BAB 3 Rumus-Rumus Trigonometri

7/30/2019 BAB 3 Rumus-Rumus Trigonometri

1/30

Kunci Penyelesaian Matematika SMA Jl.2A- Sukino Bab 3 | page 51

Latihan Kompetensi Siswa 1

A. Evaluasi Pengertian atau Ingatan

1. C. xx sincos32

1

30cos x = 30sinsin30coscos xx

=2

1.sin3

2

1.cos xx

= xx sincos32

1

2. A. xx sincos22

1

45cos x = 45sinsin45coscos xx

= xx sincos22

1

3. A. 1321 75cos = 4530cos

= 45sin30sin45cos30cos

= 221.

212

21.3

21

= 1321

4. A.22

13

15cos = 4560cos= 45sin60sin45cos60cos

= 21

.32

12

1.

2

1

= 2

231241

=22

13

5. C. yx22

sincos

yxyx coscos = yxyx sinsincoscos yxyx sinsincoscos

= yxyx2222

sinsincoscos

= yxyx 2222 sincos1sin1cos = yx

22sincos

6. A. 60cos x

xx cos2

1sin3

2

1 = xx cos60cossin60sin

= 60cos x

7. B. -0,5

cos = sinsincoscos

=14

35.

34

14

11.

7

1

= 5,098

49

8

6011

8. D.2

12

13

2

1aa

3cos

=

3sinsin

3coscos

= 3

2

1.

2

1.1

2aa

9. A. 1

35sin125sin35cos125cos = 35125cos= 90cos = 1

10. B.5

3

cos = sinsincoscos

=5

3.

25

7

5

4.

25

24

= 5.25

75

= 5

3

11. E.125

117

cos = sinsincoscos

=25

7.

5

3

5

4.

25

24 =

125

117

12. D.2

1

BA cos = BABA sinsincoscos

= 0.2

31.

2

1 =

2

1

13. E. x7cos

xxxx 3cos4cos3sin.4sin = xx 34cos = x7cos

BAB 3

RUMUS RUMUS

TRIGONOMETRI


2/30


14. B. 3122

1

22

1cos P 45P

32

1cos Q 30Q

R = 105180 QP

105coscosR = 4560cos= 45sin60sin45cos60cos

= 22

1.3

12

2

1.

2

1

= 3124

1

15. B.5

13

Ccos = BA 180cos= BABA sin180sincos180cos

= BABA

sin.0cos.1= BABA sinsincoscos

=

17

15.

5

3

17

8.

5

4

=85

13

B. Evaluasi Pemahaman dan PenguasaanMateri

1. a. ba 32cos = baba 3sin2sin3cos2cosb. yx3cos = yxyx sin3sincos3cos

c. 45cos x = 45sinsin45coscos xx

= xx sincos22

1

d. 15cos x = 15sinsin15coscos xx

e.

3cos2

x = 2

3sinsin

3coscos

xx

= 2

xx sin3

2

1cos

2

1

= xx sin3cos22

1

f.

6cos32 x = 32

6sinsin

6coscos xx

= 32

xx sin

2

1cos3

2

1

= 3 xx sincos3

g.

BA

2

1

3

1cos =

= BABA2

1sin

3

1sin

2

1cos

3

1cos

h.

BA

4

1

3

2cos =

= BABA1

sin3

2sin

1cos

2cos

2. a. x90cos = xx sin90sincos90cos = xx sin.1cos.0

= xsin

b. x270cos = xx sin270sincos270cos = xx sin.1cos.0= xsin

c. x270cos = xx sin270sincos270cos = xx sin.1cos.0= xsin

d. 90cos x = 90sinsin90coscos xx = xx sin.1cos.0

= xsin

e. 180cos x = 180sinsin180coscos xx = xx sin.0cos.1

= xcos

f.

5

2cos =

=

5sin2

sin5cos2

cos

= 5sin.15cos.0 = 5sin

g.

2

3cos =

2

3sinsin

2

3coscos

= sin.1cos.0 = sin

h. 2cos = sin2sincos2cos = sin.0cos.1

= cos

3. a.3

sin3

sin3

cos3

cos

=

33

cos

= 1

b.6

sin3

sin6

cos3

cos

=

63

cos

=

2cos

= 0

c.4

sin3

2sin

4cos

3

2cos

=

43

2cos

=

12

11

cos

d.3

sin5

sin3

cos3

5cos

=

4

3

3

5cos

=

12

29cos

e. 60sin50sin60cos50cos = 110cos

f. 30sin40sin30cos40cos = 70cos


3/30

Kunci Penyelesaian Matematika SMA Jl.2A- Sukino Bab3 | page 53

4. a. 60sin180sin60cos180cos == 60180cos

=2

1

b. 90sin150sin90cos150cos == 240cos

= 240cos= 60180cos

= -2

1

c. 10sin140sin10cos140cos =

= 10140cos= 150cos

= -2

13

d. 20sin200sin20cos200cos =

= 20200cos= 180cos= -1

5. a. cos = sinsincoscos

=13

5.

5

3

13

12.

5

4

=5

63

b. cos = sinsincoscos

=

13

5.

5

3

13

12.

5

4

=5

33

c. cos = sinsincoscos

=

13

5.

5

3

13

12.

5

4

=5

33

d. cos = sinsincoscos

=

13

5.

5

3

13

12.

5

4

=5

33

e.

2cos =

sin

2sincos

2cos

= sin.1cos.0

= sin

f.

2cos =

sin

2sincoscos

= sin.1cos.0

= sin

g.

4cos =

sinsincoscos

= sin.22

1cos.2

2

1

= sincos22

1

h.

6cos = sin

6sincos

6cos

= sin.2

1cos.3

2

1

= sincos32

1

7. a. BA cos = BABA sinsincoscos = 32

1

=2

1coscos BA = 3

2

1

= BAcoscos = 132

1

b. BA cos = BABA sinsincoscos

= 2

113

2

1

= 2321

8. a. BA cos = BABA sinsincoscos

=10

9.

10

1

10

19.

10

99

=100

91881


=10

9.

10

1

10

19.

10

99

=100

91881

c. BA cos = BABA sinsincoscos

=10

9.

10

1

10

19.

10

99

=100

91881

d. BA cos = BABA sinsincoscos

=10

9.

10

1

10

19.

10

99

=100

91881


4/30


e. BA cos = BABA sinsincoscos

=10

9.

10

1

10

19.

10

99

=100

91881

f.

BAcos = BABA sinsincoscos

=10

9.

10

1

10

19.

10

99

=100

91881


=13

5.

25

24

13

12.

25

7

=325

204


=13

5.

25

24

13

12.

25

7

=325

36


=13

5.

25

24

13

12.

25

7

=325

204

d. BA

cos = BABA sinsincoscos

=13

5.

25

24

13

12.

25

7

=325

36


=5

3.

13

12

5

4.

13

5

=65

16


=5

3.

13

12

5

4.

13

5

=5

56

11. a. 22 sinsincoscos BABA

=BABA

BABA

sinsin2sinsin

coscos2coscos

22

22

= BABA

BABA

sinsincoscos2

sinsincoscos 2222

= BA cos22

b. BA cos . BA cos

= BABA

BABA

sinsincoscos

sinsincoscos

= BABA 2222 sinsincoscos

= BABA 2222 cos1sincossin1 = AB 22 sincos

12. 222

1QP = 22 coscossinsin

2

1yxyx

=

yxyx

yxyx

sinsincoscos2

coscossinsin

2

1 2222

= yx cos222

1

= 1 + cos(x y)

13. 45cos2 x = 45cos x= 45sinsin45coscos2 xx =

45sinsin45coscos xx

= 45coscos x = 45sinsin3 x

a.x

xx

cos

sintan =

45sin3

45cos=

3

1

b.x

xsin

coscot =

1

3= 3

c. x30cos = xx sin30sincos30cos

= xx sin2

1cos3

2

1

= xx sincos32

1

d. 30cos x = 30sinsin30coscos xx

= xx sin2

1cos3

2

1

= xx sincos32

1

14.

t

t

30cos

30cos=

t

tt

sin30sincos30cos

sin30sincos30cos

tt sin2

1cos3

2

1 =

ttn sin

2

1cos3

2

1

nt 1cos321

= 1sin21

nt

tt

cos

sintan =

1

13

21

21

n

n

= n

n

1

13


5/30


15. a.

3

4cos

3

2coscos AAA

=

3

4sinsin

3

4coscos

3

2sinsin

3

2coscoscos

AA

AAA

=

AA

AAA

sin32

1cos

2

1

sin321cos

21cos

= 0

b. cos + cos = sinsincoscos +

sinsincoscos

= coscos2

c. cos - cos = sinsincoscos -

sinsincoscos = sinsin2

C. Evaluasi Kemampuan Analisis

1. cos = 2 cos= sinsincoscos =

2 sinsincoscos = sinsin3coscos

=

cos

sin.

cos

sintan.tan =

3

1

2.

3cos

=

3sinsin

3coscos

= sin32

1cos

2

1 =

1..(1)

3cos

=

3sinsin

3coscos

= sin32

1cos

2

1 =

14

11..(2)

(1) dan (2)

sin321cos

21 = 7

1

sin32

1cos

2

1 =

14

11

+

14

13cos

3. cos + cos = sinsincoscoscos

= sinsincoscos180cos = sinsincoscoscos

=

sinsincoscos

sinsincoscos

= sinsin2

4. a.

BA

BA

coscos

cos=

BA

BABA

coscos

sinsincoscos

= BA tantan1

b. A

BA

sinsin

cos=

BA

BABA

sinsin

sinsincoscos

= 1cotcot BA

c.

BA

BA

coscos

cos=

A

BABA

coscos

sinsincoscos

= BA tantan1

d. A

BAsinsin

cos =BA

BABAsinsin

sinsincoscos

= 1cotcot BA

e.

BA

BA

sinsin

cos

=

A

BABA

sinsin

sinsincoscos

= BA cotcot1

f.A

BA

cossin

cos =

A

BABA

cossin

sinsincoscos

= BA tan11.cot

= BA tancot

g.

BA

BA

cossin

cos

=

BA

BABA

cosin

sinsincoscos

= BA tancot= AB cottan

h. BA

sinsin

cos

=

A

BABA

sinsin

sinsincoscos

= 1cotcot BA

= - BAcotcot1

8. a.

3

cos4

x =

3

sinsin3

coscos4

xx

=

xx sin3

2

1cos

2

14

= xx sin32cos2

jadi, a = 2 dan b = 32

b.

4cos2

x =

4sinsin

4coscos2

xx

=

xx sin2

2

1cos2

2

12

= xx sincos jadi, a = 1 dan b = 1


6/30


9. a. cos.cos6sin8 r= sinsincoscos r

6cos r cos

6r

8sin r sin

8r

cos

6=

sin

8

cos

sintan =3

48

b. cos.cos3sin4 r= sinsincoscos r

3cos r cos

3r

4sin r sin

4r

cos

3=

sin

4

cos

sintan =

3

4

c. cos.sin7cos4 r= sinsincoscos r

4cos r cos

4r

7sin r sin

7r

cos

4=

sin

7

cos

sintan =

7

d. cos.cos5sin12 r= sinsincoscos r

5cos r cos

5r

12sin r sin

12r

cos

5=

sin

12

cos

sintan =

5

12

10. a.

3cos

3coscos 222

=

3cos2cos 22

=

2

2

2

12cos

= 2cos2

1

b. 222 sin60cos60cos

= 260sinsin60coscos +

2sin60sincos60cos - 2sin

=

2

sin32

1cos2

1

+

2

sin32

1cos2

1

-

2sin

= 222 sinsin2

3cos

2

1

= 22 sin2

1cos

2

1 =

2

1

c. 15sin75cos 22 =dari soal no.11b bagian B diperoleh :

15sin75cos22

= 1575cos1575cos= 60cos90cos

= 02

1.0



1. E. 1

11sin79cos11cos79sin = 1179sin

= 90sin = 1

2. A. 31


= 31

3. E. 32

1


= 31

4. A. xx cossin21 45sin x = 45sincos45cossin xx

= xx cossin22

1

5. A. 1626

1

30sin x = 30sincos30cossin xx

= xx cos2

1sin3

2

1

=1

.

2

1

3

22.3

2

1

= 1626

1

6. A. 30sin x

xx cos2

1sin3

2

1 = xx cos30sinsin30cos

= 30sin x

7. A.2

1 a

140sin = 5090sin= 50sin90cos50cos90sin

= 50sin.01.1 2 a

=2

1 a


7/30


8. A.22

13


=

2

13

2

12

2

1

= 2

2132

4

1

=22

13

9. C.22

31


=

2

13

2

12

2

1

= 2

2132

4

1

=22

13

10. A. xy 22 coscos

yxyx sinsin = yxyx sincoscossin yxyx sincoscossin

= yxyx 2222 sincoscossin

= yxyx 2222 cos1coscoscos1 = xy 22 coscos

11. D. sin2

4sin

4sin =

sincoscos

4sin

sin

4coscos

4sin

=

sincos2

= sin22

1.2 = sin2

12. A.6

1

sin = sincoscossin = 30sin

=2

1sincos

3

1

6

1

sincos

13. E.125

44

BAsin = BABA sincoscossin

=25

7.

5

4

25

24.

5

3

=

125

44

14. A.221

21

BAsin = BABA sincoscossin

=17

8.

13

12

17

15.

13

5

=221

21

15. D.25

24

Csin = BA 180sin= BABA sin180coscos180sin= BA sin1= BA sin= BABA sincoscossin

=5

3.

5

4

5

4.

5

3 =

25

24


1. a. BA sin = 6030sin = 1

BA sin = 6030sin =2

130sin

b. BA sin = 3060sin = 1

BA sin = 3060sin =2

130sin

c. BA sin =

46sin

= sin6

cos4

cos6

sin

=

22

31

BA sin =

46sin

= sin6

coscos6

sin

=

22

31

d. BA sin = 4030sin = 70sin BA sin = 4030sin = 10sin

= 10sin


8/30


e. BA sin =

312sin

=

12

5sin

BA sin =

312sin

=

12

3sin

=

4sin

= 2

2

1

f. BA sin =

44sin =

2sin

= 1

BA sin =

44sin = 2sin

2. a. BA sin = BABA sincoscossin

=13

5.

5

3

13

12.

5

4 =

65

63

b. BA sin = BABA sincoscossin

=13

5.

5

3

13

12.

5

4 =

65

33

c. BA sin + BA sin =65

63+

65

33=

65

98

d. BA sin - BA sin =65

63-

65

33=

65

30

e. BA sin . BA sin =65

63.

5

33=

2425

2079

f. BA

BA

sin

sin=

65

63x

33

65=

33

63

3. a. xsin = xx sincoscossin

= xx sin1cos.0 = xsinb. xsin = xx sincoscossin

= xx sin1cos.0 = xsin

c.

x

2

3sin

= xx sin

2

3coscos

2

3sin

= xx sin.0cos.1 = cos

d.

x

2

3sin

= xcos

e.

2sin

x =

2sincos

2cossin

xx

= xx cos.10.sin = xosf. xsin = sincoscossin xx

= xx cos.0sin.1 = xsin

g.

x

2sin

= xx sin

2coscos

2sin

= xx sin.0cos.1 = xos

h.

2

3sin

x =

2

3sincos

3cossin

xx

= xx cos.1sin.0 = xcos

4. a. BA

BA

sin

sin=

BABA

BABA

sincoscossin

sincoscossin

=

BA

BABABA

BABA

coscos

sincoscossincoscos

sincoscossin

= A

BA

anan

tantan

b. BA

sinsin

sin=

BABA

sinsin

sincoscossin

= AB cotcot

c.

BA

BA

sinin

sin =

BA

BABA

sinsin

sincoscossin

= AB cotcot

d.

BA

BA

coscos

sin=

BA

ABBA

coscos

cossincossin

= BA tantan

5. a. 105sin = 4560sin= 45sin60cos45cos60sin

=

2

13

2

12

2

1

= 2

2132

4

1

=22

31

b. 105sin = 105sin

=

22

31

c. 195sin = 45150sin = 45sin150cos45cos150sin

= 6090cos6090sin22

1

=4

62

d. 255sin = 45210sin= 45sin210cos45cos210sin

= 30180cos30180sin221

=

3

2

1

2

12

2

1

=62

6. a.

sinsin

coscos

=

sincoscossinsincoscossin

sinsincoscossinsincoscos

=

cossin2

coscos2= cot


9/30


b. 2

1sin

2

1cos =

2

1sin

2

1cos

2

1cos

2

1sin

2

1cos

=

2

1sin2

1cos

2

1cos

2

1sin180

2

1cos

= 2

1sin

2

1cos

2

1cos

2

1sin

2

1sin

=

2

1sin

2

1cos

2

1cos

2

1sin

2

1sin

2

1cos

2

1cos

2

1sin

= 2

1cos

2

1sin2

7. a.

5cos

cos

5sin

sin =

5cos5in

5sincos5cossin

=

5cos5sin

6sin

b. cos + cos =

sinsincoscos + sinsincoscos

= coscos2

c. cos - cos = cossin2d. sin + sin =

sincoscossin + sincoscossin

= cossin2

e. sin - sin = sincos2

8. a. BAcossin2

dari soal no.7 (d) diperoleh :

BAcossin2 = BAsin + BA sin= 0,7 + 0,5 = 1,2

b. BAsincos2

dari soal no.7 (e) diperoleh :

BAsincos2 = BAsin - BA sin= 0,7 - 0,5 = 0,2

c. BA cos = BABA sinsincoscos = BABA coscos

2

1-

BABA coscos2

1

= BA cos = 32

1

d. BAsin = 0,5

9. cos = sinsincoscos =3

cos

= 0,1 - sinsin =21

sinsin = -0,4

a. cos = sinsincoscos = 0,1 + (-0,4) = -0,3

b. cos = sinsincoscos = 0,1 + (-0,4) = -0,3

c. sin = -0,1d. sin = -0,1

10.2

sinsincoscos BABA +2

cossinsincos BABA

=

BABABA

BABA

BABABA

22

2222

22

cossinsinsincoscos2

sincossinsin

sinsincoscos2coscos

= BBABBA 222222 cossincoscossinsin = BBAA 2222 cossincossin = 1


1. a. CBA sin = CBA sin= CBACBA sincoscossin

= CBCBA

CBCBA

sincoscossincos

sinsincoscossin

=CBACBA

CBACBA

sinsinsinsincoscos

cossincoscoscossin

b. CBA cos =

CBACBA sinsincoscos=

CBACBA

CBACBA

sincossincossinsin

sinsincoscoscoscos

3. Asincos = Asincos sincoscoscossincos AA =

AA sincoscoscossincos

=

sincoscossincos

coscos2sin

A

A

= Atan =

coscos2

sincossin

AA

4. BA cos = BABA sinsincoscos =5

3

BA tantan =B

BA

coscos

sinsin= 2

BAsinsin = BAcoscos2

BA cos = BAcoscos3 =5

3

b. BAcoscos =

5

1

a. BAsinsin =5

1.2 =

5

2


10/30



=5

1-

5

2=

5

1

5. a. 15sin75sin 22 = 15sin1590sin 22

= 15sin15cos 22

dari soal no.11(b) bagian B := 1515cos1515cos = 0cos30cos

= 1.2

1=

2

1

b. 75cos15cos 22 = 1590cos15cos2

= 15sin15cos 22

=2

1


=17

15.

13

5

17

15.

13

12

=

221

255

b. BA sin = BABA sincoscossin

=17

15.

13

12

17

15.

13

5

=

21

105

c. BA sin =221

255

d. BA cos =221

105

8. a. BA sin = BABA sincoscossin

=

13

12.

17

8

13

5.

17

15=

21

21


=

13

12.

17

15

13

5.

17

8

=221

140

c. kuadran III, karena BAsin dan BA cosbernilai negatif.

9. BftaBftay sin2coscos2sin

Bfta 2sin.

10.21

yy =

tAtA cos

2cos

21

=

2sinsin

2coscos

1

ttA +

sinsincoscos2 ttA = tA sin1 + tA cos2

= tAtA sincos 12

11. a. sinr = sincoscossin rr = cos6sin8

8cosr os

8r

6sin r sin

6r

cos

8=

in

6

cossintan = 4

3

b. sinr = sincoscossin rr

= cos3sin4

4cosr os

4r

3sin rsin

3r

cos

4=

sin

3

cos

sintan =

4

3

c. 2cosr cos

2r

3sin r sin

3r

cos

2=

sin

3

cos

sintan =

2

3

d. 12cosr os

12r

5sin r sin

5r

cos

12=

sin

5

cos

sintan =

12

5

12. a.

4sin2

x =

4sincos

4cossin2

xx

= xx cossin

b.

4sin2

x =

4sincos

4cossin2

xx

= xx cossin

c.

6sin2

x =

6sincos

6cossin2

xx

=

xx cos

2

1sin3

2

12

= xx cossin3

d.

3sin

x -

6cos

x =

=3

sincos3

cossin

xx -

6sinsin

6coscos

xx

= xxxx sincos32

1cos3sin

2

1

= xsin


11/30


15. BABABABA cossin2cossin2= BABA sincoscossin2

BABA sinsincoscos - BABA sincoscossin2 BABA sinsincoscos

= BAABBA 22 sincossin4cossinsin4 = BAsinsin4 BABA sincoscossin = BABA sinsinsin4= CBA 180sinsinsin4= CBA sinsinsin4



1. C. tan (-x)

4

4

tan1

tan1

x

x=

44

44

tan.tan1

tantan

x

x

=

44tan

x

= tan (-x)

2. B. 2tan

452tan1

452tan1

=

452tan.45tan1

452tan45tan

= 45245tan

= 2tan

3. A. CBA tan4. C. CBA tan5. A. -1

66an.69an1

66tan69tan= 6669tan

= 135tan

= 45180tan = -1

6. D. 54tan

9sin9cos

9sin9cos=

9an1

9tan1=

9an.45tan1

9tan45tan

= 945tan= 54tan

7. B.

tan =

tan.tan1

tantan

= 1.1

6661

66

61

111

65

111

6

5

= 45

8. B. 34tan

11sin11cos

11sin11cos=

11an1

11tan1

=

11tan.45tan1

11tan45tan

= 1145tan= 34tan

9. B.11

BA cot = BAtantan

tan.tan1

=BA

BA antan

tan.tan

anan

1

=BAa tan

1tan

1

11

=

A cotcot

11

=11

10. D. 1

4cot .

4cot =

tantan

tan.tan1

4

4 x

tantan

tan.tan1

4

4

=

2

4

2

2

4

2

tantan

tan.tan1

=

2

2

an1

tan1

= 1


1. a.

115tan.155an1

115tan155tan= 115155tan

= 40tan

= 1525tan

=

15tan.25an1

15tan25tan

= 13

1313

13

.1

a

a

= 113

113

aa

aa

b.

150an.175tan1

150tan175tan= 150175tan

= 25tan = a

c. 15tan.10an1

15tan10tan= 25tan = a

d. 5,10tan.5,14tan1

5,10tan5,14tan= 25tan = a


12/30


2. a. 15tan = 6045tan

=

60tan.5an1

60tan45tan

=31

31

b. 75tan = 75tan= 3045tan

=

30tan.45tan1

30tan45tan

=

31

31

31

31

=

33

33

c. 255tan = 75180tan

=5an180an1

75tan180tan

= 75tan

=33

33

d. 105tan = 105tan= 4560tan

=45tan.60an1

45tan60tan

=

31

13

e. 195tan = 195tan

=15tan.180an1

15tan180tan

= 15tan =

31

13=

31

31

f. 285tan = 105180tan

=105an180an1

105tan180tan

= 105tan

=31

31

g. 105tan =31

31

h. 195tan =31

13

i. 105cot =105an

1=

31

31

j. 195cot =195an

1=

13

31

3. a. BA tan =BA

BA

an.an1

tantan

= 33

56

.11533

1556

512

34

512

34

BA tan =63

16

15

63

1516

b. BA tan = 63

16

.115

63

1516

34

512

34

512

BA tan =33

56

15

33

1556

c. BA tan =6214

8213

.1214

6214

214

8213

21

243

21

243

BA tan =6214

8213

d. BA tan = 63

16

.1125

43

125

43

BA tan =33

56

4. a.

A4tan

= A

A

tan.tan1

tantan

4

4

=

A

an1

tan1

=AA

AA

sincos

sincos

b.

A

4tan

=

A

A

tan.tan1

tantan

4

4

=

A

A

an1

tan1

=AA

AA

sincos

sincos

c. A45tan . A45tan =

=AA

AA

sincos

sincos

.

AA

AA

sincos

sincos

= 1

d.

6tan =

6

6

tan.tan1tantan

=

tan.31tan3

31

31

=

tan.33

tan33

=

tan3

tan31

3

3

=

tan3

tan31

5. a. BA tan =BA

BA

an.an1

tantan

= 0,75

B

B

an21

tan2

= 0,75

BB tan5,175,0tan2

25,1tan5,2 B

Btan 0,5

b. BA tan =BA

BA

an.an1

tantan

=25

25

.171

43

71

4

3

= 1

A + B =4


13/30


c. BA tan =A

BA

an.an1

tantan

=

3

4

21

21

.tan1

tan

A

A

=

3

4

AA tan2

3

4

2

1tan

65tan

35 A

5

3

6

5tan A = 0,5

6. a. 70tan c. 60tanb. 30tan d. 20tan

7. a. 1tan . 1tan =1tantantantan

= 1tantan

tantan1

= tantan1

2

b.

tan

tan=

tan.tan1

tantan

.

tantan

tantan1

=11

tantan

.

tantan

11

= 0

8. a.

BBAABA

tan.tan1

tantan

= ABA tan = Btan

b. BAA

BAA

tantan1

tantan= BAA tan = Btan

c. BA cot =A

BA

anan

tan.tan1

=BABA

BABA

incoscossin

sinsincoscos

x

BA

BA

sinsin1

sinsin1

=A

BA

cotot

1cot.cot

d. BA cot =BA

BA

anan

tan.tan1

= BABA

BABA

sincoscossin

sinsincoscos

x

BA

BA

sinsin1

sinsin

1

=B

BA

cotcot

1cot.cot

9. a. BA tan = 323

36

.1 512

724

512

724

b. BA tan =

253204

.1 512

724

512

724

c. sudut (A + B) di kuadran III

d. sudut (A - B) di kuadran III

10. a. BA tan =A

BA

anan1

tantan

= 21

220

.18

15125

815

125

b. BA tan = 171

140

.18

15125

815

125

c. sudut (A - B) di kuadran IIId. sudut (A + B) di kuadran IV


1. A45tan = A45tan4

45an.tan1

tan45tan

A

A

=A

tan.45tan1

tan45tan.4

2tan1 A = 2tan14 A

AA tan2tan1 2 = AA tan8tan44 2

3tan10tan3 2 AA = 0

3tan1tan3 AA = 0

3

1tan A atau 3tan A

2. yx tan =yx

yx

tan.tan1

tantan

=

yx tan =

yx

yx

tan.tan1

tantan

= b

yx tantan = yxa tantan = byx tantan

=2

tantan2ba

xxba

a. x2tan =

4

2 2

1tan1

tan2ba

ba

x

x

=bba

ba

2

4422

yx tan =yx

yx

tan.tan1

tantan

=

=ya

yba

Ba

tan

tan

2

2

= a

= yba

aayba

tan2

tan2

2

2

1tan

ba

ba

a

ay

b. y2tan =

2

2

2

2

2

2

11

1.2

tan1

tan2

ba

ba

ba

ba

a

a

a

a

y

y


14/30


3. tan =

tan.tan1

tantan

= QPQP

PQ

QP

QP

QP

121.1 11

11

= cotcot

4. a. CBA tantantan =

BABA 180tantantan

= BA tantan BA

BA

tan.180tan1

tan180tan

= BA tantanA

BA

an.an1

tantan

=BA

BABA

BABA

an.an1

tantantantan

tantantantan

2

2

= CBA tantantan

b. Ctan = BA180tan= BA tan

=A

BA

tan.an1

tantan

= 5,14,2.5,11

4,25,1

tan A = tan Cjadi, A = C

5. 0152 2 xx

0112 xx

2

11

x ; 12

x

2

1tan ; 1tan

a. tan = 31.1

1

21

2

3

21

21

b. tan =3

1

1.1

1

23

21

21

21

6. a. BA tan = BA 180tan= Ctan

=7

5

b. BA tan =B

BA

an.an1

tantan

=

7

5

=81

tantan

BA=

5

= BA tantan = 75

= 5

7. a. tan =

tan.tan1

tantan

=21

21

.1 mm

mm

b. tan =

11

21 mm

90

9. BA cot = BABA

antan

tan.tan1

=xy

xy

x

y

x

1



1. C. 25

7

x2cos = x2sin21

=25

321

5

421

2

=

25

7

2. B.89

240

x2sin = xx cossin2

=17

8.

17

15.2 =

289

240

3. C.169

119

x2cos = x2

sin21

=169

501

13

521

2

=

169

119

4. E.7

24

x2tan = 167

23

2

43

43

21

.2

tan1

tan2

x

x

= 241623

5. C.27

8

x3sin = xx cos2sin2

= xx 2cossin4

=

2

3

2.

3

1.4

=

27

8


15/30


6. E. -1

x3cos = xx2cos= xxxx sin2sincos2cos

= xxxx cossin2cos1cos2 22

=2

1.

2

3.2

2

1.1

2

1.2

22

=4

31 = -1

7. A. xtan

x

x

2cos1

2sin

=

1cos21

cossin22

xx= xtan

8. E.

22

223

QP

PQP

xQxP 2sin2cos

=

x

xxxQ

xxP

2

2

cos

1cos

1

22

cossin2

sincos

=

x

xQxP

2cos

1

2tan2tan1

=

2

2

2

1

21

Q

QP

Q

P QP

= 232 2PQPPQ = 323 PPQ

= 223 PQP

=

22

223

QP

PQP

9. D. xtan

x

x

2cos1

2cos1

=

x

x

x

x2

2

2

2

cos2

sin2

1cos21

sin211

= xtan

B. Evaluasi Pemahaman dan Penguasaan

Materi

1. A3cos = AA 2cos= AAAA sin2sincos2cos

= AAxA cossin2cossin21 22

= AAA cossin4cos2

= AA 2sin41cos

A4cos = AA 22cos = AAAA 2sin2sin2cos2cos

= AAxA cossin41cos2sin21 222 = AAAA 2222 cossin81cossin2 = AA

22cossin81

= A2sin212

A3tan = AA 2tan

=

A

AA

an.2an1

2tan2tan

=A

A

A

A

A

A

tan.1

2tan

2

2

tan1

tan2

tan1

tan2

=AA

AA22

3

tan2an1

tantan3

=A

AA2

3

an31

tantan3

A4tan = AA 22tan

=A

A

2an1

2tan22

= 2tan1

tan2tan1

tan4

2

2

1A

AA

A

=

22222

2

tan1

tan4tan1

tan1

tan4

A

AA

A

A

=

AAAA

222

2

tan4tan1

tan1tan4

2. a. A2sin = AAcossin2

=

17

15.

17

8.2 =

289

240

b. A2cos = A2sin21

=

2

7

8.21

=

289

161

c. A2tan =A

A2

an1

tan2

= 61

240

61

225

15

16

1

.2

225

61

1516

2

158

158

3. a. A2sin = AAcossin2

=

25

24.

25

7.2 =

625

336


16/30


b. A2cos = A2sin21

=

2

25

7.21

=

625

527

c. A2tan =A2

tan1

tan2

=

527336

527

576

12

7

1

.2

576527

127

2247

247

4. a. AA 2cos2sin4 = AA 2cos2sin22= A4sin2

b. xx22

cossin4 = xx cossin2 xx cossin2

= x2sin2

c.B

B

3an1

3tan22

= BB 6tan3.2tan

d. 8sin162 A = 1sin28 2 A = A2sin218

= A2cos8

e.0sin

70cos1 = 35cos35sin235sin2

35cos35in35sin211

22

= 35tan

f.2

210cos1 = 105cos

5. a. x2tan =x

x2

an1

tan2

= 4

3

8

9

3

2

1

.22

31

31

b. y2tan =y

y

2tan1

tan2

= 3

41

1

.2

4

32

21

21

c. yx 2tan =yx

yx

2tan.tan1

2tantan

=5

9

3

5

.13

4

3

1

34

31

= 3

d. yx 22tan =yx

yx

2tan.2tan1

2tan2tan

= 34

43

3

4

4

3

.1

6. tan = 180tan= tan

=

tan.tan1

tantan

=

25.41

254

=2201

254

7. 62

tan

t

cos1

cos1

cos1 = cos22 tt

b. cos =2

2

1

1 t

c. tan =

21

2t

a. sin =2

1

2t

8. a.A

sin

2cos1 =

AA

A

AA

A

cossin2

sin2

cossin2

sin211 22

= Atan

b.A

2cos1

2sin

=

1cos21cossin22 A

AA= Atan

c.

2

an1

tan2

=

2

22

cos

sincos

tan2

=

2cos.

cos

sin2

= 2sincossin2

d.

2

2

tan1

tan1

=

2cos

sincos

sincos22

22

9. a.

4tan =

tan.tan1

tantan

4

4

=

an1

tan1

=

sincos

sincos

x

sincos

sincos

=

22

22

sincos

cossin2sincos

=

2cos

2sin1

b.

4tan =

tan.tan1

tantan

4

4

=

an1

tan1

=

sincos

sincos

=

22

22

sincos

cossin2cossin

=

2cos

2sin1

c.

2

2

1

sin2

1

coscos2

=

sin

2

cos1

2

cos1cos2

= sin1cos2 = cossin2cos2

= 2sincos2

d. 2cot =2tan

1=

2

2

tan

1

tan

12

tan2

tan1

=

cot2

1cot2


17/30


e. BBB 2cos2sintan

B

BBBB

cos

2coscos2sinsin =

B

BB

cos

2cos

=B

B

cos

cos

=B

B

cos

cos= 1

f. 1cos21cos2 AA = 1cos4 2 A

= 11cos22 2 A= 12cos2 A

g.CC

C

sincos

2cos

=

C

CC

inos

sincos 22

=

CCCCCC

sincos

sincossincos

= CC cossin

h.1cos

cos2cos

x

xx=

1cos

cos1cos2 2

x

xx

= 1cos

1cos1cos2

x

xx

= 1cos2 x

10. a.

2cos

2sin1 =

22

22

sincos

cossin2cossin

=

sincossincos

sincos2

=

cos1

cos

1

sincos

sincos

=

an1

tan1

b.

2

2sin

2cos

= 2cos1 2

cos1 sin

= sin1

c. 2

2

tan1

tan1

=

2

22

2

22

cos

cossin2sincos

cos

sincos

=2sin1

1

d.A

A

2cos1

2cos1

=

A

A

A

A2

2

2

2

sin

cos

sin211

1cos21

e.xx

sin2sin

2coscos1

=

x

xx

sincossin2

1cos2cos1 2

= xx

xx

cos21sin

cos21cos

= xcot

f.

cossin1

cossin1

=

212

21

21

212

21

21

sin21cossin21

1cos2cossin21

=

21

21

21

21

21

21

cos2sin2sin

cos2sin2cos

= 1

cot

g.

2

1

4tan =

2

24

24

tan.tan1

tantan

=

2

2

2

tan1

tan1

=

sin1

sin1

h. AAA 2coscos22sin 22

= AAAA 2coscos2cossin4 222 = AAA 2cossin2cos2 22 = AAA 222 sin21sin2cos2 = A2cos2

i.CC

CC

ancot

cottan

=

C

C

C

C

tan

tan

tan1

tan1

=

C

C2

2

an1

1tan

=CC

CC

2cos

1

sincos

cossin22

22

= sec2C

j. cosec2B + cot2B =B

B

2sin

2cos

2sin

1

=

BB

B

cossin2

1cos212

= BB

Bcot

sin

cos

11. a. 0sin2sin xx

0sincossin2 xxx

01cos2sin xx

0sin x ;2

1cos x

x = 0 ; 120 ; 180

b. 0cos2cos xx

0cos1cos22 xx

01cos1cos2 xx

2

1cos x ; 1cos x

x = 60 ; 300 ; 180 ; 360 ; 270

c. 05sin32cos xx

05sin3sin212

xx

06sin3sin22 xx

2.2

62.433sin

2

2,1

x

=573

x = arcsin573

; atau

x = arcsin573


18/30


12. a. 8cos2222

= 14cos22222 2

= 4cos422 2

= 4cos222 = 12cos2222 2

= 2cos42 2

= 2cos22

= 1cos222 2 = cos2

b. 8cos2222 = 2cos22

= 2sin2122 = sin2

c. 8cos222sin22

= 2cos2sin22

= 22 sin212sin2 = 2sin212

= 2

cos.2 = 2cos

d. 8cos222cos2 2

= 2cos2cos2

2

= 1cos22cos2 22

= 2cos212 = sin2


6. 32222 yxyx= 22 2coscos2sinsin AAAA

32coscos2sinsin 22 AAAA

=

AAAA

AAAA

2coscos22sinsin2

2cos2sincossin 2222x

32coscos22sinsin2

2cos2sincossin 2222

AAAA

AAAA

= AAAA 2coscos22sinsin22 x 12coscos22sinsin22 AAAA

= 12cos22cos22 AAAA

= AAA cos22cos4cos4 2

= 2cos2cos4 2 AA

= AA cos21cos222

= AA cos22cos2

= AA 2coscos2 = 2y

8. b. 30tan15tan30tan15tan

= 30tan15tan30tan15tan11530tan= 1

9. a. CBA 2sin2sin2sin

= BABA 1802sin2sin2sin= BABA 2sin2sin2sin= BABABA 2sin2cos2cos2sin2sin2sin

= ABBA 2cos12sin2cos12sin

= ABBA 22 cos2.2sincos2.2sin

= ABBBAA 22 cos2.cossin2cos2.cossin2

= BABABA sincoscossincoscos4 = BABA sincoscos4= BABA 180sincoscos4= CBA sincoscos4

b. CBA 2cos2cos2cos

= BABA 1802cos2cos2cos= BABA 2cos2cos2cos= BABABA 2sin2sin2cos2cos2cos2cos

=

B

BBA

coscossinsin4

2cos2cos12cos

=

BABA

BBA

coscossinsin4

1cos2cos21cos2222

= 1sinsincoscoscoscos4 BABABA= 1coscoscos4 BABA= 1180coscoscos4 BABA

= CBA coscoscos41

c. CBA 2cos2cos2cos

= BABA 1802cos2cos2cos= BABA 2cos2cos2cos= BABABA 2sin2sin2cos2cos2cos2cos = BABBA 2sin2sin2cos2cos12cos

=

BABA

BBA

coscossinsin4

2cossin22cos2

=

BABA

BBA

coscossinsin4

sin21sin2sin21222

= 1sinsincoscossinsin4 BABABA= 1cossinsin4 BABA= 1180cossinsin4 BABA= 1cossinsin4 CBA


19/30




1. D.22

1

2

cos1

2sinxx

=2

143

=8

1 = 2

1 =

22

1

2. C.3

1

x

xx

cos1

cos1

2tan

=54

54

11

=9

1 =

3

1

3. D.2

22

2

45sin

2

122sin =

2

45cos1

=2

2121

=4

22

=2

22

4. C. 12

2

45tan3022tan =

45cos1

45cos1

=21

21

21

21

=22

22

= 12

5. B. xcot

2tan

tan

1

2

1

2

xx

=

2

2

2

tan

tan1

2

1x

x

=

x

x

xx

cos1

cos1

cos1cos11

2

1

=

x

x2

cos1

cos2

2

1

=x

x

sin

cos= xcot

7. B. cos

2

2

2

2

tan1

tan1

=

cos1cos1

cos1

cos1

1

1

=

2

cos2 = cos

9. C. -3

cos1

cos1

2sin

=8,01

8,01

=2,0

8,1 = 9 = 3


Materi

1. a.2

cos1

2cos

AA

=2

11715

=34

32= 17

17

4

b.A

AA

cos1

cos1

2tan

=1715

1715

1

1

=32

2=

1

c.2

cos1

2sin

AA

=

2

11715

=34

2= 17

17

1


20/30


2. a.2

cos1

2sin

AA

=2

1419

= 41.2

32

= 411

4

b.2

cos1

2cos

AA

=2

1419

= 411

5

c.A

AA

cos1

cos1

2tan

= 419

419

1

1

=50

32 =

5

4

3. a.2

tanA

=

2524

2524

1

1

= 7

b.2

sinA

=2

12524

= 210

7

c.

2

cosA

=

2

12524

=

100

2 = 2

10

1

4. a.A

AA

cos1

cos1

2tan

=

257

257

1

1

=

9

16 =

3

4

b.2

cos1

2cos

AA

=2

1257

=5

3

c.2

cos12

sin AA

=2

1257

=5

4

5. a.2

45tan

1

2

122cot

=

45cos1

45cos1

=21

21

21

21

=2222

= 22

22

=2

222 = 21

g.2

45sin2122sin =

245cos1

=2

2121

=4

22

=2

22

h.2

45cos

2

122cos =

2

22

6. a. AA cotseccos =A

A

A sin

cos

sin

1

=A

A

sin

cos1

=

22

2

2

cossin2

sin211AA

A

=2

2

cos

sinA

A

=2

tanA

b. 1cos2

tan AA

=

2

2

cos

cos1sinA

A A

=

2

2

2

2

cos

1cos21sinA

AA

=2

2

2

2

cos

cossin2

A

AA

=2

cos2

sin2AA

= sinA

c.

2

2sin2cos

xx

= cos2sin22cos2sin

22 xxxx

= xsin1


21/30


d. BB

cos2

cos2 2 = 1cos12

cos2 2

B

B

= 1coscos BB

= 1

e.2

cos

1

2sec

x

x =

2cos1

1

=

cos1

2

f. cotseccos =

sin

cos

sin

1

=

sin

cos1

=

22

2

2

cossin2

1cos21

=2

2

sin

cos

=2

cot

7.

22

sin

sin

A

DA

n

=

2cos1

2

cos1

A

DA

=

A

DA

cos1

cos1

8.

cos1

sin1

=

1cos21cossin21

2

2

22

=2

2

2

2cos

sin

cos2

1

=2

tancos

cossin

2

1

2

2

2

2

2

2

=2

tan12

tan2

1 2

=

2tan21

2tan

21 2

=

2

12

tan2

1

9. a.

sin2

coscos2

sin2

1sin

=13

5

2

12524

+13

12

2

12524

= 2130

5+ 2

130

84

= 213089

b.2

sincos2

cossin2

1sin

=2

1.

25

7 135

+2

1.

25

24 135

= 1326

6.

25

7+ 13

26

4.

25

24

= 13650

138

c.

sin2

sincos2

cos2

1cos

=13

5

2

12524

+13

12

2

12524

=13

5.2

10

7+

13

12.2

10

1

= 2130

47

d.

2

1cos =

13

5

2

1 2524

-13

12

2

1 2524

=13

5.2

10

7-

13

12.2

10

1

= 2130

23

e.

2

1tan =

2tan.tan1

2tantan

=

135

135

135

13

5

1

1

247

1

1

247

.1

=

58

49

.164

247

64

247

f.

2

1tan =

512

1

1

5

12

1

1

..12524

2524

2524

2524

=

23

89

.15

1271

512

71

g.

2

1tan =

512

1

1

512

1

1

..125

24

2524

2524

2524

=

47

79

.15

1271

512

71

h.

2

1cos =

2

1.

25

24 135

+2

1.

25

7 135

= 1326

6.

25

24+ 13

26

4.

25

7

= 13650

172

10. a. 15sin21 2 = 30cos = 0,866

b. 2

122sin

2

122cos

22= 45cos = 0,7071

c. 15cos.15sin =21.

2130sin

21 = 0,25

d.

212

21

22tan1

22tan2= 45tan = 1


22/30



1. a. CBA sinsinsin

= BABA 180sinsinsin= BABA sinsinsin= BABABA sincoscossinsinsin

= ABBA cos1sincos1sin

=2

cos2.2

cos2

sin22 BAA

+2

cos2.2

cos2

sin22 ABB

=

2sin

2cos

2cos

2sin

2cos

2cos4

BABABA

=

22sin

2cos

2cos4

BABA

=

BA

BA180

2

1cos

2cos

2cos4

=

2

cos

2

cos

2

cos4CBA

b. CBA sinsinsin

= BABA 180sinsinsin= BABA sinsinsin= BABABA sincoscossinsinsin = ABBA cos1sincos1sin

=2

sin2.2

cos2

sin22 BAA

+2

sin2.2

cos2

sin22 ABB

=

2cos

2sin

2cos

2sin

2sin

2sin4

BAABBA

=

22sin2sin2sin4BABA

=

BA

BA180

2

1cos

2sin

2sin4

= cos2

sin2

sin4CBA

3. a.

b

aarccos

2

1

4tan

=

24tan

=2

2

tan1

tan1

=

b

ab

a

b

ab

a

1

1

1

1

1

1

b

aarccos

2

1

4tan

=

b

ab

a

b

ab

a

1

1

1

1

1

1

b

aarccos

2

1

4tan

+

b

aarccos

2

1

4tan

=

ba

ba

ba

b

a

1

1

1

1

1

1

+

ba

ba

ba

b

a

1

1

1

1

1

1

= a

b

ba

b

ab

a

ba

b

a

2

2

4

1

22

1

1

1

1



1. A. 13

15cos45sin4 = 30sin60sin2

=

2

13

2

12 = 31

2. A.23

15cos75sin = 60sin90sin2

1

=

3

2

11

2

1

=

32

1

3. E.1

15cos75cos2 = 60cos90cos

=1

0 =2

1

4. A.2

23

75sin105sin = 30cos180cos

=

3

211

=2

32

5. D.12

2

137cos

2

17sin = 30sin45sin

2

1

=

2

12

2

1

2

1

=4

12


23/30


8. D. 2cos

45cos45cos2 = 2cos90cos = 2cos

9. A. 2sin

2sin

2cos2 = 2sinsin

= 2sin0

= 2sin


1. a. cos4sin = 3sin5sin2

1

b. BABA cossin2 = BA 2sin2sin

c. BABA 2

1sin

2

1cos = BA sinsin

d. 90sin90cos2 = 2sin180sin = 2sin

e. 85cos15sin2 = 70sin100sin

= 70sin100sin

f. 2

1sin

2

3cos2 = sin2sin

= 0

2. a. baba coscos2 = ba 2cos2cos

b. baba 2

1cos

2

1cos2 = ba coscos

c. baba 2sin2sin = ba 2cos4cos2

1

d. cbacba sinsin2 = cba 2cos2cos

e. 4

1cos

2

1cos =

4

1cos

4

3cos

2

1

f. 20sin200sin2 = 180cos220cos

3. a. 110cos70cos2 = 40cos180cos = 40cos1

b. 50sin40sin2 = 10cos90cos = 10cos0 = 10cos

c. 60cos80sin2 = 20sin140sin

d. 50sin110cos2 = 60sin160sin

= 32

1160sin

4. a. BAsinsin = BABA sinsin2

1

b. xx sin5cos3

= xx 4sin6sin3

c. xx2

3sin

2

5cos5 = xx sin4sin

2

5

d. yxyx cos53cos =

yxyx 42cos64cos2

1

e. CBACBA sincos =

CBA 2sin2sin2

1

f. AA 3cos5sin = AA 2sin8sin2

1

g. 40cos200cos = 160cos240cos2

1

h.

4cos

4sin10

=

2sin2sin5

i. BABA 2

1sin

1sin

2

1= BA coscos1

j.

BABA 3

2

13sin

2

7cos =

BABA 23sin410sin1

k. AA 8cos7cos3 = AA cos15cos2

3

l. AA4cossin8

= AA3sin5sin4

5. a. 15cos75sin = 60sin90sin2

1

=

3

2

11

2

1

=4

32

b. 15sin75cos = 60sin90sin2

1

=

3

2

11

2

1

=4

32

c. 15cos75cos = 60cos90cos2

1

=

2

10

2

1

=1

d. 15sin75sin = 60cos90cos2

1

=

210

21 =

41

e.


24/30


7. a. 3cos2cos5cos24sin3sin2 =

3cos3cos7coscos7cos = cos

b.

AA

AAAA

2

7cos

2

3sin2

2

3cos

2

5sin2

2

3cos

2

1sin2

=

AAAAAA

2sin

5sinsin4sinsin2sin

= AA 5sin4sin

9. a. 4sin12sin2sin6sinsin3sin

32sin96sin16sin48sin8sin24sin

=

64cos128cos32cos64cos

16cos32cos8cos16cos

4cos8cos2cos4cos

2

1

= 2cos128cos21

=

2

2128sin

2

2128sin

= 63sin65sin

b. 4cos12cos2cos6coscos3cos 8cos24cos

=

16cos32cos8cos

16cos4cos8cos2cos4cos

= 32cos2cos

= 2232sin

2232sin

= 15sin17sin



1. B. 1

20sin0sin

70cos20cos=

25sin5cos2

25sin45sin2= 1

4. E.2

1

15cos105cos = 45cos.60cos2

=

2

2

1

.2

12

= 22

1=

2

1

6. B. BA tan

AB

AB

2sin2sin

2cos2cos

=

BABA

BABA

cossin2

sinsin2

= BA tan

7. E. cot

3coscos

3sinsin

=

sin2sin2

cos2sin2= cot

8. E. x2cot

x

xx

3coscos

sin3sin

=

xx

xx

in2in

sin2cos2= x2cot

9. A.

2tan

coscos

sinsin

=

22

22

coscos2

cossin2

=

2tan

10. D. xtan

xx

xx

5coscos

5sin7sin

=

xx

xx

cos6os2

sin6cos2= xtan

15. D. Atan

AA

2cossin1

2cos2sin1

=

1cos2cossin21

sin21cossin212

2

AAA

AAA

=AA

AAA2

2

cos2cossin2

sin2cossin2

= AAA

AAA

cos2sin2cos

sin2cos2sin

= Atan


Materi

1. a. 3sinsin = cos2sin2

b. 2coscos =

2

cos

2

3cos2

c. 3coscos = sin2sin2

d. 2sinsin =

2cos

2

3cos2

e. AA 2cos4cos = AAsin3sin2

f. yxyx coscos = yx coscos2g. BABA 62sin24sin =

BABA 2sin43cos2 h. qpqp sin53sin =

qpqp 3cos22sin2 i.

yxyx 34cos2cos =

yxyx 2cos3cos2 j. BABA 2cos2cos = BA 2sinsin2


25/30


2. a. 50sin40sin = 5cos45sin2 = 5cos2

b. 10cos70cos = 30cos40cos2 = 40cos3

c. 40cos80cos = 20cos60cos2 = 20cos

d. 10sin70sin = 30sin40cos2 = 40cos

e. 35sin55sin = 10cos45sin2 = 10cos2

f. 52cos68cos = 8cos60cos2 = 8cosg. 35cos95cos = 30cos65cos2 =

65cos3

h. 30cos70cos = 20sin50sin2

i. 40sin80sin = 20sin60cos2 = 20sin

4. a. 15sin105sin = 45cos60sin2

= 22

1.3

2

1.2

= 6

2

1

b. 165cos465cos = 150sin315sin2

=1

.22

1.2

= 22

1

c. 15cos105cos = 45cos60cos2

= 22

1.

2

1.2

= 22

1

f. 15sin75sin = 30sin45cos2

=2

1.2

2

1.2

= 22

1

g.

15sin70sin

15cos70cos=

5,27sin5,42cos2

5,27sin5,42sin2

= 5,42tan

h.

15sin75sin

15cos75cos=

30sin45cos2

30cos45cos2

= 30cot= 3

i. 15sin5sin

15cos75cos=

30cos5sin2

30sin45sin2

= 33

1

5. a.10

3sin

5sin

=

20sin

4cos2

=

20

sin2

b.12

11sin

12sin

=

24

10cos

2sin2

d.

2

3sin2

3sin =

2sin

6cos2

= 2sin.32

1.2

= 2sin3

e.10

3cos

5cos

=

10cos

2cos2

= 0

g.

12cos

12

5cos =

2

3cos

2cos2

7. a.

3sin7in

3cos7cos

= cos5sin2

2cos5cos2

= 5cot

b.

coscos

sin5sin

=

2coscos2

2cos3sin2

= 3tan

c.

3sin5sin

3cos5cos

=

sincos2

cos4cos2

= cot

d.

sin6sin

4cos6cos

=

cos5sin2

sin5sin2

= tan

e.

6os8cos6sin8sin

=

cos7cos2sin7cos2

= tan

f.

os3cos

sin3sin

=

sin2sin2

sin2cos2

= 2cot

g.

sinsin

coscos

=

22

22

sincos2

sinsin2

=

2

tan

h.

2cos2cos

2sin2sin

=

coscos2

sincos2

= tan

j.ts

ts

coscos

sinsin

=

22

22

coscos2

cossin2

tsts

tsts

=

2tan

ts

8. a. 20sin80sin = 30sin50cos2

= 50cos


26/30


b. 25sin65sin = 20cos45sin2

= .22

1.2 20cos

= 2 20cos

d.

12

sin

12

5sin

=

6

cossin2

= 32

1.2

2

1.2

= 62

1

e.

33cos3cos

33sin3sin=

15cos18cos2

15cos18sin2

= 18tan

f. 10cos50cos

50sin10sin=

20sin30sin2

20sin30cos2

= 3

g. 40cos20cos

20sin40sin = 10sin30sin210sin30cos2

= 3

=

10cos50cos

50sin10sin

h.

20sin70sin

20cos70cos=

25cos5sin

25cos45cos2= 1

9. a. CBA 2sin2sin2sin

= BABA 1802sin2sin2sin= BABA 2sin2sin2sin

= BABABA 2sin2cos2cos2sin2sin2sin = ABBA 2cos12sin2cos12sin = BABABA cossinsincossinsin4 = BABA sinsinsin4= BABA 180sinsinsin4= CBA sinsinsin4

b. CBA coscoscos

= BABA 180coscoscos= BABA coscoscos= BABABA sinsincoscoscoscos = BABBA sinsincoscos1cos

=

2

sin21 2A

2sin2 2

B+

2

sin21 2B

+

2cos

2cossin

2sin4

BABA

=2

sin2

sin2

sin41CBA

c. CBA sinsinsin

= BABA 180sinsinsin

= BABA sinsinsin

= BABABA sincoscossinsinsin

= ABBA cos1sincos1sin

=2

cos2.2

cos2

sin2 2BAA

+2

cos2.2

cos2

sin2 2ABB

=

2sin

2cos

2cos

2sin

2cos

2cos4

BABABA

=2

cos2

cos2

cos4CBA

d. CBA sinsinsin = BABA 180sinsinsin= BABA sinsinsin

= BABABA sincoscossinsinsin = ABBA cos1sincos1sin

=2

sin2.2

cos2

sin2 2BAA

+2

sin2.cos2

sin2 2ABB

=2

cossin2

sin4CBA


1. a.

2cos

A+

2cos

B+

2cos

C

=

2cos

A+

2cos

B+

2

180cos

BA

=

2cos

A+

2cos

B+

2sin

BA

=

2cos

A+

2cos

B+

2sin

2cos

2cos

2sin

BABA

=

2cos

A

2sin1

B+

2cos

B

2sin1

A

=

4cos4

A

4cos

B

4cos

C

4. a.C

CBA

sinsinsin

sinsinsin

=

222

222

cossinsin4

coscoscos4

CBA

CBA

=

2

cotA

2

cotB

b.CA

CBA

sinsinsin

2sin2sin2sin

=

222coscoscos4

sinsinsin4

CBA

CBA

=222

222222

coscoscos

cossin2.cossin2.cossin2

CBA

CCBBAA

=2

sin2

sin2

sin8CBA


27/30


Ujian Kompetensi Akhir Bab 3

A. Pilihan Ganda

1. A.p

q1

qBABABA sincoscossinsin

qpBA sincos

p

qp

BA

BABA

cossin

sincostancot =

p

q1

2. E. 22

1

12cos

12

7cos

= coscos2

= .2

1.2 2

2

1= 2

2

1

3. E.65

63

sin = sincoscossin

=5

4.

13

12

5

3.

13

5

=65

63

4. B. m-n

sin = sincoscossin = n sincos = m-n

5. A. 0

BA sin = BABA sincoscossin = 30sin =2

1

BAsincos =4

1

BA sin =

4

1

4

1= 0

6. B. sin4cos2

4sin 4sin = sin4cos2

7. E. 6cos

3cos4 3sin 4 =

3sin3cos 22 3sin3cos 22 = 6cos

8. A. 32

660tan840tan =

300360tan120360.2tan = 300tan120tan

= 120180tan120tan = 120tan120tan

= 120tan2 = 60180tan2 = 60tan2 = 32

9. D. 3152

1

yxcos = yxyx sinsincoscos = 51

ycoscos = 51

- 34

1

yxyxyx

coscossinsintantan =

35341

41

4

1

= 3152

1

10. B. 12

2

135tan5,67tan =

135cos1

135cos1

=21

21

21

21

=22

22

= 22

22

= 12

11. E.2

2

1

1

p

p

22 sincos2cos

= 22 coscos p

= 22 cos1

p

= 2

2

1

11

pp

=

2

2

1

1

p

p

dimana :

2

2

2

cos

sinp

2

2

2

cos

cos1p

2

21

cos

1p

2

21

cos

1p

2

2

1

1cos

p

12. E. 22 11 pqqp

sin = sincoscossin

= qpqp .11 22

13. C. 12

x

A2

1cos =

x

x

2

1A

1cos

Asin 12 x


28/30


14. B. 2

50cos0os

10cos=

10cos90cos10cos

21

= 2

15. A. 152

1

x

xx 2an1

tan22tan = 2

xx 2tan22tan2

02tan2tan2 2 xx

01tantan 2 xx

1.2

1.1.411tan

x =

2

51

1tanx = 51

2

1 ;

2tanx =

2

15

16. A.

2

cosA

2sin

CB=

2

180sin

A

=2

sin90cos2

cos90sinAA

=2

cosA

17. B.

2sin2

A

2cos

CB

CB coscos = CBCB coscos2

=

2cos

2

180cos2

CBA

=

2cos

290cos2

CBA

=2

sin2A

2cos

CB

24. D. 29

4

BA 32sin = BABA 3sin2cos3cos2sin

= 3

1

BA 32cos = BABA 3sin2sin3cos2cos

= 23

2

BA 32sin BA 32cos =

BA 6sin4sin2

1 = 2

2

BA 6sin4sin = 29

4

25. A. 32

1

75cos4 75sin 4 = 150cos = 32

1

26. D. 2cos22

22 yx =

cos3cos2cos3cos

sin3sin2sin3sin

22

22

= 2cos22

31. A.22

13

k

k

20sin10sin

35cos25cosk =

10cos30cos

10cos60cos

2

1

21

10cos2

110cos3

2

1

k

32

1

2

1110cos kk

1

3110cos 2

1

k

k

=22

13

k

33. E. 62

15os

1

75cos

1=

75cos15cos

75cos15cos

= 60cos90cos

30cos45cos2

2

1

=21

21

21

21

.

3.2.2

= 62

36. A.

tantan1

tantan

cos

sin=

coscos1

coscos1

sinsincoscos

sincoscossin

=

tantan1

tantan

38. D. Asin2

1

28sin 2 A -

28sin2 A

=

28

sin28

sinAA

.

28

sin28

sinAA

=

2cos

8sin2

A

2sin

8cos2

A

= Asin.sin

= Asin2

1


29/30


39. C.

15tan1

15tan15tan32

3

= 1530tan45tan

=

15tan30an1

15tan30tan

=

15tan.1

15tan

15tan115tan2

15tan1

15tan2

2

2

=

15an31

15tan15tan32

3

40. A. AA 4cos8

1cos

2

13

A4

sin = 22sin A

=

2

2

cos1

A

=4

coscos21 2 AA

=4

cos212

14cos AA

= AA 4cos8

1cos

2

1

8

3

B. Bentuk Uraian

1. a. ts sin = tsts sincoscossin

= 21

.5

42

2

1.

5

3

= 210

1

b. ts cos = tsts sinsincoscos

= 22

1.

5

32

2

1.

5

4

= 210

1

2. a.

h

xfhxf=

xhx sinsin

=xxx sinsinhcoscoshsin

=

hx

hx

sinhcos

1coshsin

b. xghxg

= xhx coscos

=xxx cossinhsincoshcos

=

hxhxsinh

sin1cosh

cos

c. xThxT

= xhx tantan

=h

xx

x tantanhtan1tanhtan

= tanh.tan1

tanhtantantanhtan2

xh

xxx

= tanh.tan1tan1tanh

2

xhx

=anhan1

sectanh2

x

x

3. a. BA cot =BA

BA

BA

BA

tantan1

tantan1

tantan

tantan1

=AB

BA

cotcot

1cotcot

b. BA cot =BA

BA

tanan

tantan1

=AB

BAcotot

1cotcot

4.

24tan

=

24

24

tantan1

tantan

=

2

2

tan1

tan1

x

x

x

x

x

x

x

x

1

2

1

2

1

2

1

2

1

11

1

11

=x

x

x

x

x

111

1

1

1

1

5. a.

1

1

1

1

sin

sin

sin

sin

BA

BA

BA

BA

c

c=

B

A

BA

BA

tan

tan

sincos2

cossin2

Atan = Bc

ctan

1

1

6. a.

2sin2tan

2sin2tan

y

x

=

12cos

12sin

12cos

12sin

=

2cos1

2cos1

= 2tan

b. 222 2sin2tan2sin2tan = 22sin2tan4

= 2sin2tan16 22

=

2sin.

2cos

2sin16 2

2

2


30/30

=

2sin

2cos

2cos116 2

2

2

=

2sin12cos

116 2

2

= 2sin2tan16 22 = 2sin2tan16 2sin2tan = xy16

9. mx

xx

2cos1

2cos1tan

xmmx 2cos2cos1 22

c.2

2

1

12cos

m

mx

a.

22

2222

1

2

1

112sin

mmmx

b. xxx 2sin3sin= xxxx sin2coscos2sin

=22

2

22

1.

1

1

1

1.

1

2

m

m

m

m

mm

m

=

323

1

3

m

mm

d. x3cos = 22

2

1.

1

1

m

m

m

m

- 22

1

1.

1

2

mm

m

=

322

1

31

m

m

10. a. x5sin = xx 32sin = xxxx 3sin2cos3cos2sin

= xxxxx sin2sincos2cos2sin2

xxxxx sin2coscos2cos2sin2

= xxxxx cos2cos2sin24cossin

b. x5cos =

xx 32cos

= xxxx 3sin2sin3cos2cos

= xxxxx sin2sin2coscos2cos2

xxxxx sin2cos2sincos2sin2

= xxxxx 2cos2sinsin24coscos

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BAB 3 Rumus-Rumus Trigonometri