2 2

cos x sin yP cot x cot y

sin x sin y

. 20/ 20

1 sin x cos x cotxP

sin x 1 cotx 1 tan x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

21/ 21

1 cosx 1 sinxP tanx cotx

sinx cosx

. 22/ 22

1 1P 1 cotx 1 cotx

sinx sinx

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

23/ 2 223P 1 tanx cos x 1 cotx sin x . 24/ 24 2 2

1P , 0 x

2sinx cot x cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài7. Biên đô i ca c biê u thưc sau thanh ti ch sô

1/ 2A 2cos x 1 .

...................................................................................................................................................

2/ 2B 3 4 sin x .

...................................................................................................................................................

3/ 2C sinx cos x cos x 1 .

...................................................................................................................................................

4/ 2D sin x sin x cos x 1 .

...................................................................................................................................................



5/ E 1 sin x cos x tan x .

...................................................................................................................................................

6/ F tan x cot x sin x cos x .

...................................................................................................................................................

7/ 2G cos x tan x 1 cos x .

...................................................................................................................................................

8/ 2H 3 4cos x sin x 2 sin x 1 .

...................................................................................................................................................

...................................................................................................................................................

9/ 2 2I sin x 3 cos x 6 cos x 2 sin x .

...................................................................................................................................................

...................................................................................................................................................

10/ 3 3J cos x sin x sinx cos x .

...................................................................................................................................................

...................................................................................................................................................

11/ 3 2K cos x cos x 2sin x 2 .

...................................................................................................................................................

...................................................................................................................................................

12/ 2 3L cos x sin x cos x .

...................................................................................................................................................

13/ 2M 1 cos x cos x sin x 1 cos x .

...................................................................................................................................................

14/ 3 2N 2cos x 2 cos x sin x 1 .

...................................................................................................................................................

15/ 3 3 2O cos x sin x 2sin x 1 .

...................................................................................................................................................

16/ 2P 2sin x 1 2cos x 2 sin x 1 3 4cos x .

...................................................................................................................................................



17/ Q 2cos x 1 sin x cos x 1 .

...................................................................................................................................................

18/ 3 3 2R 4sin x 3cos x 3sin x sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

19/ 2S 1 sin x tan x 1 cos x .

...................................................................................................................................................

20/ 2T 2 5sin x 3 1 sin x tan x .

...................................................................................................................................................

21/ 2U 2sinx cos x 2sin x 3sinx cos x 1 .

...................................................................................................................................................

22/ V tan x 3 cot x 4 sin x 3 cos x .

...................................................................................................................................................

...................................................................................................................................................

23/ X 3 sin x 2 cos x 3 tan x 2 .

...................................................................................................................................................

24/ Y 2 tan x sin x 3 cot x cos x 5 .

...................................................................................................................................................

25/ Z 3 cot x cos x 5 tan x sin x 2 .

...................................................................................................................................................

...................................................................................................................................................

Bài8. Chưng minh ca c biêu thưc sau không phu thuô c va o biê n x

1/ 4 4 2A cos x sin x 2sin x .

...................................................................................................................................................

...................................................................................................................................................

2/ 4 2 2 2B sin x sin x cos x cos x .

...................................................................................................................................................

...................................................................................................................................................



3/ 4 2 2 2C cos x sin x cos x sin x .

...................................................................................................................................................

...................................................................................................................................................

4/ 4 2 4 2D cos x 2 cos x 3 sin x 2 sin x 3 .

...................................................................................................................................................

...................................................................................................................................................

5/ 6 6 4 4 2E sin x cos x 2sin x cos x sin x .

...................................................................................................................................................

...................................................................................................................................................

6/ 1 1F sin x. , 0 x

1 cos x 1 cos x 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 4 2 4 2G sin x 4 cos x cos x 4 sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ 2 2 2 2 2H cos x cot x 5cos x cot x 4 sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 3 3I 1 cotx sin x 1 tan x cos x sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

10/ 4 4 2 2J sin x cos x 1 tan x cot x 2 .



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 8 8 6 6 4K 3 sin x cos x 4 cos x 2 sin x 6 sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

12/ 4 2 4 2 2 2L sin x 1 sin x cos x 1 cos x 5 sin x cos x 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 2

4 4 2 2 8 8M 2 sin x cos x sin x cos x sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài9. Chưng minh cac biêu thưc sau không phu thuô c va o biê n x

1/ 2 cotx 1A

tan x 1 cot x 1

.

...................................................................................................................................................

...................................................................................................................................................

2/ 22

2 2 2

1 tan x 1B

4 tan x 4 sin x cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



3/ 2

2 21 tan xC 1 tan x 1 cot x

tan x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 22

2 2 2

1 cot x 1D

cot x sin x cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 6 2

6 2

1 sin x 3 tan xE

cos x cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ 2 2 2 2

2 2

tan x cos x cot x sin xF

sin x cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 2 2

2

cot x cos x sin x cos xG

cot xcot x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ 4 4

6 6

sin x cos x 1H

sin x cos x

.



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



1. Đường tròn lượng giác và dấu góc lượng giác

2.Cung liên kết

Cung đối nhau Cung bù nhau Cung phụ nhau

cos( ) cosa a sin( ) sina a sin cos2

a a

sin( ) sina a cos( ) cosa a cos sin2

a a

tan( ) tana a tan( ) tana a tan cot2

a a

cot( ) cota a cot( ) cota a cot tan2

a a

Cung hơn kém Cung hơn kém 2

sin( ) sina a sin cos2

a a

cos( ) cosa a cos sin2

a a

tan( ) tana a tan cot2

a a

cot( ) cota a cot tan2

a a

2π0

O

-1

-1

1

1

3π

2

π

π

2

sinx

cosx

(IV)(III)

(II) (I)

B – ĐƯỜNG TRÒN LƯỢNG GIÁC – CUNG GÓC LƯỢNG GIÁC

CUNG LIÊN KẾT

Cung phần tư Giá trị LG

I II III IV

+ + – –

+ – – +

+ – + –

+ – + – (Nhất cả – Nhì sin – Tam tan – Tứ cos)



BA I TÂP A P DUNG

MÔ I LIÊN HÊ GIƯA ĐƠN VI ĐÔ VA RADIAN

Bài10. Đô i sô đo ca c cung sau ra đô , phut, giây:

2 3 5 7 13 3 15, , , , , , , 2 , ,

3 4 9 5 3 3 4 7 15

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài11. Đô i sô đo ca c goc sau ra radian rad :

0 0 0 0 0 0 0 0 090 , 36 , 15 , 72 , 270 , 240 , 540 , 750 , 21 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài12. Đô i sô đo đô cua ca c cung tron sau thanh sô đo radian (chinh xac đê n phâ n nghin): 0 0 021 30 ', 75 54 ', 12 36 ' 0 0 0, 15 30 ', 105 45 ' 30 '', 27 38 ' 49 '' .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài13. Điê n cac gia tri thich hơ p vao ô trô ng

00 015 030 045 060 075 090 0120 0150 0180

16

8

6 5

12

2 2

3 3

4 5

6

43



Radian 74

12 5

6

2 2

3 3

4

Đô 0135 050 0810

TINH ĐÔ DAI CUNG TRON

Bài14. Môt đương tron co ba n kinh 10 cm . Tinh đô dai ca c cung co sô đo la

0 02 3 7 13 3 3, , , , , 2 , , 45 , rad , 75

3 18 5 3 6 4 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài15. Kim giơ cua đô ng hô da i 7 cm , kim phut da i 10 cm . Ti nh quang đương kim phut,

kim giơ đi đươc trong 30 phut.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài16. Ba nh xe co đương kinh (tinh ca lô p) la 55 cm . Nêu xe chay vơ i vân tô c /40 km h

thi trong môt giây banh xe quay đươc bao nhiê u vo ng ?

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài17. Kim phut va kim giơ cu a đô ng hô lơ n nha Bưu Điê n Ha Nô i theo thư tư dai

1,75 m va 1,26 m . Ho i trong 15 phut, mu i kim phut va mui kim giơ vach trên

cung tron co đô dai băng bao nhiêu met ?

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

BIÊ U DIÊ N NGON CUNG LƯƠNG GIAC

Bài18. Biêu diê n cac goc sau lên đương tron lươ ng giac gô c A: 0 0 0 0 030 , 45 , 120 , 120 , 330 ,

0 0 0 0 07 4 15 2010630 , 750 , 1250 , , , 750 , 1125 , ,

4 3 2 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài19. Xa c đi nh điê m cuô i cua cung co sô đo: k2 , k , k , k , k , k2 3 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài20. Xa c đi nh điê m ngon cua ca c ho nghiê m sau đây trên đươ ng tron lương giac vơ i k .

1/ x k . 2/ x k22

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



3/ x 4k . 4/ x k3

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ x k26

. 6/ x k4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ kx

3

. 8/ k2x

3 3

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ k3x

4 4

. 10/ kx

6 3

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài21. Cho hai điêm M va N sao cho đs AM6

va đ ks AN

798

vơ i k . Ti m k đê :

1/ M tru ng vơi N. 2/ M đôi xưng vơ i N qua tâm O.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



Bài22. Hai goc lương giac co sô đo radian 353 va m

5 vơ i m la sô nguyên co thê co cung

tia đâu va tia cuô i hay không ?

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

GIA TRI LƯƠNG GIAC CUA MÔ T CUNG (GOC)

Bài23. Xa c đi nh dâ u cua ca c gia tri lươ ng giac hoă c biêu thưc

1/ sin x, cos x, tan x, cot x vơ i 3x

2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 3sin x , cos x , tan x , cot x

4 2 2 2

vơ i 0 x

2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 0 0A sin 40 .cos 290 . 4/ 0 0B sin 25 .cos170 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 0 0 0C sin225 . tan130 .cot 175 . 6/ 0 0 0D cos195 . tan269 .cot 98 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



7/ 0 0E sin50 .cos 300 . 8/ 0 21F sin 215 . tan

7

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 3 2G cot .sin

5 3

. 10/ os 4 4 9

H c .sin .tan .cot5 3 3 5

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài24. Ti nh gia tri con la i cua go c x, biêt

1/ 1sin x

2 vơ i 0 090 x 180 . 2/ 4

sin x5

vơ i 0 0270 x 360 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 3sin x

5 vơ i 3

x2

. 4/ 1cos x

4 vơ i 0 x

2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 3cos x

5 vơ i 00 x 90 . 6/ 5

cos x13

vơ i 0 0180 x 270 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 2cos x

5 vơ i x 0

2

. 8/ 4cos x

5 vơ i 0 0270 x 360 .



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 5sin x

13 vơ i x

2 . 10/ 1

sin x3

vơ i 0 0180 x 270 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ tan x 3 vơ i 3x

2

. 12/ tan x 2 vơ i x2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 1tan x

2 vơ i x

2 . 14/ cotx 3 vơ i 3

x2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

15/ 3tan x

4 vơ i 3

x2

. 16/ tan x 2 vơ i x2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

17/ 2cotx

3 vơ i 0 x

2

. 18/ cot x 3 vơ i x2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



19/ cotx 3 vơ i 3x 2

2 . 20/ 0cot15 2 3

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài25. Ti nh gia tri cu a ca c biêu thư c lươ ng giac sau

1/ Cho tan x 2 . Ti nh: 1 2

5 cot x 4 tan x 2 sin x cos xA , A

5cotx 4 tan x cos x 3 sin x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ Cho cot x 2 . Ti nh: 1 2

3 sin x cos x sin x 3cos xB , B

sin x cos x sin x 3cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ Cho cotx 2 . Ti nh: 1 2 2

2 sin x 3cos x 2C , C

3sin x 2cos x cos x sin x cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ Cho tan x 2 . Ti nh: 1

2 sin x 3 cos xD ,

4 sin x 5 cos x

2 3 3

3 sin x 2cos xD ,

5 sin x 4 cos x

3 3

3 2

sin x cos xD ,

sin x sin x cos x

4 3 3

sin x 5 cos xD ,

sin x 2 cos x

3 3

5 3

8 cos x 2 sin x cos xD ,

2 cos x sin x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ Cho 3sin x , 0 x

5 2

. Ti nh cot x tan xE

cot x tan x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ Cho 0 01sin x , 90 x 180

3 . Ti nh

28 tan x 3cot x 1F

tan x cot x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ Cho 2cos x

3 . Ti nh cot x 3 tan x

G2cot x tan x

.



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ Cho 02sin x , 0 x 90

3 . Ti nh

1 2 2

tan x cos x tan x cos xH , H cos x cot x

cot x sin x

. ........

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ Cho 4cos x , x

5 2

. Ti nh 1 2

cot x tan x sin xI , I cot x

cotx tan x 1 cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài26. Cho 5sin x cos x

4 . Hay ti nh gia tri cu a biêu thưc sau

1/ A sin x.cos x . ĐS: 9A

32 .

...................................................................................................................................................

...................................................................................................................................................

2/ B sin x cos x . ĐS: 7B

4 .

...................................................................................................................................................

...................................................................................................................................................



3/ 3 3C sin x cos x . ĐS: 41 7C

128 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài27. Cho tan x cotx 3 . Hay ti nh gia tri cu a biêu thưc sau

1/ 2 2A tan x cot x . ĐS: A 11 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ B tan x cotx . ĐS: B 13 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 4 4C tan x cot x . ĐS: C 33 13 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài28. Cho sin x cos x m . Hay ti nh theo m gia tri ca c biêu thư c

1/ A sin x cos x . 2/ 3 3B sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 4 4C sin x cos x . 4/ D sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

5/ 2 2E tan x cot x . 6/ 6 6F sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài29. Ti nh sin x, cos x, tan x, cot x . Biêt ră ng

1/ sin x cos x 2 . 2/ sin x cos x 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 1sin x cos x

5 . ĐS: 4 3 4 3

; ; ;5 5 3 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ tan x cotx 4 . ĐS: 1 2 3; ; 2 3; 2 3

22 2 3

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài30. Cho tan x 2cotx 1 . Hay ti nh

1/ 2 2A tan x cot x . 2/ 3 3B tan x cot x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 4 4C tan x 2 cot x . 4/ 5 5D tan x 3cot x .



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài31. Ti nh gia tri cu a ca c biêu thư c sau đây khi

1/ Cho 4 4 33 sin x cos x

4 . Ti nh 4 4A sin x 3 cos x . ĐS: A 7

4 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ Cho 4 4 13 sin x cos x

2 . Ti nh 4 4B sin x 3 cos x . ĐS: B 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ Cho 4 4 74 sin x 3 cos x

4 . Ti nh 4 4C 3sin x 4 cos x . ĐS:

7 57C C

4 28 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



CUNG GOC LIÊN KÊT

Bài32. Biêu diê n cac sô đo cu a cung dươi da ng k2 vơ i k .

3 5 11 14 21, , , ,

2 4 3 5 6

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài33. Biêu diê n cac sô đo cu a cung dươi da ng 0k360 vơ i k .

0 0 0 0 0370 , 512 , 765 , 1000 , 1234 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài34. Biêu diê n cac sô đo cu a cung dươi da ng k2 vơ i k .

8 11 7 18 27 25, , , , ,

3 5 2 7 10 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài35. Dung cung liên kêt (không dung may ti nh), hay ti nh cac gia tri sau

1/ 0sin150 . 2/ 0cot135 . 3/ 0cos225 . 4/ 0tan210 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 0cot225 . 6/ 0sin240 . 7/ 0cos 315 . 8/ 0tan 300 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



9/ 0cot 1380 . 10/ cos11 . 11/ sin13 . 12/ tan10 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 7cot

6 . 14/ 11

cos3 . 15/ 25

cot4 . 16/ 17

sin3 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

17/ 29cos

6 . 18/ 45

tan4 . 19/ 16

cos3

. 20/ 31

sin2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

21/ 19cot

4

. 22/ 26

cos3

. 23/ 159

tan4

. 24/ 115

sin6

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài36. Diê n ta gia tri lương giac cu a cac goc sau bă ng gia tri lương giac cu a goc x

1/ 0sin x 90 . 2/ 0cos 180 x . 3/ 0sin 270 x . 4/ 0sin x 180 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 0cos x 540 . 6/ 0cot 180 x . 7/ 0sin x 540 . 8/ 0tan 360 x .

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 0cos 450 x . 10/ 2 0sin 270 x . 11/ 3 0cos 90 x . 12/ 5 0cot 180 x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài37. Diê n ta gia tri lương giac cu a cac goc sau bă ng gia tri lương giac cu a goc x

1/ cot x . 2/ sin x . 3/ tan 2 x . 4/ cot 3 x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ sin x 7 . 6/ tan x 5 . 7/ 5sin x

2

. 8/ 3

cos x2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 3cot x

2

. 10/ 5

cos x2

. 11/ 11

tan x2

. 12/ 7

sin x2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 2sin x . 14/ 9cos x . 15/ 11cot x 3 . 16/ 4cos x 3 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



17/ 2cot x 5 . 18/ 6cos x . 19/ 8cos x2

. 20/ 5cos x

2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

21/ 2012sin x2

. 22/ 2 9

tan x2

. 23/ 2013 7

cos x2

. 24/ 1991 5

sin x2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

25/ 2015 11cos x

2

. 26/ 2 9

cot x2

. 27/ 11 11

tan x2

. 28/ cos x 2012 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài38. Ru t gon (đơn gian) ca c biê u thưc

1/ A cos x sin x2

.

...................................................................................................................................................

...................................................................................................................................................

2/ B cos x sin x cos x sin x2 2 2 2

.

...................................................................................................................................................

...................................................................................................................................................

3/ 7 3C 2cos x 3 cos x sin x tan x

2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

4/ 3D 2sin x sin(5 x) sin x cos x

2 2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 7 3E 2cos x 3cos( x) 5 sin x cot x

2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ 3F sin 5 x cos x cot 3 x tan x

2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 3 11G cos 15 x sin x tan x cot x

2 2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ 3H sin x cos x cot 2 x tan x

2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 3 3I cos 5 x sin x tan x cot 3 x

2 2

.



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

10/ 0 0 0 0J cos 270 x 2sin x 450 cos x 900 2sin 270 x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 2 2 2 23K sin x sin x sin x sin x

4 2 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

12/ 2 2 2 2 2 22 5 7L sin sin sin sin sin sin

3 6 9 9 8 8

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 2017 2017 2016 2015M cos x cos x .sin x sin x2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

14/ 6 6 4 4 23N sin x cos x 2 sin x 2 sin x cos x

2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

15/

19tan x .cos 36 x .sin x 5

2O

9sin x .cos x 99

2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

16/ 2 285 3P sin x cos 207 x sin 33 x sin x

2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài39. Ru t gon va ti nh gia tri cu a biêu thưc (không dung ma y ti nh)

1/ 0 0 0 0A cos 0 cos20 cos 40 ...... cos180 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 0 0 0 0B cos20 cos 40 cos60 ...... cos180 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 0 0 0 0C cos10 cos 40 cos70 ...... cos170 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

4/ 0 0 0 0D tan20 tan 40 tan 60 ...... tan180 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 0 0 0 0E cot15 cot30 cot45 ...... cot165 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ 0 0 0 0F sin 5 sin10 sin15 ...... sin 360 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 0 0 0 0G cot195 cot210 cot225 ...... cot 345 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ 0 0 0 0H cot15 .cot35 .cot55 .cot75 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 0 0 0 0I tan10 .tan20 .tan 30 ......tan 80 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

10/ 0 0 0 0J tan1 .tan2 .tan3 ...... tan 89 .



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 2 0 2 0 2 0 2 0K sin 28 sin 36 sin 54 cos 152 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

12/ 2 0 2 0 2 0 2 0L cos 2 cos 4 cos 6 ...... cos 88 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 2 0 2 0 2 0 2 0M sin 10 sin 20 sin 30 ...... sin 90 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

14/ 2 0 2 0 2 0 2 0N cos 10 cos 20 cos 30 ... cos 180 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

15/ 0 0 0 0 0O sin20 sin 40 sin 60 ... sin 340 sin 360 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài40. Ru t gon va ti nh gia tri cu a biêu thưc (không dung ma y ti nh)

1/ 0 0A cos 315 .sin765 .

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 0 0 0 0B sin 32 .sin148 sin 302 .sin122 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 0 0 0 0C sin 810 .cos540 tan135 .cot585 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 0 0 0 0D sin 825 .cos 15 cos75 .sin 555 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 0 0 0E 2 tan540 2 cos1170 4 sin 990 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ 0 0

0

0 0

sin 234 cos216F .tan36

sin144 cos126

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 0 0

0

0 0

sin 324 cos 666G .cot36

sin1206 cos 36

.

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/

0 0 0 0

0 0

sin 328 .sin 958 cos 508 .cos 1022H

cot572 tan 212

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/

0 0

0

0 0

cos 288 .cot72I tan18

tan 142 .sin108

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

10/ 0 0 0 0J 2sin 790 x cos 1260 x tan 630 x .tan 1260 x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 0 0

0 0 0

2 sin 2550 .cos 1881K

tan368 2 cos 638 cos98

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

12/ 0 0 0

0 0

0

cos 44 tan226 .cos 406L cos72 .cot18

cos 316

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

13/ 0 0 0 0

0 0

0

tan 46 .sin 44 cot 136 .sin 404M tan 36 .tan54

cos316

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

14/

0 0 0 0

0 0

sin 328 .sin 958 cos 508 .cos 1022N

cos572 tan 212

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

15/

4 17sin cos

3 6O23 10

sin cos6 3

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

16/

sin 4,8 .sin 5,7 cos 6,7 .cos 5,8

Pcot 5,2 tan 6,2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài41. Chưng minh ră ng nê u A, B, C la ba goc cu a mô t tam gia c thi

1/ sin B sin A C . 2/ cos A B cosC .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

3/ A B Csin cos

2 2

. 4/ cos B C cos A 2C .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ cos A B C cos2C . 6/ sin A 2B C sin B .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ cot A B C cot2B . 8/ 3A B Ccos sin2A

2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ A B 3Csin cosC

2

. 10/ A B 2C 3Ctan cot

2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ A 3B Ccos sin B

2

. 12/ A 2B C 3Bcot tan

2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



sin a b sina.cos b sin b.cos a sin a b sin a.cosb sin b.cos a .

cos a b cosa.cos b sina.sin b cos a b cosa.cos b sina.sin b .

tana tan btan a b

1 tana.tan b

. tana tan btan a b

1 tana.tan b

.

Hê qua:

1 tantan

4 1 tan

. 1 tan

tan4 1 tan

.

BA I TÂP A P DUNG

Bài42. Không dung may ti nh. Hay ti nh gia tri cu a ca c biê u thư c

1/ 0 0 0 0A sin12 .cos48 cos12 .sin 48 .

...................................................................................................................................................

...................................................................................................................................................

2/ 0 0 0 0B cos 38 .cos22 sin 38 .sin 22 .

...................................................................................................................................................

...................................................................................................................................................

3/ 0 0 0 0C sin10 .cos55 cos10 .sin 55 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 0 0 0 0D sin 36 .cos 6 sin126 .cos 84 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 0 0 0 0E cos112 .cos23 sin112 .sin23 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ 0 0 0 0F sin200 .sin 310 cos340 .cos50 .

...................................................................................................................................................

C – CÔNG THỨC CỘNG



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 0 0 0 0 0G cos11 .cos21 cos69 .cos79 cos10 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ 0 0 0 0 0H cos68 .cos78 cos22 .cos12 sin100 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 0 0 0 0I cos 53 .sin 337 sin 307 .sin113 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài43. Ti nh gia tri lươ ng giac cua ca c cung go c sau

1/ 015 . 2/ 075 . 3/ 0105 . 4/ 0285 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 1912 . 6/ 5

12 . 7/ 7

12 . 8/ 13

12 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài44. Ti nh gia tri cu a ca c biêu thư c sau



1/ 0

0

1 tan15A

1 tan15

. 2/

0 0

0 0

tan25 tan20B

1 tan25 tan20

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 0 0 0 0

0 0 0 0

sin10 cos20 sin20 cos10C

cos17 cos13 sin17 sin13

. 4/

0 0 0

0 0

tan225 cot81 cot69D

cot261 tan201

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 0 0 0 0

0 0 0 0

sin 73 cos 3 sin 87 cos17E

cos132 cos 62 cos 42 cos28

. 6/

0 o0 0

0 0

cot225 cot79 ot71F

cot259 cot251

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 2 0 2 0G cos 75 sin 75 . 8/ 2 0 2 0 2 0H sin 20 sin 100 sin 140 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 2 0 0 2 0I cos 10 cos110 cos 130 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

10/ 0 0 0 0 0 0J tan20 .tan80 tan80 .tan140 tan140 .tan20 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

11/ 0 0 0 0 0 0K tan10 .tan70 tan70 .tan130 tan130 .tan190 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

12/ 0 0 0 0 0 0L sin160 cos110 sin250 cos340 tan110 tan340 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 0 0 0 0 0 0 0 0M cos70 cos50 cos 310 cos290 cos 40 cos160 cos320 cos380 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài45. Ti nh gia tri cu a ca c biêu thư c sau

1/ A cos x x3

biêt 1

sin x3

va 0 x2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ B sin x3

biêt 12

cos x13

va 3x

2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 0C cos x 30 biêt tan x 2 va 00 x 90 .

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ D tan x3

biêt 3

sin x5

va x2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ E cos x3

biêt 12

sin x13

va 3x 2

2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ F cot x4

biêt 4

sin x5

va 3x

2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ G tan x4

biêt 5

cot x 22

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ 7H sin 2x

4

biêt 2

cotx3

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



9/ I cos a b .cos a b biêt 1cosa

3 va 1

cosb4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài46. Biêt 0 04sin , 0 90

5 va 0 08

sin , 90 18017

. Hay ti nh gia tri cu a

biê u thư c A cos va B sin .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài47. Biêt 8 5sin , tanb

17 12 va , la ca c goc nhon. Hay ti nh gia tri cu a ca c biêu

thư c A sin , B cos va C tan .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài48. Cho 0 , ,2 4

va tan .tan 3 2 2 .

1/ Hay ti nh tan ; tan tan . 2/ Tinh tan ; tan . Suy ra va .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài49. Cho 3

. Tinh gia tri cu a cac biêu thư c

1/ 2 2A cos cos sin sin . 2/ 2 2

B cos sin cos sin .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

Bài50. Ru t gon cac biêu thư c

1/ A sin x 3 cos x . 2/ B 3 sin 7x cos 7x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 2 2C a sin x b cos x, a b 0 . 4/ D 3 sin x sin x3 6

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ E cos7x.cos5x 3 sin2x sin7x.sin5x . 6/ F 3 cos2x sin 2x 2 sin 2x6

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ G 2sin 2x 4 sin x 16

. 8/ H sin2x 2 2 cosx 2sin x 3

4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ A sin x cos5x cos x sin5x .

...................................................................................................................................................

2/ B sin 4x cot2x cos4x .

...................................................................................................................................................

3/ C cos6x tan 3x sin6x .

...................................................................................................................................................



4/ D sin x y cos x y sin x y cos x y .

...................................................................................................................................................

5/ 0 0 0 0E cos 40 x cos x 20 sin 40 x sin x 20 .

...................................................................................................................................................

6/ 0 0 0 0F sin 14 2x cos 16 2x cos 14 2x sin 16 2x .

...................................................................................................................................................

7/ 0 0 0 0G sin x 10 cos 2x 80 sin x 100 cos 2x 10 .

...................................................................................................................................................

...................................................................................................................................................

8/ H sin x cos x sin x cos x3 4 4 3

.

...................................................................................................................................................

...................................................................................................................................................

9/ 3I cos x cos x cos x cos x

3 4 6 4

.

...................................................................................................................................................

...................................................................................................................................................

10/ 9 5 5J sin x cos x sin x cos x

3 4 4 3

.

...................................................................................................................................................

...................................................................................................................................................

11/ K cos x cos x cos x cos x3 4 6 4

.

...................................................................................................................................................

...................................................................................................................................................

12/ 13 13 3L cos x cos x cos x cos x

3 4 6 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................




1/ tan 3x tan xA

1 tan x tan 3x

. 2/ tan2x 1B

1 tan2x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/

0

0

tan2x cot 90 xC

1 cot 90 2x tan x

. 4/

2 2

2 2

tan 2x tan xD

1 tan 2x tan x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ A sin a b sin a sin b2

.

...................................................................................................................................................

...................................................................................................................................................

2/ B cos a cos b cos a b2 2

.

...................................................................................................................................................

...................................................................................................................................................

3/ 21C cos a cos a sin a

4 4 2

.

...................................................................................................................................................

...................................................................................................................................................

4/ 2 2 2 2D sin a sin b cos a cos b .

...................................................................................................................................................

...................................................................................................................................................

5/

2 sin a b

E tan bcos a b cos a b

.



...................................................................................................................................................

...................................................................................................................................................


1/ 2 2A cos x 3 sin2x sin x . 2/ 3B 4 sin x 3 sin x 3 cos 3x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 0 0C sin x 45 cos x 45 . 4/ D tan3x tan x sin2x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ sin2x 2sin x cos x .

...................................................................................................................................................

2/ 2 2cos2x cos x sin x .

...................................................................................................................................................

3/ 2

2 tan xtan2x

1 tan x

.

...................................................................................................................................................

4/ 3sin 3x 3 sin x 4 sin x .

...................................................................................................................................................

5/ 3cos 3x 4 cos x 3 cos x .

...................................................................................................................................................

6/ cos x sin x 2 cos x 2 sin x4 4

.

...................................................................................................................................................

...................................................................................................................................................

7/ cos x sin x 2 cos x 2 sin x4 4

.



...................................................................................................................................................

...................................................................................................................................................

8/ 2 2 2 2sin x y sin x y sin x sin y cos y cos x .

...................................................................................................................................................

...................................................................................................................................................

9/ 2 2 2 2cos x y cos x y cos x sin y cos y sin x .

...................................................................................................................................................

...................................................................................................................................................


1/ sin x sin x 2 sin x4 4

.

...................................................................................................................................................

...................................................................................................................................................

2/ 24 sin x sin x 4 sin x 33 3

.

...................................................................................................................................................

...................................................................................................................................................

3/ sin x sin y z sin y sin z x sin z sin x y 0 .

...................................................................................................................................................

...................................................................................................................................................

4/ cos x sin y z cos y sin z x cos z sin x y 0 .

...................................................................................................................................................

...................................................................................................................................................

5/ tan x y tan x tan y tan x y tan x tan y .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ tan2x tan x tan2x tan x tan x tan x 16 3 3 6

.

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

7/ 2 2tan x. tan x tan x .tan x tan x . tan x 3

3 3 3 3

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ 3 2cos x .cos x cos x .cos x 1 3

3 4 6 4 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ o o o o o o o ocos70 cos50 cos230 cos290 cos40 cos160 cos320 cos380 0 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/

cos a b cota cotb 1cota cotb 1cos a b

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ sin a b sin b c sin c a

0cosa cosb cosb cos c cos c cosa

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



3/ 2 2

2 2

sin a b sin a btan a tan b

cos a cos b

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 2 2

2 2

cos a b cos a b1 tan a tan b

cos a cos b

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài58. Chứng minh các hệ thức sau, với điều kiện cho trước

1/ Nê u cos a b 0 thi sin a 2b sin a .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ Nê u sin 2a b 3 sin b thi tan a b 2 tana . HD:

b a b a

2a b a b a

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ Nê u tana 2 tanb thi sin a b 3 sin a b .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ Nê u 1tan a.tanb

3 thi cos a b 2 cos a b . HD: Khai triển gia thiêt.



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ Nê u 5 sin b sin 2a b thi 3tan a b tana

2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ Nê u sin b sin a cos a b thi 2 tana tan a b . HD: b a b a .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ Nê u cos 2a b 1 thi btan a b tana 2 tan

2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ Nê u cos a b k cos a b thi 1 ktana tan b

1 k

. HD:

a 2b a b b

a a b b

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ sin a b c

tana tan b tan c tana tan b tan ccos a cos b cos c

.



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 2 2 2cos x cos x y cos y 2cos x cos y cos x y 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài60. Chưng minh cac biê u thưc sau đôc lâ p vơ i biê n x

1/ 2A sin x cos x cos x3 3

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 2 2 2B cos x cos x cos x3 3

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 2 2 22 2C sin x sin x sin x

3 3

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 2 2 22 2D cos x cos x cos x

3 3

.

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 3 3a cos x cos 3x a sin x sin 3x

F , a constcos x sin x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài61. Chưng minh ră ng: tan x tan x tan x tan 3x3 3

. Tư đo ti nh gia tri cu a biê u

thư c 0 0 0P tan10 tan 50 tan110 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài62. Cho tam giác ABC vơi A,B,C lâ n lươt la ba go c cua tam gia c. Chứng minh

1/ sinC sin A.cosB sin B.cos A .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ sin A sin BcosC sinC cos B .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ cosA sinBsinC cosBcosC .

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 0sinCtanA tanB, A,B 90

cosA.cosB .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 0tan A tan B tanC tan A.tan B.tanC, A,B,C 90 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ cotA.cotB cotB.cotC cotC.cotA 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ A B C B Csin cos cos sin sin

2 2 2 2 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ A B C B Ccos sin cos cos sin

2 2 2 2 2 .

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ A B B C C Atan .tan tan .tan tan .tan 1

2 2 2 2 2 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

10/ A B C A B Ccot cot cot cot .cot .cot

2 2 2 2 2 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 0cosC cosBcotB cotC , A 90

sinB.cosA sinC.cosA .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

12/ A B C A B C A B C A B Ccos .cos .cos sin sin cos sin cos sin cos sin sin

2 2 2 2 2 2 2 2 2 2 2 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 2 2 2A B C A B Csin sin sin 1 2 sin sin sin

2 2 2 2 2 2 .

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

14/ sin A cosBcosC sin BcosC cosA sinC cos A cosB sin A sin B sinC .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài63. Cho tam giác ABC vơi A,B,C lâ n lươt la ba go c cua tam gia c. Chứng minh

1/ tan A tan B tanC 3 3, A,B,C nhon.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 2 2 2tan A tan B tan C 9, A,B,C nhon.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 6 6 6tan A tan B tan C 81, A,B,C nhon.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 2 2 2A B Ctan tan tan 1

2 2 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

5/ A B Ctan tan tan 3

2 2 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

HD:

1/, 2/, 3/, sư dung tan A tan B tanC tan A. tanB.tanC va bât đă ng thư c

Cauchy.

4 / sư dung bâ t đăng thưc bô đê 2 2 2a b c ab bc ca va

A B B C C Atan .tan tan .tan tan .tan 1

2 2 2 2 2 2 .

5 / khai triển 2

A B Ctan tan tan

2 2 2

va sư du ng câu 3 / .



Công thưc nhân đôi

sin2 2sin .cos .

2 2 2 2cos2 cos sin 2 cos 1 1 2 sin .

2

2 tantan2

1 tan

.

2cot 1

cot22cot

.

Công thưc ha bâc Công thưc nhân ba (mơ rô ng)

2 1 cos2sin

2

. 3sin 3 3 sin 4 sin .

2 1 cos2tan

1 cos2

. 3

2

3 tan tantan 3

1 3 tan

.

BA I TÂP A P DUNG

RUT GO N VA TINH GIA TRI CUA BIÊ U THƯ C

Bài64. Ti nh gia tri cu a ca c biêu thư c sau (không dung ma y tinh bo tu i)

1/ 2 2A cos sin8 8

. 2/ 0 3 0B 3cos10 4 cos 10 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 0 2 0C sin20 1 4 cos 20 . 4/ 3 0 0D 4sin 40 3 sin130 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 3 0 0E 4sin 50 3cos140 . 6/ 0

2 0

tan15F

1 tan 15

.

...................................................................................................................................................

...................................................................................................................................................

D – CÔNG THỨC NHÂN – HẠ BẬC



...................................................................................................................................................

7/ 2

7tan

8G1 tan

8

. 8/

0 0

1 3H

sin10 cos10 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 00

1I 4 sin 70

sin10 . 10/ 2 2 5

J tan tan12 12

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 2 0 2 0K tan 36 tan 72 . 12/ 2 0 2 0 0L cos 70 sin 40 sin100 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 2 0 2 0 0M cos 20 2 sin 55 2 sin 65 . 14/ 0 0 0 0N sin 6 .sin 42 .sin 66 .sin 78 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ 0 0A cos36 cos72 . 2/ 2B cos cos

5 5

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ C sin .cos cos8 8 4

. 4/ D sin .cos .cos16 16 8

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



5/ 0 0 0E sin10 sin50 sin70 . 6/ 2 4F cos cos cos

7 7 7

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 4 5G cos cos cos

7 7 7

. 8/ 0 0 0 0H sin 6 cos12 cos24 cos 48 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 0 0 0 0I sin6 sin 42 sin66 sin78 . 10/ 0 0 0 0 0J sin5 .sin15 .sin25 .... sin75 .sin85 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 0 0 0 0 0K cos10 .cos20 .cos 30 ...cos70 .cos 80 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

12/ 0 0 0 0L 8tan18 cos18 cos 36 cos72 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 0 0 0 0M cos20 cos 40 cos 60 cos 80 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

14/ 2 3 4N cos cos cos cos

9 9 9 9

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



15/ O 96 3 sin cos cos cos cos48 48 24 12 6

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

16/ 2 4 8 16 32P cos .cos .cos .cos .cos

31 31 31 31 31

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

17/ 2 4 8 16Q cos cos cos cos cos

33 33 33 33 33

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

18/ 2 3 4 5 6 7R cos .cos .cos .cos .cos .cos .cos

15 15 15 15 15 15 15

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ 0 0

0 0 0 0

cos 80 cos20A

cos 35 cos15 sin 35 sin15

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 0

0 04 0 4 0

sin 60B 3 sin15 cos15

sin 15 cos 15

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 0 0 0 0C tan9 tan27 tan63 tan81 .

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

4/ D 4tan 2tan tan cot8 16 32 32

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 2 0 0 0 2 0

2 0 2 0

cos 696 tan 260 tan530 cos 156E

tan 252 cot 342

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ 2A sin x cos x . 2/ 2 2B 1 4 sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ C sin x cos x cos2x . 4/ 4 4D cos 2x sin 2x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 2 2E cos x sin x2 2

. 6/ F sin x cos x cos2x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ G 4 sin x sin x sin 2x2 2

. 8/ 2 2x x

H sin sin8 2 8 2

.

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

9/ 2 0I sin 8x 2 cos 45 4x . 10/ J sin2x cos2x 2cosx sinx cosx 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ 22cos x 1

Asin x cos x

. 2/

21 2 sin 2xB

cos2x sin 2x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/

2C

1 tan x 1 cotx

. 4/ 2D 1 tan x cot x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ sin2x cos2xE

sin x cos x . 6/ cotx tan x

Fcos2x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/

31 sin x

2G

1 sin x2

. 8/ 2 2

2 2

sin 2x 4 sin xH

sin 2x 4 sin x 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ sin 4x cos2xI .

1 cos 4x 1 cos2x

. 10/ sin 3x cos 3x

Jsin x cos x

.



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 1 cos x

2xK tan .

4 2sin x

2

. 12/ 1L cot2x tan x

2sin2x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 1 cos2x sin2xM .cot x

1 cos2x sin2x

. 14/ 22 sin2x 2cos x 1

Ncos x sin x cos 3x sin 3x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

15/ O 1 sinx 1 sinx, 0 x4

. 16/ 1 1 1

P cot8xsin2x sin 4x sin 8x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

17/ O 1 sin2x 1 sin 2x, x4 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



Bài69. Tính giá trị của biểu thức lượng giác, khi biết

1/ 3sin2x, cos2x khi sin x , x

5 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ sin2x, cos2x khi sin x cos x 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 5 3cos2x, sin2x, tan2x khi cos x , x

13 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ cos2x, sin2x, tan2x khi tan x 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 4 3sin x, cos x khi sin2x , x

5 2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ 7cos2x, sin2x, tan2x khi tan x

8 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ sin2x, cos2x, tan2x khi tan x 2 3, 0 x2

.



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài70. Tính giá trị của biểu thức lượng giác, khi biết

1/ 5 5A sin x cos x cos x sin x khi biêt x16

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 4 3 3 4B sin x sin x cos x cos x sinx cos x khi biêt x48

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ C cos4x khi biêt tan x cotx 3 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 0D cos 270 4x khi biêt 0cot 45 x 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 4 4E sin x cos x khi biêt sin x cos x m vơi m .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ tan x cot xF

tan x cot x

khi biêt 3sin x cos x , 0 x

4 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài71. Ti nh gia tri lươ ng giac cua (không dung ma y ti nh bo tu i)



1/ 0cos15 . 2/ 0cos18 . 3/ 0sin 36 . 4/ 0cos 36 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài72. Ti nh xtan

2 khi biêt

1/ 4cos x , 0 x

5 2

. 2/ 24 3tan x , x

7 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 7sin x cos x , 0 x

2 6

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài73. Cho 1tan x , 0 x

2 2

va 1

tan y , 0 y3 2

.

1/ Tinh x y . 2/ sin x 2y va cos 2x y .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài74. Ti m x khi biêt

1/ 0 x

2tan x 2 1

. 2/ 0 x

26 2

cos x4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



Bài75. Ti nh theo cos2x ca c biêu thưc sau

1/ 2A 1 cos x . 2/ 2 2B sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 2

2

1 sin xC

cos x

. 4/

2

2

1 cot xD

1 cot x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 6 6E sin x cos x . 6/ 6 2 6 2F sin x cos x cos x sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài76. Ti nh theo xt tan

2 ca c biêu thưc sau

1/ A sin x . 2/ B cos x .

...................................................................................................................................................

...................................................................................................................................................

3/ sin xC

3 2cos x

. 4/ D tan x cot x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 1 tan xE

1 cot x

. 6/ tan x sin xF

tanx sin x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



CHƯNG MINH ĐĂ NG THƯC


1/ 4 4cos x sin x cos2x . 2/ 2sin 4x 4 sin x cos x 1 2 sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 2 2cos 2x sin x cos x cos 3x . 4/ 4 2cos4x 8 cos x 8 cos x 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 48 sin x 3 4cos2x cos4x . 6/ 4 4 3 1sin cos x cos4x

4 4 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 4 4 2 2sin cos x 6cos x sin x cos4x . 8/ 6 6 5 3sin x cos x cos4x

8 8 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 6 6 15 1sin x cos x cos2x cos6x

16 16 . 10/ 6 6 2x x 1

sin cos cos x sin x 42 2 4 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 8 8 7 1cos x sin x cos2x cos6x

8 8 . 13/ 3 3 4 4cos xcos3x sin xsin3x 3 sin x cos x 2 .

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài78. Chưng minh ră ng nê u a b4

thi 1 tan a 1 tan b 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ 1 cos2xtan x

sin2x

. 2/ 2tan x cotx

sin2x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ cotx tan x 2cot2x . 4/ cos x xcot

1 sin x 4 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 21 cos x x.tan 1

1 cos x 2

. 6/ 1cot2x cot x

sin2x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ x 2tan cot

4 2 4 2 cos x

. 8/ tan2x tan x cos2x tan x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ tan x tan x 2 tan2x4 4

. 10/ 3 3 1

cos x sin x sin x cos x sin4x2

.

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

11/ 3 3 3cos3x sin x sin 3x cos x sin4x

4 . 12/ tan x tan x .tan x tan3x

3 3

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 2tanx tan x tan x 3tan3x

3 3

. 14/ cotx tanx 2tan2x 4tan4x 8cot8x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ 1 sin x xcot

cos x 4 2

. 2/

21 1 2 sin xtan2x

cos2x 1 sin 2x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/

2 sin2x sin 4xtan2x cos x

2 cos x cos 3x

. 4/ 2 26 2cos4x

tan x cot x1 cos4x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



5/ 21 2 sin x 1 tan x

1 sin2x 1 tan x

. 6/ 1 x1 tan tan x

cos x 2

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 1 sin2xtan x

cos2x 4

. 8/ 22 sin x sin2x x

tan2 sin x sin2x 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 2

2

1 2 sin x1

2 tan x cos x4 4

. 10/

xsin x sin x2 tan

x 21 cos x cos

2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 1 cos x cos2xcot x

sin2x sin x

. 12/ 1 cos 4x 1sin 4x

cotx tan x 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/

sin2x cos x xtan

21 cos2x 1 cos x

. 14/

4 4 2

2sin x cos x cos x xcos

21 1 cos x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



15/ 2 2 6 2cos4xcot x tan x

1 cos 4x

. 16/ 2

x x 4cot tan

2 2 1 2 tan x cot2x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

17/ 2cot 2x 1

cos 8x cot4x sin 8x2cot2x

. 18/

21 1 2 sin xtan2x

cos2x 1 sin 2x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

19/ 1 sin2x cos2xtan 4x

cos 4x sin2x cos2x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

20/ cosx sinx cosx sinx2tan2x

cosx sinx cosx sinx

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

21/

x xcos sin 12 2 tan x

x x cos xcos sin

2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



22/ 2

1 cos x cos2x cos 3x2cos x

2cos x cos x 1

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

23/ 1 x x2 cot2x cot tan

sin2x 2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

24/ 4 4sin x sin2x cos x

cos2xtan2x 1

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

25/ 2 2

42 2

sin 2x 4 sin xtan x

sin 2x 4 sin x 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

26/ 2 2

2 2

sin 3x cos 3x8cos2x

sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

27/ 2 3cos x cos 3x sin x sin 3x

3cos x sin x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

28/ 2

1 1 sin 2x1 tan x 1 tan x

cos x cos x cos x

.

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài81. Chưng minh cac biê u thưc sau đây không phu thuô c va o biê n sô

1/ 2sin2x 2 sin x xA tan

sin2x 2 sin x 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 3 3cos x cos 3x sin x sin 3x

Bcos x sin x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 4 1C 4sin x 2cos2x cos4x

2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 4 2 2 4D 3cos2x 5sin x 4sin x cos x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 2tan x 1

E cotx cos 4x cot2x sin 4x2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ 4 4 4 4 3F sin x sin x sin x sin x

4 2 4

.

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

BIÊ N ĐÔI THANH TICH

Bài82. Biên đô i thanh ti ch cac biê u thư c sau

1/ A 2sin x 1 2 cos x sin x sin2x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ B 2cos x 1 2 sin x cos x sin 2x sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 2C 2sin x 1 2cos2x 2 sin x 3 4 sin x 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 2D 2cos x 1 2cos x 2 sin x 1 4 sin x 3 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 2E 2sin x 1 3cos 4x 2 sin x 4 4cos x 3 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ F 1 sin x cos x sin2x cos2x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 2G 1 cos x cos x sin x 1 cos x .

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

8/ 2 2H 1 sin x cos x sin 2x 1 cos x sin x 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ 3 3A sin x cos x cos x sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 3 3B sin x cos x sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 3 3C sin x cos x cos2x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 2 3D cos x sin x cos x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 3 2E cos x cos x 2sin x 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ 3 3 2F cos x sin x 2sin x 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................




1/ A sin 2x cos2x cos x sin x .

...................................................................................................................................................

2/ B sin2x cos2x 3 sin x cos x 2 .

...................................................................................................................................................

3/ C cos2x sin2x 3 sin x cos x 2 .

...................................................................................................................................................

4/ D sin2x cos2x 2 cos x sin x 3 .

...................................................................................................................................................

5/ E 9 sin x 6 cos x 3 sin2x cos2x .

...................................................................................................................................................

...................................................................................................................................................

6/ F sin 2x 2 cos2x 4 cos x sin x 1 .

...................................................................................................................................................

...................................................................................................................................................

7/ G 2cos x 1 sin x cos x 1 .

...................................................................................................................................................

...................................................................................................................................................

8/ H 2 sin 2x cos2x 7 sin x 2 cos x 4 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



Công thưc biên đôi tô ng thanh ti ch

● a b a bcosa cosb 2 cos .cos

2 2

. ● sin a b

tana tan bcos a.cosb

.

● a b a bcosa cosb 2sin .sin

2 2

. ● sin a b

tana tan bcosa.cos b

.

● a b a bsin a sin b 2sin .cos

2 2

. ● sin a b

cota cotbsin a.sin b

.

● a b a bsin a sin b 2cos .sin

2 2

. ● sin b a

cota cotbsin a.sin b

.

Hê qua

● sin cos 2.sin 2.cos4 4

.

● sin cos 2 sin 2 cos4 4

.

Công thưc biên đôi ti ch tha nh tô ng

● 1cosa.cosb cos a b cos a b

2 .

● 1sina.sin b cos a b cos a b

2 .

● 1sina.cosb sin a b sin a b

2 .

BA I TÂP A P DUNG

CÔNG THƯC BIÊ N ĐÔ I TÔNG THANH TICH

Bài85. Ti nh gia tri cu a biê u thư c

1/ 0cos2x cos 4xA khi x 20

sin 4x sin2x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

E – CÔNG THỨC BIẾN ĐỔI LƯỢNG GIÁC



2/ cos x.cos13xB khi x

cos 3x cos 5x 17

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ cos x.cos10xC khi x

cos2x cos 4x 13

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ tan2x sin2x 2D khi tan x

tan2x sin2x 15

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ sin x sin2x sin 3x 1E khi sin x , x

cos x cos2x cos 3x 3 4 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài86. Ru t gon biêu thư c

1/ 1 2cos xA

1 2cos x

. 2/ 3 2cos3xB

3 2cos3x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 1 2cos2xC

3 2 sin 2x

. 4/ 2 2 sin2x

D2 2sin2x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ sin 5x sin 3xE

2cos4x

. 6/ cos 4x cos2xF

sin 4x sin2x

.



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ sin x y

Gsin x sin y

. 8/ sin x sin y

Hcos x cos y

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ cos x sin xI

cos x sin x

. 10/ 2 2

2 2

sin 4x sin 2xJ

cos x cos 2x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 2sin 4x

K2cos x cos 3x cos5x

. 12/ sin2xL

tan x cot2x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ tan 3x tan5xM

cot3x cot5x

. 14/ tan2x cot2xN

1 tan2x. tan 4x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

15/ 1 sin2x cos2xO

1 sin2x cos2x

. 16/ 1 sin 4x cos 4xP

1 cos 4x sin 4x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

17/ sin2x 2 sin 3x sin 4xQ

cos 3x 2cos 4x cos5x

. 18/ sin x sin 4x sin7xR

cos x cos 4x cos7x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



19/ cos2x sin 4x cos6xS

cos2x sin 4x cos 6x

. 20/ 2

1 cos x cos2x cos 3xT

2cos x cos x 1

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

21/ 22 sin2x 2cos x 1

Ucos x sin x cos 3x sin 3x

. 22/

sin x y sinx cos x y cosxV

sin x y sinx cos x y cosx

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

23/ cos7x cos 8x cos9x cos10xX

sin7x sin 8x sin 9x sin10x

. 24/ sin 4x sin5x sin 6xY

cos 4x cos 5x cos 6x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài87. Biên đô i thanh ti ch cac biê u thư c sau đây

1/ A cos 3x cos x . 2/ B sin 3x sin2x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ C cos4x cos x . 4/ D sin 5x sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ E 1 sin2x . 6/ F 1 sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ G 1 2 cos x . 8/ H 2 sin2x 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



9/ I 3 2 cos2x . 10/ J sin a b sin a b .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 2 2K cos x cos y . 12/ L 1 sin x cos2x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ M 1 sin x cos x . 14/ N cos x sin2x cos 3x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

15/ O sin 3x sin x sin2x . 16/ Q cos x cos2x sin 3x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

17/ R sin x sin 2x sin 3x . 18/ S cos x cos2x cos 3x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

19/ T 2 sin 2x cos 5x cos9x . 20/ U sin 3x 2 sin 2x sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

21/ V cos x cos 3x 2 cos 5x . 22/ 0 0 0X cos46 cos22 2cos78 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

23/ 2 3Y cos cos cos

7 7 7

. 24/ 2 4 6Z cos cos cos

7 7 7

.

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

25/ 0 0 0W sin70 sin20 sin50 . 26/ cos 5x cos7x cos 6x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

27/ cos 5x sin x cos 3x2

. 28/ x y

sin x sin y sin2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

29/ sin x sin y sin x y . 30/ cos x cos y sin x y .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

31/ cos x cos y cos x y 1 . 32/ 0 0cos 60 x cos 60 x cos3x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

33/ 1 cos2x cos 4x cos6x . 34/ sin2x sin 4x sin 6x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

35/ sin 5x sin 6x sin 7x sin 8x . 36/ cos5x cos 8x cos9x cos12x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ 1 cos x cos2x cos 3x . 2/ sin x sin 3x sin 7x sin 5x .

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

3/ sin x sin2x sin 5x sin 8x . 4/ cos7x sin 3x sin2x cos 3x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ cos9x cos7x cos 3x cos x . 6/ cos10x cos 8x cos 6x 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 0 0 0 0sin 35 cos 40 sin55 cos20 . 8/ 0 0 0 0sin57 sin59 sin93 sin 61 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 0 0 0 0sin 47 sin 61 sin11 sin25 . 10/ cos5x 3 cos 7x 3 cos 9x cos11x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 2 3x1 sin x cos5x sin7x 2cos

2 . 12/ sin3x sinx sin2x 2 1 cosx cosx .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ sin x sin2x sin 3x 1 cos x cos2x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

14/ 1 sin x cos 3x cos x sin 2x cos2x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

15/ sin x sin2x sin 3x cos x cos2x cos 3x .

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................


1/ 2 2 2cos x cos 2x cos 3x 1 . 2/ 2 2 2 3sin x sin 2x sin 3x

2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 2 2 2sin 3x sin 2x sin x . 4/ 2 2 2sin x cos 2x cos 3x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 2 2 2sin x 2sin 2x sin 3x . 6/ 2 2sin 4x cos 6x sin 10,5 10x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 2 2 2 2sin 3x cos 4x sin 5x cos 6x . 8/ 2 2 2 2 3cos x cos 2x cos 3x cos 4x

2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 2 2 2 2cos x cos 2x cos 3x cos 4x 2 . 10/ 2 25 9xcos3x sin7x 2sin 2cos

4 2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ sin 2x cot3x sin 2x 2cos5x2

. 12/

cos3xcos 5x2

2sin2x cos 3x2 cos5x

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



Bài90. Chưng minh

1/ 0 0 6sin75 cos 75

2 . 2/ 0 0 0cos12 cos 48 sin18 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 0 0 0sin 65 sin 55 3 cos 5 . 4/ 0 0tan267 tan93 0 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 0 0 0cos85 cos35 cos25 0 . 6/ 0 0 0 0tan9 tan27 tan63 tan81 4 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 0 0 0 0 1cos24 cos48 cos84 cos12

2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ 0 0 0 0 0 0tan9 tan15 cot15 cot9 tan27 cot27 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 2 31 1cos x cos 3x cos5x 8 sin x cos x

2 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

10/ sin x 1 2 cos2x 2 cos 4x 2 cos 6x sin 7x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



11/ 4 x1 4 cos x 6 sin2x 4 sin x 16sin2x sin

2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

12/ 2 0 08 sin x sin x 60 sin x 60 cos 4x cos2x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 2 0 0sin x cos x cos 4x 4 sin 2x sin x 15 cos x 15 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài91. Cho a b c . Chưng minh: a b csin a sin b sin c 4 cos cos sin

2 2 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài92. Ti nh cac gia tri cua biê u thư c

1/ 6 6A sin cos24 24

.

...................................................................................................................................................

2/ 2 2B tan cot12 12

.

...................................................................................................................................................

3/ 3 5 7 9C cos cos cos cos cos

11 11 11 11 11

.

...................................................................................................................................................

...................................................................................................................................................

4/ 2 4 6 8 10D cos cos cos cos cos

11 11 11 11 11

.

...................................................................................................................................................



...................................................................................................................................................

5/ 0 2 3 4 5 6E cos 0 cos cos cos cos cos cos

7 7 7 7 7 7

.

...................................................................................................................................................

...................................................................................................................................................

6/ 2 2

0 0 0 0 0F sin 40 cos10 cos 40 sin10 cos140 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài93. Trong ΔABC co ba go c lan lươ t la A,B,C . Chưng minh ră ng

1/ A B CsinA sinB sinC 4 cos cos cos

2 2 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ A B CsinA sinB sinC 4sin sin cos

2 2 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ A B CcosA cosB cosC 1 4 sin sin sin

2 2 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ sin 2A sin2B sin2C 4 sin A sin B sinC .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 1 cos2A cos2B cos2C 4 cos A cosB cosC .

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

6/ 2 2 2sin A sin B sin C 2 1 cosAcosBcosC .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 2 2 2cos A cos B cos C 1 2cosAcosBcosC .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

CÔNG THƯC BIÊ N ĐÔ I TICH THANH TÔ NG

Bài94. Biên đô i thanh tô ng cac biê u thư c sau

1/ 2A sin sin

5 5

. 2/ B sin5x cos 3x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 3C sin cos

4 6

. 4/ 7D sin cos

12 12

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ E sin x y cos x y . 6/ 0 0F sin x 30 cos x 30 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ G 2sin x sin2x sin 3x . 8/ H 8cos x sin2x sin 3x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ I sin x sin x cos2x6 6

. 10/ J 4 cos a b cos b c cos c a .



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài95. Ti nh gia tri cu a biê u thư c

1/ 0 0A cos75 cos15 .

...................................................................................................................................................

2/ 5B sin sin

12 12

.

...................................................................................................................................................

3/ 11 5C sin cos

12 12

.

...................................................................................................................................................

4/ 0 0 0D sin20 .sin 40 .sin 80 .

...................................................................................................................................................

5/ 0x 5xE sin .sin , khi x 60

4 4 .

...................................................................................................................................................

6/ 0 0 0 0F sin 20 sin 40 sin 60 sin 80 .

...................................................................................................................................................

...................................................................................................................................................

7/ 4 5 7 11G 2 .sin .sin .sin .sin

24 24 24 24

.

...................................................................................................................................................

...................................................................................................................................................

8/ 0 0 0 0 0 0H sin5 .sin15 .sin25 .......sin65 sin75 .sin 85 .

...................................................................................................................................................

...................................................................................................................................................

9/ 2 0 0 0I sin 10 cos70 cos50 .

...................................................................................................................................................

...................................................................................................................................................

10/ 0 0 0 0 01J cos10 cos50 cos5 cos25 cos10

2 .



...................................................................................................................................................

...................................................................................................................................................

11/ 2 0 2 0 0 0K sin 50 sin 10 sin 50 sin10 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

12/ 2 0 2 0 0 0L cos 73 cos 47 cos73 .cos 47 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 2 0 2 0 0 0M sin 50 sin 70 cos50 cos70 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

14/ 0 0 0 0 01N cos10 cos50 cos5 cos25 sin10

2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

15/ 0

0 00

sin 55O 2 cos22 cos 44

sin11 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

16/ 0

0 04 0 4 0

sin 60P 3sin15 sin 75

sin 15 sin 75

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

17/ 0

0 00 0

1 cos 40Q 2 2 sin10 2 sin 35

2cos5 sin 5

.

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

18/ 00

1R 4 sin70

sin10 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

19/ 0 0

1 3S

sin10 cos10 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

20/ 2 3T cos cos cos

7 7 7

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

21/ 2 4 5 2 4 6 8U cos cos cos cos cos cos cos

7 7 7 5 5 5 5

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

22/ 2 4 6 8V cos cos cos cos

9 9 9 9

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài96. Ru t gon biêu thư c A 2sin x cos x cos 3x cos 5x . Tư đo suy ra gia tri cu a biê u

thư c 3 5B cos cos cos

7 7 7

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................


1/ A cos11x cos 3x cos17x cos9x .

...................................................................................................................................................

...................................................................................................................................................

2/ B sin18x cos13x sin9x cos 4x .

...................................................................................................................................................

...................................................................................................................................................

3/ C sin x sin 3x sin 4x sin 8x .

...................................................................................................................................................

...................................................................................................................................................

4/ D sin2x sin6x cos x cos 3x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ E cos 3x cos6x cos 4x cos7x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ 0 0F sin x sin 60 x sin 60 x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 0 0G 8cos x cos 60 x cos 60 x 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ 1H cos x cos2x sin 3x sin12x

4 .

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

9/ I 4 sin2x sin 5x sin 7x sin 4x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

10/ 1J sin2x sin 6x cos 4x cos12x

4 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 1K sin x sin2x sin 3x sin 4x

4 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

12/ L 4 cos x sin x sin x cos2x6 6

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ M sin 4x sin10x sin11x sin 3x sin 7x sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

14/ x 3xN cos cos sin x sin 3x sin2x sin 3x

2 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

15/ 7x 3x x 5xO sin cos sin cos sin2x cos7x

2 2 2 2 .

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

16/ P cos2x cos 4x cos 6x 4 cos x cos2x cos 3x 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

17/ Q sin x 1 2 cos2x 2 cos 4x 2 cos6x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

18/ sin5xR 2 cos2x cos 4x

sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

19/ 2 2S sin x sin x sin cos 2x4 4 12 12

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................


1/ 0 0 0 0 0 3 1cos12 cos18 4 cos15 cos21 cos24

2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 0 0 0 0tan9 tan63 tan81 tan27 4 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 0 0 0 0 08 3tan 30 tan 40 tan50 tan60 cos20

3 .



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 0 0 1 1cos 2x 60 cos 2x 60 cos4x

2 4 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 2 0 0sin x cos x cos 4x 4 sin2x sin x 15 cos x 15 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ 2 2 3sin x sin x sin x sin x

3 3 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ sin a sin b c sin b sin c a sin c sin a b 0 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ 2 2 2cos x 2cosa cos x cos a x cos a x sin a .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

9/ 8 8 7 1sin x cos x cos2x cos6x

8 8 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

10/ 8 8 35 7 1sin x cos x cos4x cos 8x

64 16 64 .



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

11/ 3 2tan 4 tan 11

11 11 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

12/ 0 0 0tan20 tan 40 tan 80 3 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

13/ 3 0 2 08 sin 18 8sin 18 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

14/ 4 4 4 43 5 6 3sin sin sin sin

16 16 16 16 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

15/ 0 0 0 0 0 0tan10 tan25 tan25 tan55 tan55 tan10 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

16/ 0 0 0 0tan15 tan25 tan 35 tan 85 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

17/ 0 0 0tan20 tan 40 tan 80 3 3 .

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

18/ 0 o 0 0tan10 tan 50 tan 60 tan 70 2 3 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

19/ 0 0 0 0 0tan20 tan 40 tan 80 tan 60 8 sin 40 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

20/ 6 0 4 0 2 0tan 20 33 tan 20 27 tan 20 3 0 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài99. Ti nh cac goc cu a ΔABC biê t ră ng

1/ B C3

va 1sinB.sinC

2 . ĐS: B , C , A

2 6 3

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 2B C

3

va 1 3sinB.cosC

4

. ĐS: 5A , B , C

3 12 4

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài100. Chứng minh điều kiện cần và đủ đê ΔABC vuông

1/ cos2A cos2B cos2C 1 . 2/ tan 2A tan 2B tan2C 0 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

3/ b c acosB cosC sinB.sinC

. 4/ B a ccot

2 b

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài101. Chứng minh điều kiện cần và đủ đê ΔABC cân:

1/ A Ba tanA b tanB a b tan

2

. 2/ 22 tanB tanC tan B.tanC .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ sinA sinB 1tanA tanB

cosA cosB 2

. 4/ C 2sinA.sinBcot

2 sinC .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài102. Chứng minh bất đẳng thức, từ đó suy ra điều kiện cần và đủ đê ΔABC đều:

1/ 3 3sinA sinB sinC

2 . HD: Cộng sin

3 vào VT.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



2/ 3cosA cosB cosC

2 . HD: Cộng cos

3 vào VT.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ tanA tanB tanC 3 3 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 1cosA.cosB.cosC

8 .

HD: Biên đô i 1cosA.cosB.cosC

8 vê dang hă ng đăng thưc.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài103. Cho cos cos a cos b vơi a,b k2

. Chưng minh: 2a a btan tan tan

2 2 2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài104. Cho sin 2a b 5 sin b . Chưng minh: 2 tan a b

3tana

vơi

a k2

a b l2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài105. Cho tan a b 3 tan a . Chưng minh: sin 2a 2b sin2a 2 sin2b .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................




1/ A cos cos 3 cos5 ... cos 2n 1 . ĐS: sin2nA

2sin

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ n 12 3

B sin sin sin ... sinn n n n

. ĐS: B cot

2n

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 2n 13 5

C cos cos cos ... cosn n n n

. ĐS: C cos

n

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ 1 1 1D ... ,

cosa.cos2a cos2a.cos 3a cos4a.cos5a a

5

. ĐS: D 1 5 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ n 1

1 1 1 1E 1 1 1 ... 1

cos x cos2x cos 3x cos2 x

. ĐS:

n 1tan2 xE

xtan

2

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

6/ n

1 1 1 1F ...

sin2x sin 4x sin 8x sin2 x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ in3 3 2 3 n 1 3n

x x x xG sin 3s 3 sin ...... 3 sin

3 3 3 3 .



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài107. Ti m phân nguyên cua biê u thư c: 4 4 2 23 3 2M tan tan 2 tan tan

4 4 7 7

.

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

Bài108. Chưng minh cac bâ t đăng thư c sau

1/ sin 3x cos x cos 3x sin x cos2x 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 3 sin 3x cos2x cos x sin2x sin x 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3/ 0 0 1sin x sin x 60 sin x 60

4 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

4/ cos x cos 3x sin2x sin 4x 1 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

5/ 2cos2x cos x sin x sin 3x sin x cos 3x 1 .

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

Bài109. Ti m gia tri lơ n nhâ t va gia tri nho nhât cu a ca c biê u thư c sau

1/ y 3 sin x cos x 2 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2/ 1y cos x , x

cos x 2 2

.

...................................................................................................................................................

...................................................................................................................................................

3/ y sin x cos x cos2x cos 4x .

...................................................................................................................................................

...................................................................................................................................................

4/ 4 4y sin x cos x, 0 x2

.

...................................................................................................................................................

...................................................................................................................................................

5/ 2y sin x 2 sin x .

...................................................................................................................................................

...................................................................................................................................................

6/ 4 4 2 2y cos x sin x cos x sin x .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

7/ 4 2 5y sin 4x cos x cos2x

4 .

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

8/ 6 6 4 4y 2sin x 2cos x sin x cos x cos2x .

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



BÀI TẬP TỔNG ÔN – BIẾN ĐỔI LƯỢNG GIÁC § 1. BIEÁN ÑOÅI LÖÔÏNG GIAÙC

1. Tính giaù trò cuûa bieåu thöùc löôïng giaùc

BT 1. Tính giá trị biểu thức lượng giác khi đề bài cho sin:

a) Tính (1 3cos2 ) (2 3cos2 ),A biết 2sin3

ĐS: 149

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

b) Tính 4 4sin cos ,A biết 2sin 23

ĐS: 79

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

c) Cho 2 và 3sin

5 Tính

2

tan1 tan

A

ĐS: 1225

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

d) Cho 2 và 4sin

5 Tính sin 2( ).A ĐS: 24

25A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

e) Cho 32

và 4sin5

Tính 1 cot1 cot

A

ĐS: 7.A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

f) Cho 02

và 3sin5

Tính cot tancot tan

A

ĐS: 257

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

g) Cho 90 180o o và 1sin3

Tính 2 tan 3cot 1tan cot

A

ĐS: 26 2 29

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

h) Cho ;2

có 1sin

3 Tính sin2 cos2 .A ĐS: 7 4 2

9A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



i) Cho góc thỏa 2

và 4sin5

Tính cos 21 cos

A

ĐS: 740

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

j) Cho 02

và 2sin3

Tính 1 sin 2 cos2sin cos

A

ĐS: 2 5

3A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

k) Tính cos ,3

A

biết rằng: 1sin3

và 02

ĐS: 6 36

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

l) Tính cos ,4

A


và .2

ĐS: 7 226

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

m) Cho thỏa tan cot 0 và 1sin5

Tính sin 2 .A ĐS: 4 625

A



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

n) Cho thỏa 02

và 1sin2

Tính tan4

A

ĐS: 2 3.A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

o) Tính tan ,3

A


và .2

ĐS: 48 25 311

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

p) Cho 2

và 1sin( )3

Tính 7tan2

A

ĐS: 2 2.A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

q) Cho 1sin3

và 4 2

Tính sin sin 2 sin 3cos cos 2 cos3

A

ĐS: 4 27

A

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

BT 2. Tính giá trị biểu thức lượng giác khi đề bài cho cos:

a) Cho 2

và 4cos5

Tính sincot1 cos

A

ĐS: 5

3A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

b) Cho 3 22 , cos2 3

Tính 2

cot1 cot

A

ĐS: 2 59

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

c) Cho 3cos5

và 32

Tính sin3

A

ĐS: 4 3 310

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

d) Cho 3cos5

và 32

Tính sin6

A

ĐS: 3 4 310

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

e) Cho 4cos5

và 32

Tính tan4

A

ĐS: 17

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

f) Cho 4cos 25

và 4 2

Tính cos 24

A

ĐS: 2

10A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

g) Cho 3 22

và 5cos13

Tính tan 2 .A ĐS: 120119

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

h) Cho 32

và 4cos5

Tính 3sin cos2 2

A ĐS: 39

50A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

i) Cho 02

và 4cos5

Tính sin cos4 4

A

ĐS: 4950

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

j) Cho 30, cos2 5

Tính 5 3tan 6 4cot .A ĐS: 4.A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

k) Cho 02

và cos3 1. Tính 32 3 sin 8cos .A ĐS: 4.A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

l) Cho 1cos 43

Tính 6 6 1cos sin4

A ĐS: 1.A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



m) Cho 23cos5

Tính 2tan 2 tan 1 4cot 1 .A a ĐS: 133

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

BT 3. Tính giá trị biểu thức lượng giác khi đề bài cho tan và cotan

a) Cho tan 2 thỏa .2 Tính 2sin 3cos

5cos 7 sinA

ĐS: 119

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

b) Cho tan 2. Tính 3 3

sinsin 3cos

A

ĐS: 1011

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

c) Cho tan 2. Tính 3

3

sin 2coscos 2sin

A

ĐS: 47

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



d) Cho tan 3. Tính 3 3

3sin 2 cos5sin 4cos

A

ĐS: 70139

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

e) Cho tan 2. Tính 3 3

3

8cos 2sin cos2cos sin

A

ĐS: 3

2A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

f) Cho tan 2. Tính sin 2cos 4 1

A

ĐS: 910

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

g) Cho 2 , tan 1.2 4

Tính cos sin .6

A

ĐS: 32

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

h) Cho tan 2

32

Tính 2 5sin sin sin 22 2

A

ĐS: 1 5

5A



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

i) Cho 3tan4

Tính 32 cos 2 sin 22

A

ĐS: 64

25A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

j) Cho 5cot 2.2

Tính tan4

A

ĐS: 3.A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

k) Cho 2

2

cot 33

Tính sin cos .6

A

ĐS: 32

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

BT 4. Tính biểu thức lượng giác khi đề bài cho dạng hỗn tạp:



a) Cho 90 0

sin cos 2

o

Tính 2 2sin 1 tan cos 1 cot .A ĐS: 2.A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

b) Cho 04

và 34

Tính 1 tan 1 tanA ĐS: 2.A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

c) Cho 1 sin 4 cos 4 31 sin 4 cos 4

02

Tính 2sin cos .6

A

ĐS: 32

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



d) Cho cos cos4 cos7 1sin sin 4 sin 7 2

x x xx x x

Tính cos8A ? ĐS: 35

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

e) Cho 5sin cos , ;4 4 2

Tính tan 2 .A ĐS: 9 7

35A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

f) Cho 5sin cos2

và 04

Tính sin cos .A ĐS: 32

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



g) Cho 4 4 33sin cos4

Tính 4 4sin 3cos .A ĐS: 74

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

h) Cho 4 4 74sin 3cos4

Tính 4 43sin 4cos .A ĐS: 7 57 4 28

A A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

i) Cho 32

và sin 2cos 1. Tính 2 tan cot .A ĐS: 16

A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

j) Cho tan cot 4. Tính sin cos .A ĐS: 62

A



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

k) Cho 90 180tan cot 1

o o

Tính tan cot .A ĐS: 5.A

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

2. Thu goïn bieåu thöùc löôïng giaùc

BT 5. Thu gọn: 3 3cos( ) 2sin tan cot(2 ).2 2

A x x x x

ĐS: cos .A x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

BT 6. Thu gọn các biểu thức sau đây:



a) 2 2

1 , 02sin cot cos

P xx x x

ĐS: sin

cos 2xPx

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

b) 7 32cos 3cos( ) sin tan2 2

P x x x x

ĐS: cot .P x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

c) 3 11cos(15 ) sin tan cot2 2 2

P x x x x

ĐS: 1.P

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

d) 0 0 0 0cos(270 ) 2sin( 450 ) cos( 900 ) 2sin(270 ).P x x x x ĐS: 3sin .P x



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

e) 2 2 2 23sin sin sin sin ( ).4 2 4

P x x x x

ĐS: 2 sin2 .P x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

f) cos7 .cos5 3 sin2 sin7 .sin 5 .P x x x x x ĐS: 2cos 23

P x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

g) 3 cos 2 sin 2 2sin 26

P x x x

ĐS: 2 2 sin 2

12P x

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

h) sin 2 2 2 cos 2sin 3

42 sin 3

x x xP

x

ĐS: 2 cos 1.P x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

i) 21cos cos sin .4 4 2

P x x x ĐS: 21 cos

2P x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

j) sin cos sin cos3 4 4 3

P x x x x ĐS: 0.P

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

k) 9 5 5sin cos sin cos3 4 4 3

P x x x x ĐS: 2 6

4P

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

l) 4 2 2 43cos2 5sin 4sin cos cosP x x x x x ĐS: 2.P

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

m) 1 sin 1 sin , 04

P x x x

ĐS: 2cos

2xP

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

n) 2tan 1 cot cos4 cot 2 sin 42xP x x x x

ĐS: 0.P

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

o) 2sin 4

2cos cos3 cos 5xP

x x x

ĐS: 24sin cos .P x x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

q) cos 2 sin 4 cos6cos 2 sin 4 cos6

x x xPx x x

ĐS: 2sin 2 12sin 2 1

xPx

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

r) 2

1 cos cos 2 cos32cos cos 1

x x xPx x

ĐS: 2cos .P x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

s) 22(sin 2 2cos 1)

cos sin cos3 sin 3x xP

x x x x

ĐS: 1sin

Px

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

t) 2 2sin sin sin sin3 3

P x x x x

ĐS: 3

4P

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

w) tan 2 tan tan 2 tan tan tan6 3 3 6

P x x x x x x ĐS: 1.P

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

3. Chöùng minh ñaúng thöùc löôïng giaùc

BT 7. Chứng minh các đẳng thức lượng giác sau:

a) 1 cos sinsin 1 cos

x xx x

b) sin sin 2 sin .

4 4x x x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

c) 2

21 sin 1 sin 4 tan .1 sin 1 sin

d)

2

2

1 cos (1 cos ) 1 2cot .sin sin

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

e) cos 1tan1 sin cos

xxx x

f) 4 4 3 1sin cos cos 4 .4 4

x x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

g) 6 6 5 3sin cos cos 4 .8 8

x x x h) 2tan cotsin 2

x xx

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

i) 1 1 1 1 1 1 cos cos , 0;2 2 2 2 2 2 8 2

xx x j)

2 24

2 2

sin 2 4sin tan .sin 2 4sin 4

x x xx x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

k) 2

1 cos cos2 cos 3 2cos .2cos cos 1

x x x xx x

l) cos sin cos sin 2 tan 2 .

cos sin cos sinx x x x xx x x x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

m) sin 2 cos tan(1 cos2 )(1 cos ) 2

x x xx x

n) 2

2

1 2sin 1.2 tan cos

4 4

x

x x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

o) tan tan 2 tan 2 .4 4

x x x p) tan tan tan tan 3 .

3 3x x x x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

q) 2tan tan tan 3tan 3 .3 3

x x x x r) 3 3 1cos sin sin cos sin 4 .

2x x x x x

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

s) 2 2 2 2 3cos cos cos3 3 2

x x x

t) 6sin75 cos75

2o o

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................

...................................................................................................................................................



MỘT SỐ BÀI TẬP TRẮC NGHIỆM – RÈN LUYỆN Câu 1. Theo định nghĩa trong sách giáo khoa,

A. Mỗi đường tròn là một đường tròn lượng giác.

B. Mỗi đường tròn có bán kính 1R là một đường tròn lượng giác.

C. Mỗi đường tròn có bán kính 1R , tâm trùng với gốc tọa độ là một đường tròn lượng giác.

D. Mỗi đường tròn định hướng có bán kính 1R , tâm trùng với gốc tọa độ là một đường tròn lượng giác.

Hướng dẫn giải

Lý thuyết : sách giáo khoa: Đường tròn lượng giác là đường tròn định hướng có tâm O, bán kính 1R .

Câu 2. Cho biết câu nào sai trong số các câu sau đây? Theo định nghĩa trong sách giáo khoa trên đường tròn lượng giác A. Mỗi góc MON với ,M N thuộc đường tròn đều là góc lượng giác.

B. Mỗi góc MON với ,M N thuộc đường tròn đều là góc lượng giác và có phân biệt

điểm M là điểm đầu, N là điểm cuối đều là góc lượng giác.

C. Mỗi góc MON với ,M N thuộc đường tròn đều là góc lượng giác và có phân biệt

tia đầu OM , tia cuối ON là điểm cuối đều là góc lượng giác.

D. Mỗi góc MON với 1;0A và N thuộc đường tròn đều là góc lượng giác

Hướng dẫn giải Góc lượng giác tạo bởi cung lượng giác nên đáp án A sai

Câu 3. Trên đường tròn cung có số đo 1 rad là? A. Cung có độ dài bằng 1 B. Cung tương ứng với góc ở tâm 060

C. Cung có độ dài bằng đường kính D. Cung có độ dài bằng nửa đường kính

Hướng dẫn giải Theo khái niệm trong sgk chọn đáp án D

Câu 4. Theo sách giáo khoa ta có:

A. 01 1rad B. 01 60rad C. 01 180rad D. 01801 rad



nl

O

B

A

nl

O

B

A


Xem lại sách giáo khoa Đại Số 10 trang 136 chọn đáp án D

Câu 5. Theo sách giáo khoa ta có:

A. 0 1rad B. 0 60rad C. 0 180rad D. 0180 rad


Do 0 0

0180 1801 . 180rad

chọn đáp án C.

Câu 6. Trên đường tròn bán kính r=5, độ dài của cung đo 8 là:

A. l .8

B. rl8

C. 5l8

D. kết quả khác

Hướng dẫn giải Độ dài cung AB có số đo cung AB bằng n độ:

l r.n 5.8

Câu 7. Trên đường tròn bán kính r=15, độ dài của cung có số đo 050 là:

A. l 750. B. 180l 15.

C. 15l .180

D. 180l 15. .50

Hướng dẫn giải Nôi dung hướng dẫn giải

0 0

0

.r.n 15.50l180 180

Câu 8. Trên đường tròn lượng giác, A. cung lượng giác có điểm đầu A và điểm cuối B chỉ có một số đo.

B. cung lượng giác có điểm đầu A và điểm cuối B chỉ có hai số đo sao cho tổng của chúng bằng 2 .

C. cung lượng giác có điểm đầu A và điểm cuối B chỉ có hai số đo hơn kém nhau 2 .

D. cung lượng giác có điểm đầu A và điểm cuối B có vô số đo sai khác nhau 2 .



Hướng dẫn giải Nôi dung hướng dẫn giải

Đáp án D

Câu 9. Trên đường tròn lượng giác, A. góc lượng giác có điểm đầu A điểm cuối B chỉ có một số đo.

B. góc lượng giác có điểm đầu A điểm cuối B chỉ có hai số đo sao cho tổng của chúng bằng 2 .

C. góc lượng giác có điểm đầu A điểm cuối B chỉ có hai số đo hơn hoặc kém nhau 2 .

D. góc lượng giác có điểm đầu A điểm cuối B có vô số số đo sai khác nhau 2 .

Câu 10. Trên đường tròn lượng giác với điểm gốc A , cung lượng giác có số đo 055 có điểm đầu A xác định A. chỉ có một điểm cuối M . B. đúng hai điểm cuối M .

C. đúng 4 điểm cuối M . D. vô số điểm cuối M .

Hướng dẫn giải Vì cung lượng giác có số đo xác định, điểm đầu A xác định nên chỉ có một điểm cuối M .

Câu 11. Trên đường tròn lượng giác với điểm gốc là A , cung AN , có điểm đầu là A , điểm cuối là N A. chỉ có một số đo. B. có đúng hai số đo.

C. có đúng 4 số đo. D. có vô số số đo.


Trên đường tròn lượng giác với điểm gốc là A , cung AN , có điểm đầu là A , điểm cuối là N có vô số số đo, các số đo này sai khác nhau 2 .

Câu 12. Lục giác ABCDEF nội tiếp đường tròn lượng giác có gốc là A , các đỉnh lấy theo thứ tự đó và các điểm ,B C có tung độ dương. Khi đó góc

lượng giác có tia đầu OA , tia cuối OC bằng: A. 0120 . B. 0240 .

C. 0120 hoặc 0240 . D. 0 0120 360 ,k k .




Theo bài ra ta có 120oAOC nên góc lượng giác có tia đầu OA , tia cuối OC có số đo bằng 0 0120 360 ,k k . Chọn D.

Câu 13. Trên đường tròn lượng giác có điểm gốc là A . Điểm M thuộc đường tròn sao cho cung lượng giác AM có số đo 045 . Gọi N là điểm đối xứng với M qua trục Ox , số đo cung lượng giác AN bằng: A. 045 . B. 0315 .

C. 045 hoặc 0315 . D. 0 045 360 ,k k .


Vì số đo cung AM bằng 045 nên 045AOM , N là điểm đối xứng với M qua trục Ox nên 045AON . Do đó số đo cung AN bằng 45o nên số đo cung lượng giác AN có số đo là 45 360 ,o ok k . Chọn D.

Câu 14. Trên đường tròn với điểm gốc là A . Điểm M thuộc đường tròn sao cho cung lượng giác AM có số đo 060 . Gọi N là điểm đối xứng với điểm M qua trục Oy , số đo

cung AN là: A. 120o . B. 0240

C. 0120 hoặc 0240 . D. 0 0120 360 ,k k .




Ta có 060AON , 060MON nên 0120AON . Khi đó số đo cung AN bằng 0120 .

Câu 15. Trên đường tròn lượng giác vớ điểm gốc là A . Điểm M thuộc đường tròn sao cho cung lượng giác AM có số đo 075 . Gọi N là điểm đối xứng với điểm M qua gốc tọa độ O , số đo cung lượng giác AN bằng: A. 0255 . B. 0105 .

C. 0105 hoặc 0255 . D. 0 0105 360 ,k k .


Ta có 075AOM , 0180MON nên cung lượng giác AN có số đo bằng 0 0105 360 ,k k

Câu 16. Trên đường tròn lượng giác với điểm gốc là A , điểm M thuộc đường tròn sao cho

cung lượng giác AM

có số đo 135O . Gọi N là điểm đối xứng của M qua trục Oy ,

số đo cung AN

là A. 45O . B. 315O . C. 45O hoặc 315O . D. 45 360O Ok , k


Vẽ sơ bộ hình biểu diễn và xác định vị trí của N ,

từ đó chọn đáp án D. 45 360O Ok , k .



Câu 17. Cho 3tan4

x và góc x thỏa mãn 90 180O Ox . Khi đó

A. 4cot3

x . B. 3cos5

x . C. 3sin5

x . D. 4sin5

x .


Do 90 180O Ox nên sin 0cos 0cot 0

xxx

. Ta thấy 3sin 05

x nên chọn đáp án C.

Câu 18. Cho 3sin5


A. 4cot3

x . B. 4cos5

x . C. 3tan4

x . D. 4cos5

x .


Do 90 180O Ox nên tan 0cos 0cot 0

xxx

. Ta thấy 4cos 05

x nên chọn đáp án D.

Câu 19. Cho 4cos5


A. 4cot3

x . B. 3sin5

x . C. 4tan5

x . D. 3sin5

x .


Do 90 180O Ox nên tan 0sin 0cot 0

xxx

. Ta thấy 3sin 05

x nên chọn đáp án B.

Câu 20. Cho 3cot4


A. 4tan3

x . B. 3cos

5x . C. 4sin

5x . D. 4sin

5x .


Do 0 90O Ox nên tan 0sin 0cos 0

xxx

. Ta thấy 4sin 05

x nên chọn đáp án C.



Câu 21. Gọi 2 2 2 2 2 2 2 2sin 10 sin 20 sin 30 sin 40 sin 50 sin 60 sin 70 sin 80O O O O O O O OM thì M bằng A. 0 . B. 2 . C. 4 . D. 8 .


Do 10 80 20 70 30 60 40 50 90O O O O O O O O O nên các cung lượng giác tương ứng đôi một phụ nhau. Áp dụng công thức sin(90 ) cosO x x , ta được

2 2 2 2 2 2 2 2sin 10 cos 10 sin 20 cos 20 sin 30 cos 30 sin 40 cos 40 1 1 1 1 4O O O O O O O OM

Chọn đáp án C.

Câu 22. Gọi 2 2 2 2 2 2 2 2cos 10 cos 20 cos 30 cos 40 cos 50 cos 60 cos 70 cos 80O O O O O O O OM thì M bằng A. 0 . B. 2 . C. 4 . D. 8 .


Do 10 80 20 70 30 60 40 50 90O O O O O O O O O nên các cung lượng giác tương ứng đôi một phụ nhau. Áp dụng công thức sin(90 ) cosO x x , ta được

2 2 2 2 2 2 2 2cos 10 sin 10 cos 20 sin 20 cos 30 sin 30 cos 40 sin 40

1 1 1 1 4

O O O O O O O OM

Chọn đáp án C.

Câu 23. Giá trị của biểu thức 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0cos 23 cos 27 cos 33 cos 37 cos 43 cos 47 cos 53 cos 57M

2 0 2 0cos 63 cos 67 bằng:

A. 1. B. 5 .

C. 10 . D. Một kết quả khác với các kết quả đã nêu.

Hướng dẫn giải Áp dụng công thức 0cos sin 90 , 2 2cos sin 1 ta có:



2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0

2 0 2 0

2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0

2 0 2 0

2 0 2 0 2 0 2 0 2

cos 23 cos 27 cos 33 cos 37 cos 43 cos 47 cos 53 cos 57cos 63 cos 67sin 67 sin 63 sin 57 sin 53 sin 47 cos 47 cos 53 cos 57cos 63 cos 67

sin 67 cos 67 sin 63 cos 63 sin

M

0 2 0 2 0 2 0

2 0 2 0

57 cos 57 sin 53 cos 53

sin 47 cos 47

5M Vậy chọn đáp án B .

Câu 24. Giá trị của biểu thức

2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0cos 10 cos 20 cos 30 cos 40 cos 50 cos 60 cos 70 cos 80M

2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0cos 90 cos 100 cos 110 cos 120 cos 130 cos 140 cos 150 cos 160

2 0 2 0cos 170 cos 180 bằng:

A. 0 . B. 8 . C. 9 . D. 18 .

Hướng dẫn giải Áp dụng công thức 0cos cos 180 , 2 2cos sin 1 ta có:

2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0

2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0

2 0 2 0

2 0 2 0 2 0 2 0 2 0

cos 10 cos 20 cos 30 cos 40 cos 50 cos 60 cos 70 cos 80cos 90 cos 100 cos 110 cos 120 cos 130 cos 140 cos 150 cos 160cos 170 cos 180cos 10 cos 20 cos 30 cos 40 cos 50 cos

M

2 0 2 0 2 0

2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0

2 0 2 0

60 cos 70 cos 80cos 90 cos 80 cos 70 cos 60 cos 50 cos 40 cos 30 cos 20cos 10 cos 90

2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0

2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0

2 cos 10 cos 20 cos 30 cos 40 cos 50 cos 60 cos 70 cos 80 cos 90

2 sin 80 sin 70 sin 60 sin 50 cos 50 cos 60 cos 70 cos 80 cos 90

8

Vậy chọn đáp án B .

Câu 25. Giá trị của biểu thức 2 0 2 0 2 0

2 0 2 0

tan 30 sin 60 cos 45cot 120 cos 150

M

bằng:

A. 27

. B. 17

. C. 5 66 3

. D. 713

.




Ta có:

2 2 2

2 0 2 0 2 0

2 22 0 2 0

3 3 23 2 2tan 30 sin 60 cos 45 7

cot 120 cos 150 133 33 2

M

Vậy chọn đáp án D .

Câu 26. Biết tan 2x , giá trị của biểu thức 3sin 2cos5cos 7sin

x xMx x

bằng:

A. 49

. B. 419

. C. 419

. D. 49

.


Cách 1:

Chia cả tử và mẫu của M cho cosx ta có:

sin3 2 3.2 2 4cossin 5 7.2 195 7cos

xxM x

x

.


Cách 2: Ta có: sintan 2 2 sin 2coscos

xx x xx

, thay sin 2 cosx x vào M :

3.2cos 2cos 4cos 45cos 7.2cos 19cos 19

x x xMx x x

.


Câu 27. Biết 1tan2

x , giá trị của biểu thức 2 2

2 2

2sin 3sin .cos 4cos5cos sin

x x x xMx x

bằng:

A. 813

. B. 219

. C. 219

. D. 819

.


Cách 1:

Chia cả tử và mẫu của M cho 2cos x ta có:

2

2 2

2

2

sin sin .cos 1 12 3 4 2. 3. 4 8cos cos 4 21sin 1955 4cos

x x xx xM

xx




Cách 2: Ta có: 1 sin 1tan cos 2sin2 cos 2

xx x xx

, thay cos 2sinx x vào M :

22 2

2 22

2sin 3sin .2sin 4. 2sin 8sin 819sin 195. 2sin sin

x x x x xMxx x


Câu 28. Biết , ,A B C là các góc của tam giác ABC , mệnh đề nào sau đây đúng:

A. sin sinA C B . B. cos cosA C B .

C. tan tanA C B . D. cot cotA C B .

Hướng dẫn giải Ta có:

sin sin sinA C B B

cos cos cosA C B B

tan tan tanA C B B

cot cot cotA C B B


Câu 29. Biết , ,A B C là các góc của tam giác ABC , mệnh đề nào sau đây đúng:

A. sin sinA C B . B. cos cosA C B .

C. tan tanA C B . D. cot cotA C B .


sin sin sinA C B B

cos cos cosA C B B

tan tan tanA C B B

cot cot cotA C B B

Vậy chọn đáp án C .

Câu 30. Biết , ,A B C là các góc của tam giác ,ABC khi đó



A. sin sin .C A B B. cos cos .C A B

C. tan tan .C A B D. cot cot .C A B

Hướng dẫn giải:

Vì , ,A B C là các góc của tam giác ABC nên 180 180 .o oA B C C A B

Do đó C và A B là 2 góc bù nhau

sin sin ; cos cos ; tan tan ; cot cot .C A B C a b C A B C A B


A. sin sin .C A B B. cos cos .C A B

C. tan tan .C A B D. cot cot .C A B


Vì , ,A B C là các góc của tam giác ABC nên 180 180 .o oA B C C A B

Do đó C và A B là 2 góc bù nhau

sin sin ; cos cos ; tan tan ; cot cot .C A B C a b C A B C A B


A. sin sin .2 2

A B C

B. sin cos .2 2

A B C

C. tan tan .2 2

A B C

D. cot cot .2 2

A B C


Vì , ,A B C là các góc của tam giác ABC nên 180 180o oA B C C A B

90 .2 2

oC A B Do đó

2C và

2A B là 2 góc phụ nhau.

sin cos ; cos sin ; tan cot ; cot tan .2 2 2 2 2 2 2 2C A B C A B C A B C A B


A. cos cos .2 2

A B C

B. cos cos .2 2

A B C



C. tan cot .2 2

A B C

D. cot cot .2 2

A B C



90 .2 2

oC A B Do đó

2C và




A. tan tan .2 2

A B C

B. tan tan .2 2

A B C

C. tan cot .2 2

A B C

D. tan cot .2 2

A B C



90 .2 2

oC A B Do đó

2C và




A. sin sin .2 2

A B C

B. sin sin .2 2

A B C

C. sin cos .2 2

A B C

D. sin cos .2 2

A B C



90 .2 2

oC A B Do đó

2C và



Câu 36. Với góc x bất kì



A. sin cos 1.x x B. 2 2sin cos 1.x x

C. 3 3sin cos 1.x x D. 4 4sin cos 1.x x

Câu 37. Với góc x bất kì. Khẳng định nào sau đây là khẳng định đúng? A. 2 2sin cos 2 1x x . B. 2 2sin cos 1x x .

C. 2 2sin cos 180 1x x . D. 2 2sin cos 180 1x x

Hướng dẫn giải Dựa vào công thức lượng giác cơ bản và cung liên quan đặc biệt. Do 2 2cos 180 cosx x nên 2 2 2 2sin cos 180 sin cos 1x x x x .

Chọn C.

Câu 38. Cho tan10 . tan 20 . tan 30 . tan 40 . tan 50 . tan 60 . tan 70 . tan 80M . Giá trị của M bằng A. 0M . B. 1M . C. 4M . D. 8M .

Hướng dẫn giải Ta có: tan .tan 90 tan .cot 1x x x x .

Vậy 1M . Chọn B.

Câu 39. Biết tan 2x và 2sin 3cos4sin 7cos

x xMx x

. Giá trị của M bằng

A. 1M . B. 115

M C. 115

M D. 29

M


Ta có: sintan sin tan .coscos

xx x x xx

.

Suy ra: 2 tan 3 14 tan 7 15

xMx

Chọn B.

Câu 40. Biết tan 2x và 2 2

2 22sin 3sin .cos 4cos

5sin 6cosx x x xM

x x

Giá trị của M bằng

A. 913

M B. 965

M C. 965

M D. 2429

M



xx x x xx

.

Suy ra: 2

22 tan 3 tan 4 9

135 tan 6x xM

x



Chọn A.

Câu 41. Biết tan 3x và 2 2

2 22sin 3sin .cos 4cos

5 tan 6cotx x x xM

x x

Giá trị của M bằng

A. 3147

M B. 93137

M C. 931370

M D. 3151

M



xx x x xx

; 221cos

tan 1x

x

và 1cot

tanx

x

Suy ra: 2 2

22

2 tan 3 tan 4 cos 936 13705 tan

tan

x x xM

xx

Chọn C.

Câu 42. Cho 2 2sin cos sin cosM x x x x . Biểu thức nào sau đây là biểu thức rút gọn

của M ? A. 1M . B. 2M . C. 4M . D. 4sin .cosM x x .


2 2 2sin cos sin cos 2sin .cos 1 2sin .cosx x x x x x x x ;

2 2 2sin cos sin cos 2sin .cos 1 2sin .cosx x x x x x x x

Suy ra: 2M . Chọn B.

Câu 43. Cho 2 2sin cos sin cosM x x x x . Biểu thức nào sau đây là biểu thức rút gọn

của M ? A. 2M . B. 4M . C. 2sin .cosM x x . D. 4sin .cosM x x .


2 2 2sin cos sin cos 2sin .cos 1 2sin .cosx x x x x x x x ;

2 2 2sin cos sin cos 2sin .cos 1 2sin .cosx x x x x x x x

Suy ra: 4sin .cosM x x . Chọn D.

Câu 31. Gọi 2tan cotM x x , ta có



A. 2M . B. 2 2

1sin .cos

Mx x

.

C. 2 2

2sin .cos

Mx x

. D. 4M .


22 22 2

2 sin cos sin cos 1tan cotcos sin cos .sin cos .sin

x x xM x xx x x x x x

.

Chọn B.

Câu 32. Cho tan cotx x m , gọi 3 3tan cotM x x . Khi đó A. 3M m . B. 3 3M m m . C. 3 3M m m . D. 2 1M m m .

Hướng dẫn giải 33 3 3tan cot tan cot 3 tan .cot tan cot 3M x x x x x x x x m m

Chọn C.

Câu 33. Cho sin cosx x m , gọi sin cosM x x . Khi đó

A. 2M m . B. 22M m . C. 2 2M m . D. 22M m .

Hướng dẫn giải Ta có: 22 2 2sin cos sin 2sin .cos cos 1 2sin .cosM x x x x x x x x .

Mặt khác: 2 22 2sin cos sin cos 4sin .cos 4sin .cosM x x x x x x m x x .

Suy ra: 2

2 11 2sin .cos 4sin .cos sin .cos2

mx x m x x x x .

Do đó: 2 2 22 2M m M m .

Chọn D.

Câu 34. Cho 25 2sinM x . Khi đó giá trị lớn nhất của M là A. 3 . B. 5 . C. 6 . D. 7 .

Hướng dẫn giải Ta có: 2 2 20 sin 1, 0 2sin 2, 5 5 2sin 3,x x x x x x .

Gía trị lớn nhất là 5 .

Chọn B.



Câu 35. Giá trị lớn nhất của biểu thức 2 27cos 2sinM x x là A. 2 . B. 5 . C. 7 . D. 16 .

Hướng dẫn giải 2 2 27 1 sin 2sin 7 9sinM x x x .

Ta có: 2 2 20 sin 1, 0 9sin 9, 7 7 2sin 2,x x x x x x .


Chọn C.

Câu 36. Cho 2 26cos 5sinM x x . Khi đó giá trị lớn nhất của M là A. 1. B. 5 . C. 6 . D. 11.

Hướng dẫn giải 2 2 2 2 26cos 5sin 6 1 sin 5sin 6 sinM x x x x x .

Ta có: 2 2 20 sin 1, 0 sin 1, 6 6 sin 5,x x x x x x .


Chọn C.

Câu 37. Cho 3sin 4 cosM x x . Chọn khẳng định đúng. A. 5M . B. 5 M . C. 5M . D. 5 5M .


3 45 sin cos 5sin5 5

M x x x

với 3 4cos , sin5 5

.

Ta có: 1 sin 1, 5 5sin 5,x x x x .

Chọn D.

Câu 38. Giá trị lớn nhất của 4 4sin cosM x x bằng : A. 1. B. 2. C. 3. D. 4.


Ta có 4 4 21sin cos 1 sin 22

M x x x

Vì 2 2 21 1 1 10 sin 2 1 sin 2 0 1 sin 2 12 2 2 2

x x x

Nên giá trị lớn nhất là 1.



Chọn đáp án A.

Câu 39. Giá trị lớn nhất của 4 4sin cosN x x bằng : A. 0. B. 1. C. 2. D. 3.

Hướng dẫn giải Ta có 4 4 2 2sin cos sin cos cos 2N x x x x x

Vì 1 cos 2 1 1 cos 2 1x x


Chọn đáp án B.

Câu 40. Giá trị lớn nhất của 6 6sin cosQ x x bằng :

A. 1. B. 2. C. 3. D. 6.


Ta có 6 6 23sin cos 1 sin 24

Q x x x

Vì 2 2 23 3 1 30 sin 2 1 sin 2 0 1 sin 2 14 4 4 4

x x x


Chọn đáp án A.

Câu 41. Giá trị lớn nhất của 6 6sin cosM x x bằng: A. 0. B. 1. C. 2. D. 3.

Hướng dẫn giải Ta có

6 6 2 2 4 2 2 4

2 2 2

2 2

sin cos (sin cos )(sin sin cos cos )1cos 2 (1 sin .cos ) cos 2 (1 sin 2 )4

3 1 3 1 3 1cos 2 cos 2 cos 2 14 4 4 4 4 4

( cos 2 1)

M x x x x x x x x

x x x x x

x x x

do x


Chọn đáp án B.

Câu 42. Giá trị của biểu thức 4 4 6 63(sin cos ) 2(sin cos )P x x x x là :

A. 1. B. 0. C. 1. D. 5.



Hướng dẫn giải Ta có 4 4 6 6 2 2 2 23(sin cos ) 2(sin cos ) 3(1 2sin cos ) 2(1 3sin cos ) 1P x x x x x x x x

Chọn đáp án C.

Câu 43. Biểu thức thu gọn của 2 2tan sinM x x là: A. 2tan .M x B. 2sin .M x

C. 2 2tan .sin .M x x D. 1.M


Ta có 2

2 2 2 2 2 22 2

sin 1tan sin sin sin 1 sin . tancos cos

xM x x x x x xx x

Chọn đáp án C.

Câu 44. Biểu thức thu gọn của 2 2cot cosM x x là: A. 2cot .M x B. 2cos .M x

C. 1M D. 2 2cot .cos .M x x


Ta có 2

2 2 2 2 2 22 2

cos 1cot cos cos cos 1 cos .cotsin sin

xM x x x x x xx x

Chọn đáp án D.

Câu 45. Nếu 2 2

2 2

cos sin , ( , )cot tan 4

x xM x k kx x

thì M bằng

A. 4tan x . B. 2sin 2 .x . C. 21 sin 2 .2

x . D. 21 sin 2 .4

x


2 2 2 2

4 42 2

2 2

2 2 2

cos sin cos sincos sincot tansin .cos1sin .cos sin 2 .4

x x x xMx xx xx x

x x x

Chọn D.

Câu 46. Giá trị của cos20 .cos40 .cos80M là

A. 116

. B. 18

. C. 14

. D. 1




sin 20 . sin 20 .cos20 .cos40 .cos801 sin 40 .cos40 .cos8021 sin80 .cos8041 sin16081 sin 20 .8

M

Suy ra: 18

M .

Chọn B.

Câu 47. Nếu 4 4sin cosM x x thì M bằng

A. 2 21 2sin .cosx x . B. 21 sin 2x .

C. 21 sin 2x . D. 211 sin 22

x .

Hướng dẫn giải 4 4

2 2 2 2 2

2

sin cos(sin cos ) 2sin .cos

11 sin 2 .2

M x xx x x x

x

Chọn D.

Câu 48. Nếu 6 6sin cosM x x thì M bằng

A. 2 21 3sin .cosx x . B. 21 3sin x .

C. 231 sin 22

x . D. 231 sin 24

x


2 3 2 3

2 2 4 4 2 2

2 2

2

sin cos(sin ) (cos )(sin cos )(sin cos sin .cos )

1 11 sin 2 sin 22 431 sin 2 .4

M x xx xx x x x x x

x x

x



Chọn D.

Câu 49. Giá trị nhỏ nhất của 4 4sin cosM x x là

A. 0 . B. 14

. C. 12

. D. 1.


2

sin cos11 sin 22112

1 .2

M x x

x

Dấu bằng xảy ra khi , .4 2

x k k Chọn C.

Câu 50. Giá trị nhỏ nhất của 6 6sin cosM x x là

A. 0 . B. 14

. C. 12

. D. 1.


2

sin cos31 sin 24314

1 .4

M x x

x

Dấu bằng xảy ra khi , .4 2

x k k Chọn B.

Câu 76. Đề sai, đã sửa lại.

Câu 51. Cho biểu thức 3

3

1 tan , ( , , )(1 tan ) 4 2

xM x k x k kx

, mệnh đề nào trong

các mệnh đề sau đúng?

A. 1M . B. 1M C. 14

M . D. 1 14

M .


Đặt tan , \ 1t x t .



Ta có: 3 2

3 2

1 1(1 ) 2 1

t t tMt t t

2( 1) (2 1) 1 0M t M t M . (*)

Với 1M thì (*) có nghiệm 0.t

Với 1M để (*) có nghiệm khác 1 thì

2 2

0(2 1) 4(M 1) 012 3 0

1 .4

MM

M

Và

2( 1)( 1) (2 1)( 1) ( 1) 1 04.

M MM

Chọn C.

Câu 52. Tính 0sin105 ta được :

A. 6 24 . B. 6 2

4

. C. 6 24 . D. 6 2

4

.

Hướng dẫn giải Có: 0 0 0sin105 sin(60 45 ) 0 0 0 0sin 60 .cos 45 cos60 .sin 45

0 3 2 1 2sin105 . .2 2 2 2

6 2

4 . Chọn: C.

Câu 53. Tính 0cos105 ta được :

A. 6 24 . B. 6 2

4

. C. 6 24 . D. 6 2

4

.

Hướng dẫn giải Có: 0cos105 0 0cos(60 45 ) 0 0 0 0cos 60 .cos 45 sin 60 .sin 45

0 1 2 3 2cos105 . .2 2 2 2

6 2

4

. Chọn: B.

Câu 54. Tính 0tan105 ta được : A. (2 3) . B. 2 3 . C. 2 3 . D. (2 3) .


Có: 0tan105 0

0

sin105cos105

6 24

6 24

6 26 2

(2 3) . Chọn: A.



Hoặc có thể làm theo cách sau:

0tan105 0 0tan(45 60 ) 0 0

0 0

tan 45 tan 601 tan 45 tan 60

1 31 3

(2 3) .

Câu 55. Tính 0sin165 ta được :

A. 6 24 . B. 6 2

4

. C. 6 24 . D. 6 2

4

.

Hướng dẫn giải Có: 0sin165 0 0sin(180 15 ) 0sin15 0 0sin(45 30 )

0sin105 0 0 0 0sin 45 .cos30 sin 30 .cos 45 2 3 1 2. .

2 2 2 2

6 24

. Chọn: A.

Câu 56. Tính 0cos165 ta được :

A. 6 24 . B. 6 2

4

. C. 6 24 . D. 6 2

4

.

Hướng dẫn giải Có: 0cos165 0 0cos(180 15 ) 0cos15 0 0cos(45 30 )

0cos165 0 0 0 0(cos 45 .cos30 sin 30 .sin 45 ) 2 3 1 2( . . )

2 2 2 2

6 24

Chọn : D.

Câu 57. Tính 0tan165 ta được : A. (2 3) . B. 2 3 . C. 2 3 . D. (2 3) .


Có: 0tan165 0

0

sin165cos165

6 24

6 24

6 26 2

(2 3) . Chọn D.

Hoặc có thể làm theo cách sau:

0tan165 0 0tan(135 30 ) 0 0

0 0

tan135 tan 301 tan135 tan 30

11311 ( 1).3

(2 3) .

Câu 58. Tính 0 0 0 0cos10 cos 20 cos 40 cos80M ta được M là :

A. 01 cos1016

M . B. 01 cos102

M . C. 01 cos104

M . D. 01 cos108

M .




Do 0sin10 0 nên: M 0 0 0 0 0

0

16sin10 cos10 cos 20 cos 40 cos8016sin10

0 0 0 0

0

8sin 20 cos 20 cos 40 cos8016sin10

M 0 0 0

0

4sin 40 cos 40 cos8016sin10

0 0

0

2sin80 cos8016sin10

0

0

sin16016sin10

M 0

0

sin 2016sin10

0 0

0

2sin10 cos1016sin10

01 cos108

.

Chọn: D.

Câu 59. Gọi 4 o 4 ocos 15 sin 15M thì:

A. 1.M B. 3 .2

M C. 1 .4

M D. 0.M


2 24 o 4 o 2 o 2 o 2 o 2 o 2 o 2 o

2 o 2 o o o

cos 15 sin 15 cos 15 sin 15 cos 15 sin 15 cos 15 sin 15

3cos 15 sin 15 cos 2.15 cos30 .2

M

Câu 60. Gọi 6 o 6 ocos 15 sin 15M thì:

A. 1.M B. 1 .2

M C. 1 .4

M D. 15 3 .32

M


6 6 2 2 4 2 2 4

22 2 2 2

2

cos sin cos sin cos cos .sin sin

cos 2 . cos sin cos .sin

1cos 2 . 1 sin 24

Vậy o 2 o1 3 1 1 15 3cos30 . 1 sin 30 . 1 . .4 2 4 4 32

M

Câu 61. Gọi 4 o 4 o 2 o 2 ocos 15 sin 15 cos 15 sin 15M thì:

A. 1.M B. 1 .2

M C. 1 .4

M D. 0.M

Hướng dẫn giải Ta có: 4 o 4 o 2 o 2 ocos 15 sin 15 cos 15 sin 15M

2 o 2 o 2 o 2 o 2 o 2 ocos 15 sin 15 cos 15 sin 15 cos 15 sin 15



2 o 2 o 2 o 2 ocos 15 sin 15 cos 15 sin 15 0.

Câu 62. Gọi 4 o 4 o 2 o 2 ocos 15 sin 15 cos 15 sin 15M thì:

A. 3.M B. 1 .2

M C. 1 .4

M D. 0.M

Hướng dẫn giải Ta có: 4 o 4 o 2 o 2 ocos 15 sin 15 cos 15 sin 15M

2 o 2 o 2 o 2 o 2 o 2 ocos 15 sin 15 cos 15 sin 15 cos 15 sin 15

2 o 2 o 2 o 2 o o ocos 15 sin 15 cos 15 sin 15 cos30 cos30 3.

Câu 63. Gọi 1 sin 2 cos 2M x x thì: A. 2cos . sin cos .M x x x B. cos . sin cos .M x x x

C. 2 cos .cos .4

M x x

D. 2 2 cos .cos .4

M x x

Hướng dẫn giải Ta có: 1 sin 2 cos 2 1 sin 2 cos 2M x x x x

2sin cos cos sin cos sinx x x x x x

sin cos sin cos cos sinx x x x x x

sin cos .2cos 2 cos .2cos4

x x x x x

Câu 64. Gọi cos cos 2 cos3M x x x thì:

A. 2cos 2 cos 1 .M x x B. 14cos 2 . cos .2

M x x

C. 2cos 2 .cos .cos .2 6 2 6x xM x

D. 4cos 2 .cos .cos .

2 6 2 6x xM x

Hướng dẫn giải Ta có: cos cos 2 cos3 cos cos3 cos 2M x x x x x x

2cos 2 .cos cos 2 cos 2 2cos 1x x x x x 12cos 2 cos2

x x

2cos 2 cos cos 2cos 2 .2cos cos3 2 6 2 6

x xx x x



Câu 65. Gọi tan tanM x y thì:

A. tan .M x y B. sin.

cos .cosx y

Mx y

C. sin.

cos .cosx y

Mx y

D. tan tan .1 tan .tan

x yMx y


Ta có: sin sin sin cos cos sintan tancos cos cos cos

x y x y x yM x yx y x y

sincos cos

x yx y

Câu 66. Gọi tan tan M x y thì:

A. tan tan M x y . B. sincos .cos

x yM

x y. C. sin

cos .cos

x y

Mx y

. D. tan tan1 tan .tan

x yM

x y

.


Ta có: tan tan M x y sin sinycos cos

xx y

sin .cos siny.coscos .cos

x y xx y

sincos .cos

x yx y

.

Câu 67. Gọi cot cot M x y thì:

A. cot M x y . B. sinsin .siny

x yM

x. C. sin

sin .sin

y x

Mx y

. D. tan tan1 tan .tan

x yM

x y

.


Ta có: cot cot M x y cos cossin siny

x yx

cos .siny sin .cosysin .siny

x xx

sin y xsin .siny

x.

Câu 68. Gọi cot cot M x y thì:

A. cot M x y . B. sinsin .siny

x yM

x. C. sin

sin .siny

y x

Mx

. D. cot .cot 1cot cot

y xMy x

.


Ta có: cot cot M x y cos cossin siny

x yx

cos .siny sin .cosysin .siny

x xx

sinsin .siny

x yx

.

Câu 69. Gọi 0 0 0 0 0 0

1 1 1cos10 .cos 20 cos 20 .cos30 cos30 .cos 40

M thì:

A. 0 0

1sin 20 .cos 40

M B. 0 0tan 40 tan 20 M



C. 0 0

12cos10 .cos 40

M D. M có kết quả khác với các kết quả nêu

trên.


Ta có: 0 0 00

00 0 0 0 0

sin 20 cos 40 cos 20sin 20sin 20 .cos10 .cos 20 cos 20 .cos30 .cos 40

M

0 0 0 0 0

0 0 0 0 0

2sin10 .cos10 2.sin 20 .cos30 .cos10cos10 .cos 20 cos 20 .cos30 .cos 40

0 0 0

0 0 0

2sin10 2.sin 20 .cos10cos 20 cos 20 .cos 40

0 0 0 0

0 0

2sin10 .cos 40 2.sin 20 .cos10cos 20 .cos 40

0 0 0 0

0 0

sin 50 sin 30 sin 30 sin10cos 20 .cos 40

0 0

0 0

sin 50 sin10cos 20 .cos 40

0 0

0 0

2.sin 30 .cos 20cos 20 .cos 40

0

1cos 40

.

0 0

1sin 20 .cos 40

M .

Câu 70. Cho A , B , C là các góc của tam giác ABC thì: A. sin 2 sin 2 sin 2 4cos .cos .cos A B C A B C .

B. sin 2 sin 2 sin 2 4cos .cos .cos A B C A B C .

C. sin 2 sin 2 sin 2 4sin .sin .sin A B C A B C .

D. sin 2 sin 2 sin 2 4sin .sin .sin A B C A B C .


sin 2 sin 2 sin 2 A B C sin 2 sin 2 sin 2 A B C 2sin .cos 2sin .cosC A B A B C

2sin .cos 2sin .cosC C A B C 2sin . cos cosC C A B

4sin .cos .cos C A B C A B C 4sin .cos .cos2 2

A B C A B CC

4sin .cos .cos2 2

C A B 4sin .sin .sin C A B .

Câu 71. Cho A , B , C là các góc của tam giác ABC (không phải tam giác vuông) thì:

A. tan tan tan tan .tan .tan2 2 2

A B CA B C .



B. tan tan tan tan .tan . tan2 2 2

A B CA B C .

C. tan tan tan tan .tan .tan A B C A B C .

D. tan tan tan tan .tan .tan A B C A B C .


Ta có: tan tan tan A B C tan tan tan A B C sin sincos .cos cos

A B CA B C

cos cos .cossin .

cos .cos .cos

A B A BC

A B Csin .sin .sincos .cos .cos

A B CA B C

tan .tan . tan A B C .

Câu 72. Cho A , B , C là các góc của tam giác ABC (không phải tam giác vuông) thì:

A. cot cot cot cot .cot .cot2 2 2 2 2 2

A B C A B C .

B. cot cot cot cot .cot .cot2 2 2 2 2 2

A B C A B C .

C. cot cot cot cot .cot .cot2 2 2

A B C A B C .

D. cot cot cot cot .cot .cot2 2 2

A B C A B C


cot cot cot2 2 2

A B C cot cot cot2 2 2

A B Csin cos2 2 2sin .sin sin

2 2 2

A B C

A B C

sin sin .sin

2 2 2cos .2 sin .sin .sin

2 2 2

C A BC

C A B

cos sin .sin2 2 2 2cos .

2 sin .sin .sin2 2 2

A B A BC

C A B

cos .cos .cos

2 2 2sin .sin .sin

2 2 2

C B A

C A B cot .cot .cot2 2 2

A B C

Câu 73. Cho , ,A B C là các góc của tam giác ABC thì tan .tan tan . tan tan . tan2 2 2 2 2 2A B B C C A

A. 1. B. 1 .



C.2

tan .tan .tan2 2 2A B C

. D. Một kết quả khác các kết quả đã nêu

trên.


Ta có: tan . tan tan . tan tan .tan2 2 2 2 2 2A B B C C A

tan . tan tan tan .tan2 2 2 2 2

tan .tan . 1 tan .tan tan .tan2 2 2 2 2 2 2

tan .tan . 1 tan .tan tan .tan2 2 2 2 2 2 2

tan .cot . 1 tan .tan tan .tan2 2 2 2 2 2

1. 1 tan .tan2

B A C C A

B A C A C C A

B B A C C A

B B A C C A

A

tan .tan 1.2 2 2C C A

Chọn A.

Câu 74. Cho , ,A B C là các góc của tam giác ABC (không là tam giác vuông) thì

cot .cot cot .cot cot .cotA B B C C A A. 1. B. 1 .

C. 2cot .cot .cotA B C . D. Một kết quả khác các kết quả đã nêu

trên.

Hướng dẫn giải Ta có : cot .cot cot .cot cot .cotA B B C C A

1 1 1tan .tan tan .tan tan .tanA B B C C A

tan tan tan

tan .tan .tanA B C

A B C

.

Mặt khác : tan tan tan tan 1 tan .tan tanA B C A B A B C

tan 1 tan .tan tanC A B C

tan 1 tan . tan tanC A B C tan tan .tanC A B .

Nên cot .cot cot .cot cot .cot 1A B B C C A .

Chọn A.

Câu 75. Cho , ,A B C là các góc của tam giác ABC thì:



A. cos cos cos 1 4sin .sin .sin2 2 2A B CA B C .

B. cos cos cos 1 4sin .sin .sin2 2 2A B CA B C .

C. cos cos cos 1 4cos .cos .cos2 2 2A B CA B C .

D. cos cos cos 1 4cos .cos .cos2 2 2A B CA B C .

Hướng dẫn giải Ta có : cos cos cosA B C

2

cos 2cos .cos2 2

cos 2cos .cos2 2

1 2sin 2sin .cos2 2 2

1 2sin . sin cos2 2 2

1 2sin . sin cos2 2 2

1 2sin . cos cos2 2 2

1 4sin .sin .sin .2 2 2

B C B CA

A B CA

A A B C

A A B C

B CA B C

A B C B C

A B C

Chọn B.

Câu 76. Cho , ,A B C là các góc của tam giác ABC thì

A. sin 2 sin 2 2sinA B C . B. sin 2 sin 2 2sinA B C .

C. sin 2 sin 2 2sinA B C . D. sin 2 sin 2 2sinA B C .


Ta có:

sin 2 sin 2 2sin .cos

2sin .cos

2sin .cos 2sin .

A B A B A B

C A B

C A B C

Dấu đẳng thức xảy ra khi cos 1A B A B .



Chọn A.

Câu 77. Gọi 2 4 6cos cos cos7 7 7

M thì:

A. 0M B. 12

M . C. 1M . D. 2M .


Ta có: 2 4 62 .sin 2cos .sin cos .sin cos .sin7 7 7 7 7 7 7

M

3 5 3 7 5sin sin sin sin sin sin7 7 7 7 7 7

sin sin sin .7 7

Nên 12

M . Chọn B.

Câu 78. Gọi 2 4 6cos cos cos7 7 7

M thì:

A. 0M . B. 12

M .

C. 1M D. Một kết quả khác các kết quả đã nêu trên.


Câu 79. Gọi cos .cos sin .sinM a b a b a b a b thì :

A. 21 2cosM a B. 21 2sinM a .

C. cos 4M a D. sin 4M a .

Hướng dẫn giải Ta có: cos .cos sin .sinM a b a b a b a b

2cos cos 2 1 2sina b a b a a .

Chọn B.

Câu 80. Gọi cos .cos sin .sinM a b a b a b a b thì :

A. 21 2sinM b . B. 21 2sinM b .



C. cos 4M b . D. sin 4M b .

Hướng dẫn giải Ta có: cos .cos sin .sinM a b a b a b a b

2cos ( ) cos 2 1 2sina b a b b b .

Chọn A.

Documents

PHẦN V: BIẾN ĐỔI LƯỢNG GIÁC · tÀi liỆu hỌc tẬp toÁn 10 – chẤt lƯỢng bÀi giẢng: biẾn ĐỔi lƯỢng giÁc admin trẦn vĂn tÀi – 0977.413.341 -