1234 BÀI TẬP TỰ LUẬN ĐIỂN HÌNH HÌNH HỌC-LƯỢNG GIÁC - LÊ HOÀNH PHÒ

8/10/2019 1234 BI TP T LUN IN HNH HNH HC-LNG GIC - L HONH PH

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8.HHHPHThc s. nh gio u t

MI

Bin son theo ni dung v hng ra thi mi ca B GD&T.Dnh cho HS 11, 12 chng trnh c bn v nng cao.

o r a _mkXHT BN I HCquc GIAH \l

WWW.FACEBOOK.COM/DAYKEM.QU

WWW.FACEBOOK.COM/BOIDUONGHOAHOCQU

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W.DAYKEMQUYNHON.UCOZ.COM

ng gp PDF bi GV. Nguyn Thanh T


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NII XT EN DI HC QUC GIA SN16 Hng Chui - Hai B Trng - H Ni

T (04) 9715013; (04) 7685236. Fax: (04) 9714899

Ch u t r c h n h i m x u t bs%:

Gim c PH N G QUC BO

Tng bin tp NGUY N B TIINII

Bin tp ni dungHU NGUYN

Sa bi

L I-IO, ANH TH

Ch bn

CNG TI ANPXA

Trnh by biaSN K

i tc lin kt xut bnCNG TI ANPI-A

1234 B TP T LUSVS M h n h hMH HC - LNG CC

M s": 1L-01H2009In 2.000 cun, ki 16 X24 C i lai Cong ti TNH n liau ) s!j'tii r'itijS'xut bn: 920-2008/CXB/0:-162/DHQCHN, ngy 02/10/200Quyt ti I1I xu L>in s": 01LK-TN/XB

n xong v up lu chiu qu nm 2009.



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e^ u^ ^ eK C cc-c thi quc gia (tt nghip / tuyh sinh...) do B

t234icMtv-. bi tp trc nghim in

bi- y b.-.tp trc nghim in hnh V t ?',^ cun sch: "2234 bi t p t lun n hnh.

^^-^Gt0jH.-^--?234 bi tp tun ih hnh Mink hc - Lng gic" o xi g i a p t hc s L H onh Ph t chc bin son.' :yNm:l^sch^bm ^st cc k nng,, kin thc chun ca cKng trnh:

%^;SC^^br|fcai rr (chng trnh chun v chng trnh nng cao). Tcgif:êcôn v gi iiu mt khi lng ln cc dng tOkr in hnh

vlt^ r^ gip-trng cc thi d B GD&T ra trong cc nm qua. Cc^:'to^| ^ c chn lc t cc dng ton cng c l thuyt v rn

n tûy..v bn.c cn tm thy nhiu dng bi tp nng cao t d h

:v: cho vic h thng ton b kin thc trng tm mn

sirih rn luyn v nng cao k nng t duy cht ch iiieo lpluntOc^hc.

nghim c mt thy gio c thi gian di trc ip ng lp i^^^ y bi dfng hc sinh gii chuyn ton, chng ti tin. rang cuh

schM s ern li nKng iu b ch cho bn c v t c nhng, kt!: ;^;trpng rihtmg k.thi sp fr.. i- o thi gian b in son c hn, mc d rt c'gng nhng cun sch ny

G0 the_c0 ring khim khuyt, Ttmong nhn c nhng ng gp chn H rhicu c c n g n g h i p v c c b n h c, s i n h g r v x a b s c h c h o n

thin/ hn trng ln ti bn sau.

}--::'T&;t gp- xin g v:- Trung tm Sch Gio dc Anpha

;v. 225C Nguyn Tr Phng, P.9, Q.5, TpJHCMT: 08.8547464 ;

- Cng ti Sch - Thit b Gio dc npha

50 Ng uy n Vn Sng, qun Tn PhyTp.HCM.T:08.2676463, 08.38107718 ;

Email: atphabokeenteryah oo.com



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D A N G 1: H T H d & B N - C N G T HC-Grs G S

1. L THUYT V PHNG PHP GII TON

- Hm s lng gic:' {ng trn lng 'gic (O; 1); im cui Mc, y ) e (; 1)' ca gc lng gic a = (Ox, OM). '. - ' /...5 ; , ^

- y ^ . sina =y; co sa = x; tana . , ' 1 ( 1 ' - X . . -V-" .' ' . 1 "1 - " ^ - - * \ o-

a 5 + k?t; cota =r , a ^ k X . X- . . ' , 2^ . X . '- C un ggoc c bit: \ ' -

\ ::: w ' \ _ ' ... V' v. -

irtrxV-- y. :

0 ' n/6 riz/4 '1Z2 "2n/3 3tc/4' *- >

- 0 30' 45 60 90* 120 , 135 .150. 180:sin .> .0 . - 1/2. V2/2 &/2 *- 1 ' \ V3 / 2 t V2 /2 -- '-

cos i \ 1 -V/2 42. /2 - 0 - -1/2 -> /2 /2 ,.. s i A ( 2 ~. -

* ta*}-' ' 0 / S \ V3 . . |v.-V5 1 -4-: -'. .

cot *' s : ' 1 ~i a /3~ ' ' l ' 1 1 ~

Chil : Du hm s"phn t I, II, HI, IV

-r: H thc co: bn: v isin2x + co s^ :^s i nx tanx = cosx

^ cosx cox =

sinx^ Cun g lin qu an c

Hai gc i nhausin (x) = -sin x tan(-x) = -tan:

Hai g c b nhau:; sin(7T - ) = sir

tan(jt a) ~ - t

lang gic tu thuc to im - cui: M thc gc'

cc iu kin xc nh-th ' Y- ~1 tanx. cotx 1

1 + tan2x = , X - - 'COS X

l + c o t 2x = --- \ ' - r . -rsin X _ " ;

b it:' ^

, cos(x) = cosx, . ' . -..7 . .. ,


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Hai g c ph nhau:

sin ( - a) = cosa, os( a) = sina ,2 2

tan(-^ a) = co ta , c o t a) = tana2 . - 2 .

Hai gc hn km 7t:s in(7T + x) = si n x , c o s ( tt + x ) = c o sx tan( + x) = tanx, cot(jr + x) = cotx

Hai gc hn km :2

sin(-^ + x) cosx, co s( + x) = sinx2 2

ta n (^ x) = cotx, cot( + x) = -tan x Z 2

Ch : "cos i sin b ph cho tan, cot 7t'

Cng thc cng:. . sin(a b) = sinaxo sb sinb.cosacos(a b) = cosa.cosb + sina.sinb

. > . TLS' t a n a i t a n btan(a b) = 1 + tan a. tan b

CS : sin x cosx = y2s in (x );

cosx sinx = V2cos(x T )4

2 . CC B TON IM HNH

Bi 1: Rt gn biu thc A = 1 + cosx

Ta c: A =1 + cosx

s i n x

Gii:V2 >

(1 - c o s x ) s

s i n 2 X

-s m x

_ 1 4- c o s x 2 c o s x

( l - c o sx )

1 - COS2 X

1 + c o s X

s i n X1 -

1 c o s x

1 + c o s x

= 2cotx.s i n X 1 -H- c o s XBi 2: Chng mnh biu thc sau khng ph hc X

B = 3(sinsx - cossx) + 4(cos6x 2 sinx) + sin4xGii:

Ta c: B 3(sin4x cos4x)(sin4x + cos4x) + 4cos6x 8sin6x5sin 6x + cos6x + 6sin4x 3sin4x cos2x + 3sin2x

+ s i n X

cos4x

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= sin 6x + cosGx + 6sin 4x6sin6x + 3sin2x cos4x - sin*x cos2x= 1 3 s i n 2x c o s 2x 3 s i n 4x c o s 2x + 3 s i n 2x c o s 4x = 1 .

Ch : Loi ton ny, kt qu bit trc bng cch chn 1 gi tr ca X,chng hn X = 0.

Bi 3: Cho tana = . Tnh c = : 2---------:----- ---------------- 24 sin a - sm cc COS a + COS a

Gi:Vi gi thit th cosa 50, nn cha c t ln mu cho co s2oc:

1 _ _______________ COS2 _______________

s i n 2 a s in a COS a COS2 a

c o s2 a c o s2 a COS2

__ ' 1 + tan 2a

t a n 2 a t a n a + 1 f l Y 1U J ~ 4 + 1

Bi 4: Cho sinx + cosx = a. Tnh D = sin5x + cos5x.Gii:

Ta Ca2(sinx + cosx)2 = 1 + 2sinx.cosx, nn

sinx.cosx = (a21) v sin2x.cos2x - (a2- I) 2 .y 4 V

D o : D = s i n 5x + c o s 5x

= ( s i n x + c o s x ) ( s i n 4x s i n 3x c o s x + s i n 2x c o s 2x s i n x c o s 3x + c o s 4x )

= (s in x + co sx) ( [ s in 4x + co s4x] sinx.cosx . [s in /Sc + co s2x] + s in 2jc co s2x)

= (sinx + cosx)([sin2x + cos 2- 2sin^.cos^K - sinx.cosx + sin2x.cos2x)= (sinx + cosx)(l - sinx.cosx - sin2x.cos2x).

= a ( l - [a2- 1 ] - ~ [a2- 1 ] *) = i ( 5 a - a5).2 4 4

Bi 5: n gin biu thcT 3 T

E = co s ( t t a) + sin(----- + a ) - t a n ( + ) , cot ( a ) .

Gii:Ta c: cos(7E ) = cosa

sin( + a ) = sin[ 2n + ( + a)] = sin( + a) = cosa

tan( + a ) = cota v

co ( a ) = cot(271 [ 4- a]) = cot( + a ) = ~ta.no..2 2 2

V y E = c o s a + c o s a + c o t a (t a n a ) = 1.



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Bi 6: om gin biu thc G = 2co s( + a) .co s( - a).4 4

Gii:

G = 2 (cos cosa - si n sina)(cos cosa + s in sina)4 4 ' 4 4

' 2 . w n/2 V2 . .= 2 ( c o sa sin a)( co sa + sina)

2 2 2 2 ;V2' V2= 2 . . ---- (cosa sin a)(c osa + sinct) = cos2ot sn 2a .

2 2Bi 7: n gin biu thc E = s in 2(30 - a)+ sin 2(30 + a) sin 2a

Gii:E - (sin30cosa - sinacos30)2 + (sin30cosa + sinacos30)2 - sin2a

_ . L___ V3 ,1 ___ ys . -2 _-_2= (--cosa -sina) + ( co sa ----sina) sin^a

2 2 22 .

= -- (cos2a + 3sin2a 2 /3 cosa.sina + cos2a + 3sin2a + 2 \3cosa.sia) sin2a4

-- (2cos2a + 6sin 2a) sin 2a = cos2a + sin2a sin2a = .4 2 2 2

Bi 8: Chng minh:cosa sin(b c) + cosb sin(c a) + COSC sin(a b) = 0 .

Gii:VT = cosa(sinbcosc sinccosb) + cosb(sinccosa sinacosc)

+ cosc(sinacosb sinbcosa)cosa sinbcosc cosasinc cosb cosbsinccosa cosbsinacosc

+ coscsinacosb coscsinbcosa = 0 = VP.Bi 9: Chng minh: nu 3sin3 = sin (2a + (3) th tan(a + P) = 2in a

cos aGii:

Ta c 3 sinp sin (2a + 3) nn 3sin((a + |3) - a) = sin ((a + 3) + a)3sm(a + 3)cosa 3sinacos(a + p)= sin(a + P)cosa + sina.cos(a + p) hay

2 s i n ( a + p) c osc t = 4 s i n a c o s ( a + P) d o s m ( a + . P) = 2 ? ln a ( p c m ) .co s(a + P) cos a

Bi 10: Cho sin(40 + ct) = b v 0 < a < 45. Tnh cos(70 + a )

Gii: T gi thit sin(40 + a ) = b v 0 < 40 + oc < 90

nn cos(40 + a ) = y1 - b2 .

Do cos(70 + a) = cos[30 + (40 + a)]

= cos30. sin(40 + a) + cos(40 + a) sin 30 = +yl b2 .



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Bi 11: Tnh tan( - a), bit sina = v TC< a < .4 13 2

Gii:_ , X _ _ '5 _ 12T gi thit suy ra cosa = => tana = .

1 3 5

1 -

Do t a n ( - a) - = J L .4 1 + ta n a 1 , 1217 1 -- ;

B i 12: Chng minh: tan o0- tan40 = 2tanl0.Gii:

tan 50 tan 40Ta c: ta n io 0= tan(5p - 40) -

1 + tan 50! tan 40

tan 50 - tan 401 + cot 40. tan 40

tan 5 0 tan 40 ^ _ ---------------- ---- nn suy ra pcm.

Bi 13: Chng minh: tan30 + tan40 + tanO0+ tan60 = cos203Gii:

Ta c:

V T = ------- - ir?-Z ---------'+ --------= 2 c o s 2 0 ( --------- - + ----- 0COS30 cos40 COSO cos60 V 3cos40 COS0

o crnO , ,0 o oa0 cos50 + -^^COs400_ 2 -c o s 50 -h V3cos4Q _ 8cos20 2_________ 2s/3 COS50 -COS40 y/s 2 s in 40 .COS40

8 c o s 20 s i n 30 COS50 + COS30 s i n 50y/s sin 40. COS40 + sin 40 .COS 40

= VP.

3. B1LUYNTP

Bi 14: Chng minh ng thcsin2a ta na + cos2a cota + 2snacosa = tana + cota

HD: Dng h thc c bn.Bi 15: Tnh tng F COS20 + cos40 + . . . + COS180.

S: 1.B i 1 6 : T m k e z b i t r ng : s i n ( 2 1 - k 9 0 ) > 0 .

S: k = 4m hoc 4m + 3 vi k nguyn.Bi 17: Cho sina + c os a = m. Tnh: sina cosa; sina cosa ; sin4a + cos4a .

HD: Bnh phng ca ng thc gi thit.

9



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/11* _________ o ' A A 0 ^ ^ . 1On0 T ' _ A COS3 a + COS a . si n 2 a - si n aBi 18: Cho tana = 2 va 90 < a < 180 . Tnh: A ------------- ---- ------------s in J a - c os a

_ 53

B i 19: Cho 3s in4x + 2cos4x = . Tnh A = 2sin4x + 3cos4x81HB: Dng h thc ca bn.

Bi 20: Tnh: sina; sinb, sinc m a, b, c tho mn:cosa = tan b, cosb = tan c, cosc = tan a

HD : Dng h thc ca bn v bnh phang.T O i - / - t u - ______7 s in x + c o s x - 1 2 c s x -Bi 21: Chng minh

Bi 23: Chng minh cc biu thc sau c lp vi bin:B = 2(sin4x + cos4x + sin2xcos2x)2- (sinsx + cossx)

BS: B = Bi 24: Chng minh cc biu thc sau c lp vi bin:

c = 3(sn8x cos8x) + 4(cosGX - 2s in6x) + 6sin4xS: c = -2

Bi 25: Chng minh: Nu ^ - 1 - th l + = . J .

a b a + b a3 b3 ( a + b )3HD: Dng h thc c bn.Bi 26: Tnh gn

D = cos(270x) 2sin(x 450) + cos(x + 000) + 2sn(720~x)S:D = cosx Ssinx

Bi 27: Tnh con E ~ sin (-4,8)i).sin(-5,7 ir) t cos(-6,7tt).cos(-5,8iQcot(-5,2it) tan(-6,27)

S : E = 1 .Bi 28:Tnh gi tr lng gic ca gc 105

HD: 105= 45+60Bi 29:Tnh gn G = co s(- 53 )sin(- 337) + sin(307)sin(l 13)

1BS: G =

2

, T ' u T T _ ta n 225 cot8 1.cot69Bi 3:Tmh gn H =C0t261 + ta n 201

BS: H =y/.

10



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Tnh sin(a - b); cos(a + ib); tan(a - b)15 , _ 2

H D: c os a = ~ ; c o s b =

Bi 32:Cho 0 < a , b < - , a + b = - v tana. tanb = 3 - 2V22 4

Bi 31:Cho a, b l cc gc nhn vi sina ; tanb = .

Tnh tana v tanb.S: tana ~ tanb = V2 .

Bi 33:Chng minh: tan75 + cot75 = 4HD: Dng h thc c bn v cng thc cng.

_ , . , tan(a b) 4- tan b cos(a + b)Bi 34: Chng minh l= - 7

ta n(a + b ) ~ ta n D co s(a - b)HD: Dng cng thc cng.

Bi 35: Chng mnh biu thc sau khng ph thuc vo x:I = cos^ 2cosa.cosx.cos(a + x) + cos2(a + x)

BS: sin2a.B i 36: Chng minh biu thc sau khng ph thuc vo x:

J = tanx.tan(x + ) + tan(x + )tan(x + ) + tan(x + }tanx3 3 3 3

BS: J = -3 .Bi 37: Tnh gn:

K = co s(x ). cos(x + ) + cos(x + ) co s(x + )

/B S K = ( l- > / 3 ) .

4Bi 38:Bin i thnh tch: L = a.snx + b .cosx

HD: Chia 2 v cho \/a 2 + b2Bi 39:B in i thnh ch: M = sinx + sin2 + sn3x + sin4x

HB: Ghp 2 nhmBi 40: Chng minh: N u sina + sinp > th cosa + CQSPI < 1

HD: S ng tng bnh phng ca 2 v tri.

B SM o f

. L THUYT V PHOMG PHP G TON __________:~Ci) |^^ :'.v ' '^;Vf.uc s2 a ~ c o s?a 4 sin2 2cosz - 1':?=: 2siri2a .

sin^^s^ar cosa; tn2a 4 :7 - ; - .. ; 'V 0;/ 1ta il a

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- Cng thc h bc hai:' 2 l + cos 2a - 2 _ 1cos2a ^ ?- - c s2aOS a ----- -; sin a ; tan a - -

, . 2 . 2 x + cos2a \ -Ket qu:1 + cos2a = 2cos2a? 1- cos2a = 2si2a

Nhn ba (m rng): , . I V. - -sin3a = 3sina 4sin 3a; cos3a = 4cos3a r- 3cosa '

, . _ 3 tan a tan3a '!, , -- ---tan3a = ---------------r------- ;~-r-1 3 ta n a > , ... , . V -V '

rx. -__ -' 3____ 3 sin a sin 3a- . - .V 3 cos a"+ COS 3aH bc ba (m rng): sin 3a = ------------ --------- ; cos3 - - --------- ---------- ' ' : 4v-:r.--/ *4 v~

GC ph t = tan(mxng): - ; . r, v ~ ' v :

_ 1 - t 2 "' ~ 2 t . ' , 2 t ' .cos a --------- - ; sin a = ---- --- tan a =

\ + t 2, y : -1 + 12 - - 1 - 12 ; , - -; i ' V ; ; Cng thc bin i t ng th nh'tch: ' ' 5- * 0 ':

,__

-+b? _ _ a - b . - -{ ' - c o sa + c o sb = 2 c o s --------- COS - ---


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Gii:r * _ 1 /A_'_ T 7Z V 7C 1 1 . TU____ 7 N__ 1 , TC 1

Ta c A = (2sin -~cos-r )cos- j = ~ (2sinCOS ) = sin = .2 8 8 4 2 2 4 4 4 2 4

~ _ ta n 15B i 4 2 : T f a h B = ^ i _ .

Gii:B = - 2 t a - r ^ ~ = - ta n 2 (1 5 ) = - ta n 3 0 = .

2 1 - tan215 2 2 4Bi 43: Chng minh cng thc

sin 3a = 3 sina 4s in3a = 4sna sin (60 + a) sin (60 a). .Gii:

Ta c sin3a = sin(a + 2a) = sinacos2a + cosasin2a = sina(l 2sin 2a) + 2sinacos2a= sina (l 2sin2a) + 2sina(l sin2a) = 3sina - 4sin3a.v 4sin a sin(60 + a) sin(60 a)= 4sina (sin60cosa + cos6sina) (sin60cosa cos60sina)

= sin a( V3 cosa + sina) (-s/3 cosa sina) = s ina (3cos2a sin 2a)= sina[3(l sin2a) sin2a] = 3sin a 4sin 3a.

B i 44: Chng minh cng thccos3a = 4cos3a 3cosa = 4cosa.cos(60 + a) cos(60 a)

Gii:Ta c cos3a = cos(a + 2a) = cosacos2a + sina sin2a

= cosa(2cos2a 1 ) + 2cosasinza= cosa(2cos2a ) + 2co sa(l cos2a) = 4cos3a 3cosa

v 4cosa.cos(60 + a) cos(60 a)= 4cosa (cos60 cosa - sin60 sina) (cos60 cosa + sin60 sina)= cosa(cosa V3 sina) (cosa + V3 sina)

= cosa (cos2a 3s in2a) = 4cossa 3cosa.Bi 45: Chng minh C0t2x = -c0- - - = (cot X tan x ) .

2 cot X 2 , 1 , a , l A_ _ a , 1 ,__a , 1 , aSuy ra tng c tana + ta n + ta n + ta n + tan

* 2 2 4 4 8 8 16 16Gii:

, c o t 2 x - 1 _ 1 , c o t 2 X 1 X _ 1 ,Ta c = ( ------------------ ) = (cotx - tanx)2 co t X 2 co t X co t X 2

_ 1 c o s X _ s i n X ^ _ c o s 2 X s i n 2 X _ c o s 2 x =

2 sin x cos x 2 sin X cos X sin 2 x

do (cotx - tanx) = cot2X tan x = cotx 2cot2x.

p dng kt qu ny vi x = , v cng v theo v th:

c = cot - 2cot2a.16 16



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Bi 46: Chng minh rng 1 +cos2x t a nx

I V . . 1 V- 1 Y- 1 "Suy ra tch D 1 + 1 + - 1 -5------ - 1 +

cosayl^ cos2a y \ c o s 4 a /1 cos8a)Gii:

sin 2x.p tan 2x _ cos2x _ sin2x GOSX

ta n x sin X COS 2x s in xGOSX

_ 2 s in XcosX co sX __ 2 eos2 X __ cos2x + 1 __cos2x s inx cos2x cos2x

p dng kt qu ny v i X = , a, 2a, 4a th:

_ ta n a ta n 2a tan4a tan 8a _ tan 8a

t a n tan a ta n2 a ta n4 a tan 2 2

Bi 47: Rt gn: E = yj2 + y2 + 2COSCX ( 0 < a < tc).

Gii:

T ac E = -y/2 + /2 + 2cos a = ^ 2 + 2 ^cos2 j

+cos2x

= 2 + 2 cos^ = ^ 2 + 2cos|^j = 2 1C O S J = 2cos^

Bi 48: Chng minh cng thc theo t = tan2

: _ 2 t 2tsina = ----- , cosa = -----5- , tana = 1 + t 2 1 + t 2 1 - t 2

Gii:

0 , 2t a n 2 tan rr r 2 t 2 a _ aTa c ---- = -------------- = ----- = 2 s in .cos = s in a1 + t l + _ 1 _ 2 2

2 COS2 *2

1 +. 2 2 ^*_2 &1 4.2 1 - ta n cos ------- sin X 1 t 0 0 o cos av --------------. = -------2 = cos a

1 "~t . X. 2 __23 - 2^ ^1 + tan. COS---(- sin 2 2 2

Suy ra tana = 2t1 - V



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Bi 49' Tnh gi tr ln nht - b nht ca hm s y =cosx 4- 2sin ?: 32c os x sin + 4

Gii:

t t = ta n-' th y = 1 " cf" (y -* 1)2- (y + 2H + 3y - 2 = 0.

Nu y = 1: phng trnh tr thnh 3 t + l = 0 = s > t = o

Nu y &1: phng trnh c nghim khi:

A 0 (y + 2)z 4(y l) (3 y - 2 ) : 0 o 115^-2^^ + 4< 0 < = > ^ - < y < 2 .11

2 2Do min gi tr , 2] nn maxy = 2 v miny = YY .

B 50: Chng minh vi a, b, c tu th:

cos(a + b + c) + cos(b + c - a) + cos(c + a b) + cos(a + b c) = 4 cosacosbcoscTa c VT = 2cos(b + c)cosa + 2cosacos(c b)

= 2cosa(cos(b + c) + cos(c b)) = 4cosa. cosc. cosb = VP. Bi 51:Bin i thnh tch so: F = sina + sin2 a + sin 3a + sin 4a

G :F = sina + sin3 a + sin 2a + sin4a = 2sin 2a cosa + sin3aco sa

. = 2c osa(sin2 a + sin3a) = 4cosa. sin . COS .

Bi 52: Bin i thnh tch s:

G = sina + sin 2 a + sin3a + cosa + cos2 a + cos3a

Ta c G = 2sin 2a cosa + sin 2a -- 2cos2a cosa + cos2a

= 2 (cosa + ).( sin2a + cos2a) = 2 - ^ 2 (cosa + COS) sin(2 a + )

= 4 V2 co s( + ) cos( - ) sin(2a + ).2 6 2 6 4

Bi 53: Chng minh tan9 tan27 - tan63 + fcan81 = 4 .Gii:

VT = (tan0+ tan81) - (tan27 + tan63)

_ sin 90________sin 90 2_______________ 2________cos9 .sin9 cos27 .s in 27 2co s9.sin9 2 COS27 .sin 27

2 2 __ s i n 54 - s in l S 0 _ . COS36 .sin 18 _ _XTT>= ------------- ----------- = 2 -------------------- = 4 ------ - 4 = VP.sin 18 sin 54 sin 18 . sin 54 sin 18. COS38

Bi 54: Tnh gn: H = 3sin l5 COS 15 + sm 60sin4 15 - COS415

15



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Gii:Ta c:

H = - s i n 3 0 + -------------------- = - _ ^ __ = 1 _ 1 = _ I2 2 (s in 215 - COS215) 4 2.co s30 4 4

Bi 55: Tnh gn I = COS 10 cos30 cos50 cos70.Gii:

1 . /I =: cos30 (coslO0. COS 50). COS 70 = -(cos(- 40) + cos60).cos702, 8

Cos40-Cos70 + cos70 = -^-c os(30) +COS 110 + ~ COS 7 08 16 16 16 ' 16

- sin 20 + ^ s i n 20 = 32 16 16 32

Ta c sin . nsin . COS7 7

1 . 71 . 1 . 3nsin + sin2 7 2 7

_ 1 . 71 Suy ra J :2

sin.7

Bi 5'S: Khng dng bng hy tnh tng: J = COS + COS + COS

Gii:

2 n , 7Z___

4tc , . 7U 67U+sin. COS-" + sin-r . COS-3-7 7 7 7 7

1 . 1 . 5n 1 5jt , 1 . 7ir-----sin- + sinsinr- + 'Sin-3-

2 7 2 7 2 7 2 7

= _ 2

Bi 57: Tnh tng vi a k2 7TK = sinh + sin(h + a) + sin(h + 2a) +sin(h + 3a) + sin(h + 4a)

Gii:

7'a c2 sin K = 2sinh. sin + 2sin(h + a),s in +2sin(h4- 2a). sin

2 2 2 2

-f*2sin(h + 3a). s in + 2s n(h + 4a) sin .2 2

== c o s (h ~ ) - c o s (h + ) 4- c o s (h + ) c o s( h + ) + c o s (h + )2 2 2 2 2

- cos(h + ) + cos(h + ) - cos(h + ) + cos(h + ) - cos(h + )2 2 2 2 2

9a

cos h - cosh +

2 s i n

16



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Bi 58: Tnh tng vi a ^ k nL = cos2a + cos22a + cos23a + cos24a + cos25a + cos26a.

Gii:

Ta c: L = (1 + cos2a) + (1+ cos4a) + (1 + cosSa)2 2 2

+ (1 + cosSa) + -(1+ coslOa) + ( 1 4 - cosl2a)2 2 2

nn: 2L 6= cos2a + cos4a + cos6a + cos8a + COS1 Oa + co sl2 a .Do : 2sna. (2L - 6) = 2sina cos2a + 2sina cos4a + 2sin a cos6a

+ 2sina cos8a + 2sina coslOa + 2sin a Csl2a= - sina + sin3a - sin3a + sina - sina + sin7a sin7a

+ sin9a sin9a + sin lla s in lla + sinl3a- T _ s in l3 a + l l s i n a

= sin l3 a - sina . Vy L = ---------- --------------- .4s ina

3. BI LUYN TP

Bi 59: Chng minh, nu sina ^ 0, th cosa cos2a cos4a cos8a COS la = .32 sin a

HD: Dng cng thc nhn i.Bi 60:Chng minh vi , b , c tu th:

. a + b a + b . c + a .sina + sinb + sinc = 4 sin ---- -----sin ---------- sin ----- + sn(a + b + c)

2HD: Bin i tng thnh tch.

Bi 61: Tnh cc gi .t lng gic ca gc 8

HD: Dng cng thc h bc

Bi 62: Cho tan a = , tan b = v i 0 < a


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B 68 : Tnh: D = sinl0sin50sin70

TT_ n 271 4 8 _ 167__32tc _ B! 69:Tmh: E = COS--COS-.COS- .COS.COSr^-.cos S: 2.65 65 65 65 65 65

1* A , * f ,__- 4 7U . 4 3tc _ 4 4 T7Bi 70:Tnh F = sin r + sin + sm + sin -f

16 16 16 16HD: H bc lng gic.

Bi 7:Tnh G = sinc~ + COS6 B$ : 24 24 16

Bi 72: Tnh H = tan2 + tan2 + tan212 12 12

HD: + ^ 1 12 12 2

Bi 73: Chng minh cng thc:

t a n 3 a = ^ t a n a a = t a n a .t a n ( 6 0 + a ). t a n (6 0 - a )_ 1 - 3 tan2aHD: 3a = 2a + a

B 74: Chng minh cng thc: COS 4a = 8cos4 a - 8cos2a + 1HD: 4a = 2 .2a

Bi 75:Chng minh cng thc: sin. 5a =16 sin 5 a - 20s in 3a + 5sin aHD: 5a = 2a + 3a

Bi 76: Chng minh: COS3 X. sin X - sin 3 X. COS X = 5" 4x4

HD: t tha cho chung v tri.

Bi 77: Chng minh: 128(sin10x + cos10x) - 5es8x +60cos4x + 63H: Dng cng thc h bcBi 78: Chng minh biu thc sau khng ph thuc vo x:

I = sin4x + sin^x + ) + sin 4(x + ) + sin^(x + ) S: -4 2 4 , 2

1 1 2 3Bi 79: Chng minh: cosx - COS 3x - cos5x = 8sin x.cos X2 2

KD: Ghp (cos 3x + cosx)2

Bi 80: Chng minh: COS12 + COS18 4cosl5 0.cos21Q.cos24 = -2

HD: Dng cng thc bin i tch thnh tng

Bi 81: Tnh tch: J = cos75.cosl5 S: .4

Bi 82: Tnh tch K = sin .COS07112 , 12

H: Dng cng thc bin i tch thnh tng

18



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20/341

Bi 83:Tnh tch L = tan20o.tan 40o.tan60o.tan80o S: V.Bi 84: Tnh tng: M = COS 85 + cos35 - cos25 BS: 0.

Bi 85:Tnh tng: N = ta n 75 - ta n 15 BS: 2V.

Bi 86: Tnh tng P = s in -^-s in -s in S: .

_ ^ 6n 8t ~ ,B i 8 7 : T n h to n g Q = COS - + COS + COS 1 + COS S : 1

5 5 5 5

Bai 88:Tinh tong^ ^ ^ 2 sn 2a sn 3a sn na sn(n +

BS:sin a. i \n(n +1)a

Bi 89: Tm gi tr 11nht v nh nht ca hm s: y = sin 4X + c o s 4 x

Bi 90: Tm gi tr ln nht v nh nht ca hm s:y = a.sin2x + b.cosxsinx + c.cos2x

HO: Dng cng thc h bc a v gc 2x.

.. D N G' 3 ' ; v NG . D N G L N G CSC

1. L THUYT V PHNG PHP GI TONua hm s lcig; g

N: : X iac vaobai tn.i s- v. t X ==sint hay X =: cost

.. -N eux --K y?^= ]^Q ;^t:X ^ E:sint;hay X r.cost ;thi^ct-X: s sirit va y ^ co st/

- Neu-x2:+;y? 2thTc;x==.r siiit va y = r.cost

rr N .l-x-^'d t.x 3 hay ' cos t .

: ;;; t x ^ t a r t h y x - c o t t 1; ..rr.N eu c +JB; c

gc nh cac%oc-;

w ab'câb'îbc^ c a ;==1 th t cc i lng an ca cc

âtangic,.::

2. CC B TON IN HNHBi 91:Cho ab * 1 ; bc 9*- 1; ca * - 1 .

,____ a b , b c , c - a_ a - b b - c c - aChng m in h --------- + ------- ; + ------- = -------. ------------------- .---

1 + ab 1 + DC 1 4- ca 1 + ab 1 + be 1 - ca

Gi:t a = tanA; b = tanB; c = tanC, th ng thc bi thnh

19



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tan A tan B fcanB tan C tan c tan A + tan A tan B 1 + tan B tan c 1 4- tan c tan A

_ tan A tan B tan B tan c tan c ta n A1 + tan A tan B 1 + tan B tan c 1 + tan c tan A *

tan(AB) + tan(B C) + tan(C A) = tan(A -B).tan (B - C).tan(C - Ta chng minh: a + (3 + Y= k o> tana + tanp + tany = tana. tanp. tany

rng (A - B) + (B - C) + (C A) = 0 => (pcm).Bi 92: Cho b < a. _ _

Chng minh: | a + b | + a - b = l a + >/a2b2 + a - Va2b2Gii:

Neu a = 0 th b = 0 nn ng thc ng.Neu a t 0, ng ic tng ng:

I I + - I + I 1 - - I = I 1 + J i ~ I + 1 1 - J i - K I (1)a a M a \ .

V |b[ < aj nn t = cost, 0

22/341

B 95: Gii bt phcmg trnh y / + X - y[ l -~xT < X.

Gii:

iu kin: - 1 < X < 1 nn t X = cos2t, t e [0, J.2

Bt phng trinh tr thnh -v/l + COS2 t -J~^ ~c s2 t cos2 t

V2 1cost - -s/2 sint < cos2t - sin2t V 2 ( c o s t s i n t ) < ( c o s t + s in t ) ( c o s t si n t )

ce> cos(t + )(cos(t - ) 1 ) 0cos(t + ) ^ 04 4 4

. < t < < 2 t < 7 t o - X < x < 0 .4 2 2

Nghim ca bt phng trnh l: - 1 X < 0.

Bi 96: Gii phng trnh: X3 + yj(1 X2)3 = Xyj2 (1 X2)

Gii:

iu kin: IXI < 1 nn t X cosu, u e [0, jc].Phng trnh tr thnh cos3u + sin3u = /2sinu cosu (X)t t = sinu + cosu, 1 1 < V2 .

(1) (sinu + cosu)(l - sinucosu) = V2sinu cosu

t ( l - = V2 o t3+ V2t2- 3t - -J2 - 02 2

{ t - )(t2+ 2V2 t + l ) = 0 t = V2 hay t = -V 2 X/2

Chon t = V2 thi c X = ----2

r

23/341

T iu kin 0 < t < , suy ra c bn nghim thch hp l:2

__ 7E __- OTZ ___. 71 , 7Tx = sm -x -;x = sxn-; X = sin- v X

18 18 14 14

Bi 98: Gii phng trnh: 4x33x = Gii:

Ta c = cos-^ = 4 c g s3 3 eos^ nn X = COS l 1 nghim ca

phng trnh.

Tng t - = cos = COS = COS nn cos5^ v c o s 7 c n g nghim.2 3 3 3 9 9 & V

V 3 nghim phn bit v phng trnh cho bc 3 nn phng trnh c1 - A TZ _ In

3 ngum X, = C0S~?3C>= cos ,x

24/341

s u y r a ( t - y ) ( t 2 + t y + y 2 + 3 ) = 0 n n t y d o y 3 - 3 y = 1 ( 1 )

Xet 2 y 2, t t = 2cosa

(1 ): 8cos3a 6cosa = 1 hay cos3a =

T gii v chn 3 nghim a l J

9 9VI (1) l phng trnh bc 3 nn c. ng 3 nghim y l

2 cos-T-:2 cos ; 2 COS suy ra 3 nghim (x,y) ca h.9 9 9

Bi 101: Cho X, y thay i tha mn X2+ y2= 1.

Tm gi tri ln nht, b nht caP = 2(x +1 + 2 x y + 2 y

(H khi B nm 2008)G:

V X2 + y2 =1 nn t X = cost, y = sint,_ 2(cos2 t + 6 cost .s int) _p = ---------- ---- ( P -6) . s in t- (P + l)cos t = 1- 2P

2 cost .s in t + 2 COS tiu kin c ngkm ca phng trnh

(P- 6)2+ ( P - I )2 > (1 - 2P) p2+ 3P - 18 < 0 6 < p < 3Vy min p = 67max p = 3.

Bi 102: Tm gi tri ln nht b nht ca biu thc p =(1 + X 2 ) + y )

Gii:

t X tancx, y = tanp vi a, |3 e (, ). p __ ( t a n a + t a n 3)(l t a n a t a n p)

(1 + tan2ct)(l + tan 2P)

= sin (a + 3).cos(a + p) = sn2(a + 3) nn < p < .2 2 2

Vy, max p = chng hn a +p = ; rain p = chng hn oc 3 = .

Bi 103: Tm gi tr ln nht, b nht ca hm s y =12x + 8x 2 - 3

(2

x2

+ l)2

tX\2 = tant, vi t

25/341

Cho t = 0 th y = 3, cho t = th y = .4 25

Vy max y = 3 v miny = .2

Bi 104: Gi X, y, z , t l nghim ca h

X + y = 4

z2 + t2 = 9 .

x t + yz 6

Tm X, y, z, t sao cho p = zx t gi tr ln nht.Gii:

t X = 2cosa, y = 2 sina; z 3cos(3, t = 3sin p,a, p e [0, 2n] th

xt + yz > 66(cosasinp + sinacosp) > 66sin(a + P)>6a+3 = 2

Lc , p = xz = 6cosacosp = 3[coss(a 3) + cos(a + [3)3 = 3 co s(a pt gi tr ln nht khi cos(a - 0) = 1 a - (3 = 0.

Suy ra a = B = x = y = V2 ; z = t = .4 2

Vy maxP = 3, khi X = y \2 ,z . t = 2

3. BI LUYN TP,Bii 105: Cho X2+ y2+ Zz + 2xyz = 1; X, y, z > 0. Chng minh:

1+ x y z =X^CL-Vxi -z? ) + y ^ /c i-^ x i-X 2) + Z y ( l - i X l - y2)

HD: t X = cosa, y = cos(3, z = cosy,0 < a, p, Y< Bi 106: Cho xy + yz + zx = 1v xyz ^ 0 .

Chng minh: (x - )(y - ) + (y - )(z - ) + (z - )(x - ) = 4.X y y z z X

HO: t X = tana , y = tanP; z = tany.Bi 107: Cho xy + yz + zx = 1.

C h ng m inh: X + y + z - 3 x y z = x (y ^ + Z2) + y ( z2 4- X2) + z ( x 2 + y 2) .

HD: t X ~ tana, y = tanP; z = tany.

B 108: Cho X2+ y2= 1, u2+ V2= 1- Chng minh: |x(u - v) + y(u + v) 1 th (1 - x)p+ (1 + x)p< 2P.b) nu 0 2P.HD: t X = COS 2a.

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Bi 11: Cho cc s x>y> z tho mn 0 < X, y, 2< 1 v xy + yz + zx = 1., , ____ - u X y z 3v3c h to g m in h _ ^ ? . + _ ^ _ ? + r ^ _ s

HD: t X = tana , y = tanp; z = tany vi cc gc thuc (0; )

Bi 111: Chng minh rng t bn s cho trc lun c th chn ra hai s X,

y sao cho: 0< 1 .1 + x y

HD: t X = tana, y = tanp;Bi 112: Gii phng trnh: 8 X 3 - 4x2- 4x + 1 = 0

TTK _ ___5 zS: xx = cos-7x2 = COS- ,x 3=cos

Bi 113: Gii phng trnh: \ j l + = x(l + 2yl - X2 )

HD: t X = sint

Bi 114: Gii phng trnh: (y( l + x)3 - 7(1 - x )3 ) = 2 + V1 ~ X2HD: t X = cost

X2Bi 115: Gii bt phng trnh: -y/l + X + -71 - X < 2 --------

HD: t X = cos2t

Bi 116: Gii h phng trnh:W i - y2 = 74

y V l - x2 = - 4

HD: t Xsina, y sinb

Bi 117: Tm gi tr ln nht, b nht ca hm s: y =1 + Xs

(1 + X2)3

S: 1 v -4

Bi 118: Tm gi tr ln nht, b nht ca hm s: y = X + -s/lX2HD: t X = sint

Bi 119: Cho a, b, c > 0 v abc + a + c b = 0., 2 2 3

Tm gi tri ln nht ca p = ----- -- 4-

a + 1 b + 1 c + 1

Bi 120: Cho a, b, c thay i tho mn a 2+ 9bz + 9c2= 16.Tm gi ln nht:ca Q = 9ab + 6bc + 9ca

HD: T gi thit vit dng tng bnh phng bng 1.

S: 3

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D N G i:


jnjfL ;

.~ J$ ?^^ '7

Gitg thc;' hjeu

A ( H r ng )

Chi/^Bin;oi:phi



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2. CC B TON IN HNH

Bi 121: Chng minh rng trong tam gic ABC a un c:: B ^ - C B _ A

s i n c o s + s in COS = c o s - i r 2

2

2

2

2Gii:

Ta C:A _ ,n , B C B , c - B c . o B

c o s = c o s( -r ( + - - ) ) = s m ( + -T-) s m ~ c o s + s m - - COS .2 2 2 2 2 2 2 2 2 2

Bi 122:Cho tam gic ABC khng vung.Chng minh rng; tanA -- tanB + tanC = tanA tanB tanC.

Ta c tan (A -f C) = tan (n B) = tan B nn tan B1 - tan A tan c

Ihay tanA + tanB + tanC = tanA tanB tanC.

Bi 123: Chng minh rngtrong tam gic ABC talun c, A . B B c c , A _ .tan-r- tan + tantan-1 + tan ta n -- = 1.

2 2 2 2 2 2Gii:

Ta C:

* / A ^ B , .7t _ c c . t a n f + t a n f 1tan(- + ) = ta n ( ------) = cot nn =

2 2 2 2 2 - . A B c1 - tan ta n tan 2

2

2

u . c /+ A B . t A B * _hay tan (tan11+ tan) I tan tan => pcm.2 2 , 2 2 2Bi 124: Chng minh rng trong tam gic ABC ta iun c:

_ A B C sinA 4- sinB + sinC = 4COS---COS COS

2 2 2

VT = 2s i n A + - COS------ -- + sin(A + B)2 2

_ . A +B __A - B - r t f A + B= 2sin ---- ----- C O S----- ----- + sn2

A -- B A - B . A + B A + B= 2sin ---- ----- COS----- ----- + 2sn ----- -----COS-----------

2 2 2 2A + B , A - B A -f- B

= 2 s i n ------ -------( c o s -------------- + COS------ ------ )2 2 2c 0 A , B A B C

2cos . 2 COS cos(------ ) = 4eos COS COS .2 2 2 2 2 2



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Bi 125: Chng minh rng trong tam gic ABC ta lun csin 2A + sin 2B + sin2C = 4sinA snB sinC

Gi:Ta c: VT = 2sin(A + B)cos(A - B) sin (2tc - 2(A + B))

= 2sin(A +B)cos(A B) sin2(A + B)= 2sin(A + B)[cos(A B) - cos(A + B)]= 2sinC.(2)sinA.sn(B)

= 4sinA sinB sinC.Bi 126: Chng minh rng trong tam gic ABC ta lun cCOS2A + cos2B + cos2C = - 1 - 4cosA cosB cosC

Gi:\TT = 2cos(A + B) cos(A B) + c o s2(tt(A + B))

= 2cos(A + B) cos(A B) + c o s(2tu2(A + B))= 2cos(A + B) cos(A B) + cos(2 (A + B))= 2cos(A + B) cos(A - B) + cos(2(A + B))= 2cos(A + B) cos(A B) + 2cosz(A + B) 1 = 1+ 2cos(A + B)(cos(A B) + cos(A -+- B))

14cosC. cosA cos(B)= 14cosA cosB cosC.Bi 127: Chng mnh rng trong tam gic ABC ta lun c

cos2A + cos2B + cos2C = 12cosA cosB cosC.Gii:

1 + co s 2A , 1 + cos 2B , _ 2/ A . T>\\tT = ---------------- + ------------- + C0S2(A + B)2 2

= 1 + (cos2A 4- cos2B) + cos2(A + B)

= 1+ cos(A + B)cos(A - B) + cos2(A + B)

= 1+ cos(A + B)[cos(A B) + cos(A + B)]= 1cosC. 2 cosA. cos(B)= 1 2cosAcosBcosC.

Bi 128: Chng minh rng trong tam gic ABC ta lun csin2A + sin2B + sin2C = 2+ 2cosAcosBcosC.

Gii:

VT = (1 cos2A) + (1cos2B) + sin2(A + B)2 2

1 (cos2A + cos2B) + 1 cos2(A + B)

~ 2cos(A + B) cos(A B) cos2(A + B)2 - cos(A + B)(cos(A - B) + cos(A + B))= 2+ cosC.(2 cosA cos(-B)) = 2+ 2cosAcosBcosC.

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30/341

Bi 129: Cho tam gic .ABC, chng minh: = 5-^-s n c c

Gii:p dng nh l sin:

4R2(sin2A sin 2B) __ sin2A sin 2B4 R 2 . s i n 2 c ~~ s i n 2 c

1 cos2A 1 cos 2B co$2B c os2A

VP =

sin2c 2 sin2 c

_ 2 sin(B + A )sin(B A) _ sinC .sin(A B) = yrji2 sin 2c ~~ sin2c

Bi 130: Cho tam gic ABC, chng minh rng: cotA + cotB + cotC =

Gii:p dig nh l sin v cng thc din tch:

2 + b2 + c2 _ T5 4R2(sin2A + sin2B + sin 2C)

a2 + + C243

VP = abc4R

= R-8RS sin A sin B sin c

sin A + sinB*2 sin B sin c 2 sin c sin A

+ sinC 2 s inAsiB

1 sinBco sC + sinCcosB 1 sin c COSA + sin A COS c2 s inBs inC 2 s inCsinA

+ 1 sin A cos B + sin B COSA2 sin A sin B

= (co te + cotB + cotA + co te + cotB + cotA) = VT.

Cch khc: p dng nh l cosin v cng thc in tch:cos A b2+ c2- a2 b2+ c2a2

cot A =sin A 2bc.sinA

Bi 131: Cho tam gic ABC, chng minh rng:4S

m* + = (a2 + b2 + c2).

p dng nh l trung tuyn:

4Gii:

_ 1 (12 , 2b 4- c

'

1

d

11 + 2 b2'c + a - + i ra2+ b2 -

~~2 2) 2 l 2J 2 l 2 J

= ( a 2+ b 2 + c2).

29



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31/341

'Bi 132:Cho tam gic ABC, chng minh nu cc cnh a, b, c ca tam giclp thnh mt cp s cng th ac 6Rj.

Gii:' , __ abc _6br c _ 3 u s

Ta c ac= 6Rr = 7- s b4R 4 2 p

2 p = 3b 2b = a + c a, b, c lp thnh cp s cng.

3. BI LUYN TPBi 133: Vi A ,B, c l ba gc mt tam gic, chng minh:

. A . B . V A B csin + sin+ sin + COS + COS + COS

4 4 4 4 4 4

= 4 -s/2 c o s ( + ). c o s ( + ). c o s ( + ).8 8 8 8 8 8

HD: sin COS = V2 sin( + )4 4 4 4Bi 134: Cho tam gic ABC, chng minh rng:

A B cc o s COS COS-T- - 1 -Z 1 2 . +_ 2 - + - 2 . . I + 1 + 1

eA eB ec 3 b c .

2bccos,HD: a = --------- 2-

b + c

S 135: Cho , B , c l ba gc mt tam gic.. A , B . c

s in s i n s in

Tnh gn 2 + C 2 a + A 2 Bc o s COS COS COS COS COS

2 2 2 2 2 2

S: 2.B 136:Cho tam gic ABC, chng minh ng thc:

cosB cosC sinB sinC + cosA 0.D: T A + B + c 7C=> A T CB + C).

Bi 137:Cho tam gic ABC, chng minh ng thc:

: A . B c . A B , c A . B . csin sin cos +COS sin + COS-sin sin 2 - 2 2 2 2 2 2 2 2

_ A c= cos . COS COS .2

2

2A B c

HD: ng ictng ng tan( + ) = cot : ng

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32/341

Bi 138: Chng minh rng trong tam gic ABC bt k ta cA B C

sinA - sinB + sinC = 4sin .COS .s in 2 2 2

HD: Ghp sinA - sinBBi 139: Chng minh rng trong tam gic ABC bt k ta c

cotA.cotB + cotB.cotC -- cotC.cotA 1H: Dng quan h A + B = 7 - c

Bi 140: Chng minh rng trong tam gic ABC bt k ta c

*. A . . B * c _ 4. A ,B . Cc o t g ^ + c o t g + c o t g = c o t g c o t g . c o t g z z z z z

- A B 71 cH: Dng quan h ^ + =

lii 141: Chng minh rng trong tam gic ABC ta lun c

A JB c , . B C A , . c A Bs in --c o s COS + sin COS C O S + sin COS COS 2

2

2 2 2 2 2 2 2

_ . A . B . c A B B c A= sin -^-sin-sin + tan tan + tantan + tan-tan

2 2 2 2 2 2 2 2 2

HD: 1 = sin = sin[( + ) + ]2 2 2 2

Bi 142: Chng minh rng trong tam gic ABC ta lun c

+ 4- 1 = ( t a n + t a n + t a n + c o t c o t c o t )sin A sin B sin c 22 222 22

HD: Ghp tan + c o t t a n + cot , tan + co t v 2 2 2 2 2 2

Bi 143: Chng minh rng trong tam gic ABC bt k ta ca = b.cosC + c.cosB

HB: Dng nh l sin hoc cosin.

Bi 144: Chng minh rng trong tam gic ABC bt k ta c a = r(cot + c o t )

HB: Chn ng vung gc t tm ni tip n cnh BC l K chia BCthnh 2on.

Bi 145: Gi H l trc tm ABC. Chng minh rng 3 ng trn ngoi tip tam gic HBC, HCA, HAB c cng bn knh vi ng trn ngoi tip RH: Dng nh sin.

Bi 146:Gi I l tm ng trn ni tip A ABC v R, R2, R3 l 3 bn knhng trn ngoi ip A IBC, IC A, IAB.Chng minh: RiRal = 2rR2HD: Dng nh l sin.

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33/341

Bi 47: Chng minh rng ixong tam gic ABC bt k ta c

HD: Dng cng thc din tch.Bi 14: Cho A ABC c 2 trung tuyn AM _!_BN.

Chng minh: cotc = 2(cotA 4- cotB)

HI): Chng minh a2+ b2 5c?Bi 149: Cho A ABC vung ti A, tm ng trn ni tip I v c tch 2____ /2

ng phn gic BK.CJ /2. Chng minh; IB .IC= 2

Hl: Chng minh s in s in = -r^ 2 2 4a

Bi 150: Chng minh rng trong tam gic A BC bt k ta c:

Bi 151:Chng minh rng trong tam gic ABC bt k ta c

HD: Dng nh i sin.

Bi 153: Cho tam gic ABC. Chng minh nu 4A = 2B c th = + -

HO:: Tnh trc cc gc t gi thit cho.Bi 154: Cho tam gic ABC.

Chng minh nu b = c, A = 20 th a 3+ b3- 3ab2= 0H 0: Dng ng cao AH chia i tam gic ABC.

Bi 155: Chng minh rng trong tam gic ABC bt k ta c

1 + -r- = cosA + cosB + cosCR

H: Dng cng thc din tchBi 156: Chng minh rng ong tam gic ABC bt k ta c

tan.tan =2 2

s = (a2sin2B + b2sin2A)

a b c

b e c o t ' + c a c o t + ab c o t =2 2 2

1 _ .a b c p

+

HB : Dng cng thc din tch.

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34/341

Bi 157: Chng minh rng trorig tam gic ABC bt k ta c a cotA + bcotB + c cotC = 2(R + r)

HD: a v cosA + cosB + cosC.Bi 158: Chng minh rng trong tam gic ABC bt k ta c

be cos2 + ca cos2 + abcos2 = p2.2 2 2

HD: Dng cng thc h bc v nh l cosin.

Bi 159: Chng minh rng trong tam gic ABC bt k ta c(b c)(p - a) cosA + (c - a)(p - b) cosB + (a b)(p c) cosC = 0.

HD: Dng nh, cosin.Bi 160: Cho A AB C c ng trn ni tip tip xc 3 cnh ti A \ B%

Cca tam gic A BC cnh a%b%c v din tch s \

Chng minh: =2 (sin s in s in )s 2 2 2

HD: Chng minh + = 2sin (sin + sin )& a b 2 2 2

D N G 2 : c p f e T R r y f e T A M g e


Bt xng thc v b'tc :cc'':tq;c-^u;phngph^^nb^gMV:chag minh

o Phng php so snh ' ' J .

Phcmg php bin i tng ng . 'Bin i bt ng thic v dng tng cc binh phng khng m hoc

tch cc tha s khng m.

Phng php'dng bt ng tic ccr bn:. - :Bt ng thc gi tr tuvt di ;

. . Vi a, b tu. thc: |a'Vb: 0,; - ' ,

a + b > c, b + c > a, c .+ \a > b , ' - . ;a b < c, b G< a, c a (b - c) (B - C) i o, (c - a) (C -A ) 2: 0, . '. : . ;(A B) (cosA cosB) :< 0?~(B - C).(cosB - cosC) < Q. r: ':. ri

33

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Bt ng thc Gsi (bt ng ic gia trmg bmrcQg v traing brih nhn)

Nu a, b > 0 th a + k > Vb , h ocvitcch khc: :

;i;^ng' ch xy m -l^-;-'chi;-iiva b;-

Nu a, b, c > 0 th -a + -k + c- > >/abc hbc vit cch khc:,

a + b + c > a +,b + c j

-Du bang chi x ra khi va

Phng php tam thc bc; hai f(x) x? + bx + c, a 560

Nu f(x) = 0 c nghim: th A >: 0Nu A < 0 i a.f(x) ^ 0, V-x.

Nu A < 0 th .f(x) > 0, Vx' . .-'

Phng php min gi tr, i.kiri c nghim, . .

Phng php to , yct,

Phng php o hm

Nu y f(x) - > 0 trn K th f(x) ng bin trn K:

. . . rx > a => f(x).> f(a); X f(x) < f(b)

- i vi yf < 0 tn.K. th ta c bt ng thc ngQ i:' :v: X > a => f(x) < f(a): X f(x) >.f(b) ^

;V;y-^Neirf t GTNN lin 0

Ni f t GTN trn P j ix t . ^Da vo bng bin thin ct 1 hm s trn D ta cng c hh gi v

hm s tng ng.Nguyntc & n'gU 'tq ' !^n h^ jn | iot iMirm hm/s:

^ tin GTLN x hani so f(^ k m ien D t xc lp 'bt ng thc dng

f(x) < A v i l hng s.cj: Bc '-2 :i xem xt d ng ic ixay r Mil nao.' au th nu tnij-hc nu chng hn gi tr X no thuc D.

Cui-cngl kt lun-GTLN'f(x)i=>A:.:'-:K h i u m a xf(x) ; nu khhg nhm lnio ghi max f(x).

' . ; . xcD .-vi ; '

. :T ong t cho vic tm G T N N . - -

34



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36/341

Nguyn: t c . tm gi r | Sm m2f, nh sali ca mt biaa thc:

E)e tm GTLN ca mi biu the T .= g(a; b; c;.) trn min D CSCiu

kin cabiri , b7 C,... ta xc lp bt ng thc dng: g(a; b; c;..) < Avi Al hng s. ,'T

Bc 2 l xem xt u ng thc xy ra khi no. Du c th rCu tntai--hpo:hcigincc gi-tr.a,-b,-c.,... no -thuc D.

' Ciii ciig kt ln GTLN g(a; b; c ;...) = A.

Tiic^g tir chd vic tm GTNN.

Ch : . . .. . :1) Gc bin i ig gic, phi hp cc cng ic ng gic.

2) Gc h thc lng gic c bn troig .tam gic phn trc.

3) V; B, G l ba gc mt tam giac th

A B c> sinA; sinB; sinC > 0, sin ; s i n ; sin > 0, - , ; - : 2 2 2

: .. A , B cCOS ; c o s ; COS > 0 .

2 2 2

4) Bt ng thc JENSEN-y hm li khng c s dng gii ton nhng li c nhng nh gi rt tt, tin li nh hng:

Cho y = f(x) c y . 0 1 du bt ng c ngc li:

- f(a ) + f(b) + f(c) ^ a.'+? + c V

;|v I 3 J_________

2.CC B1T0N IN HNH

'3 \ /3Bi 161: Chng minh vi ABC l tam gic b k: sinA + sinB + sinC <

2

G:e rng vi 0 < X, y < nth

n ^ X - y ^ n _ ^ X - y ^ - < ----- < -Z- => 0< co s------< 1nn2 2 2 2

s in x+ s i n y ____ x + y _ _X - y ^ X + y-------- ------ = sinCOS < s in

2 2 2 2

3 5

irnM!.Mwrm!Baii!aaas3asLMBHBM



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37/341

Ap dng kt qu trn:

: A Ty sinC-t-sin^ , T c + ^snA + snB ^ . A +Jd . Q + _ sin + sin -2 2 ^ 2 2

A + B + C + - _ n;. 3 ___- n V3 ,A . .

< sin-------- --------- = sin = => (pcm).

4 3 2Cch khc: du xy ra khi A = B = c = nn ta lp ghp ri

3dng bt ng thc Csi nh sau:

sinA + sinB + sinC = -=(sin A + sin B. ) + cos B + SJ COS A).& 2 2 v3 V3 /__ ' A B O 3/3

Bi 162: Chng minh vi ABC l tam gic bt k: COS + COS+ COS < -2 2 2 2

Gii:~ n ^ ^ n c o s x + c o s y ^ ____X + y

rng vi < X, y < thi ---------- 0 .Du bng xy ra ch khi sinA = sin B - sinC = 1 : v l.nn sinA + sinB + sinC sinA sinB sinB sinC sinC sinA > 0.V t ( sinA)(] sinB)(l sinC) > 0 suy ra1 sinA - sinB sinC + sinA sin B + sinB sinC + sinC sinA

+ sinA sinB sinC > 0V sinA, sinB, sinC > 0 nn c:sinA + sinB + sinC - sinAsinB - sinBsinC sinCsinA

< 1 sinA sinB sinC < 1 .Bi 164: Chng minh vi ABC l tam gic bt k:

31 < cosA + cosB + cosC < .

2



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Gii:_ A B CTa c: cosA + cosB + cosC 1 + 4sin - s i n - sin > 1 .

2

2

23

v: cosA + cosB + cosC <

A + B A B 2f + 3 2 c o s ------ r----- c o s -------------------2 c o s -------- ------ + 1 <

2 2 ^ 2 22f A + A + B _ A B 1 .

COS -------------- COS-------------- COS------- -------- + >; 0V 2 J 2 2 4

A + B 1 _ ' A - B2 1 A - ~ , [cos-------------- COS---------- ] + sin2 ----- ------ > 0: ng

2 2 2 4 { 2 J

Cch khc: du xy ra khi A = B = c = nn ta lp ghp rio

dng bt ng thc Csi nh sau:cosA + cosB + cosC = (cosA + cosB). 1 + sin A.sin B - COS A cosB.

Bi 165: Cho tam gic ABC c 3 gc nhn.Chng minh rng: tanA + tanB + tanC ^ 3 s.

Gii:Ta c h thc: tanA + tanB + tanC = tanA tanB tanC.Vi tam gic nhn nn tan A, tan B, tan c > 0, p dng bt ng thc Csi:tan A + tanB + tanC > 3 tan A tan B tan cnn (tanA + tanB + tanC)3> 27 tanA tanB tanC = 27(tanA + tanB + tanC)o (tanA + tanB + tanC)2> 27 =i> (pcm).

Bi 166: Chng minh rng, trong tam gic ABC bt k ta c:

A - B B - C C - ACOS----------- + c o s ---------- + c o s -----------2 2 2^ i f A *0 1-o 71 1< cos A + cos B + cos^r c

3 \ 3 J 3J 3^3 ,Gii:

_ I , n ^ , A - C ^ , 7 r A A C -, < --------- < => 0 < cos ---------< 1;

4 4 4 4

0 cos 1B - > COS IB ~ I > 03 3 - 4 3 2 3 3 4 3

, 1 , A - B , B - c , _ A - C A + C - 2Bne n ta CO (c o s ----------- + COS------- ) = COS----------- COS-----------------------2 2 2 4 4_ A - C 3, 71 -QN_ A - c 3 n I= cos---------COS B) = cos---------cos B _

4 4 3 4 4 3

< cos |B I= cos (B )3 3 3 3 .

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_ \ , B - C , c - A. , rc.Tng t: (cos - COS ) ^ cos^-(C-r)

2 2 ^ 3 3 '

-- (cos + COS ) ^ COS ( A ) = > pcm.2 2 2 3 3

Tng t

c o s ( A - | ) < | c o s ( A c o s | ( C - | ) < - ~ c o s ( C - I ) =>(pcm)

Bi X67: Chng minh rng nu tam gic ABC tha iu kin

cos2A + cos2B + cos2C 1

th sinA + sinB + sinC < 1+y2 .

Gii:

Ta c cos2A + COS2B + COS2C = 1 4cosAcosBcosC

nn bt ng thc bi suy ra cosAcosBcosC < 0 do tam gic ABC

khng nhn, gi s A > .2

Ta c: sinA + sinB + sinC = sinA + 2cos COS_ ^ < sinA + 2c os .2 2 2

Xt hm s f(x) = sinx + 2COS-; < X < l2 2

f (x) = cosx sin = (1 - 2sin )(1 + s in ) < 0,2 2 2

nn f(x) nghch bin trn [ ; r)9o

f(A) < f(-^ ) nn sinA + 2cos < 1 + y2 =>pcm.2 2

Bi 168: Chng minh rng trong tam gic ABC ta c

ab(a + b - 2c) + bc(b + c 2a) + ca(c +- a - 2b) > 0.

Gii:Bt ng thc bi tng ng:

a + b - 2 c , b + c - 2 a c + a - 2 b----- ------- + ------------ + ------- ----- 0c a b

( + ) + ( + ) + ( + b ) a 6 .c a c b b a

p dng bt ng thc Csi th c pcm.

B 169: Cho tam gic ABC, chng minh a4+ b4+ c42:16S 2.

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Gi:Ta c bt ng thc hin nhin

a 2 > a 2 ( b - c ) 2 = ( a 4- b - c ) ( a b + c )

b2 > b2 - (c - a)2 = (b + c - a.) ct - c + a) ,

c2s c2(a b) 2= (c + a - b)(c - a + b)=> a 2b 2c 2 > (a + b - c ) 2(b + c - a ) 2(c + a - b ) 2

=> ab c> (a + h - c)(b + c a)(c +a b)Mt khc, p ng bt ng hc Csi2 (a 2b c + a b 2c + ab c2) < a 4 + b2c2 + b 4 + c 2a 2 + c4 + a 2b 2

< a 4 -- b 4 + d 1+ (b 4 + c 4 + c 4 + a 4 + a 4 + b 4)2

= 2(a4+ b4+ C*)nn a4+ b4+ c4: abc(a + b + c) do :

a4+ b4+ c4> (a + b + c)(a + b c)(b + c a)(c + a b) = 16S2Bi 170: Trong cc tam gic ABC, tm gi tr b nht ca

T = cotA + cotB + cotCGii:

Ta chng minh: cotA + cotB + co te > V3

(cotA + cotB + cotC)2> 3cot2A + cot2B + cot2C + 2(cotAcotB +- cotB cotC + cotC cot A) > 3 (cotA - cotB ) 2+ (cotB - cotC) 2+ (cotC - cotA)2> 0

Du ng thc xy ra khi cogA = cotA = cotB = coC A = B ~ c.Vy min(cotA + cotB + cotC) = y/s.

Bi 171: Trong cc tam gic ABC khng vung, c anA, anB anC theoth t lp thnh cp s cng. Tm gi tr b nht ca gc B.

Gii:Ta c h thc anA + tanB + tanC - tanA tanB tanC, v anA, tanB, tandlp thnh cp s cng nn 2tanB = tanA + anCdo 3tanB = tanA tanB tanC suy ra tanAtanC = 3 > 0 .Vy tanA, tanB, tanC > 0 nn tam gic' ABC nhn.

Theo bt ng thc Csin _.___ . ( tan A + taaC _ A 2tt,3 = tanA tanC < ! ------------- I= tan B

=> tanB > y/s=> < B < .Vy min B = .3 2 3

B i 172: Cho tam gic ABC bt k. Tm gi ln nht ca biu thc

p = \3cosB + 3(cosA + cosC).

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Gii:

p = V cosB + 6sin cos < \/s cosB + 6sin2 2 2

- 73 (1 2sin2) + 6sin = _ 273 (sin < t .2 2 2 2 2 2

rv: M- A - c _ . , . B _ ^3Du = xy ra khi COS- = 1v sin = 2 2 2

A = c = v B = 2 . Vy max p =6 3

2_ 5 7 3

2

3. I LUYN TPBi 173: Chng minh bt ng thc vi ABC l tam gic bt k:

cos2A + cos2B + cos2C > .4 ,,

HO: Bin i tcmg ng

A 0 / 1 ^ ' ^ sin 2A + sin 2B + sin 2C! Bi 174: Cho tam gic ABC tu . Chng m in h ---- -------------------- 3.cos A + COS B + cos c

HJD: Bin i tng ngBi 175: Trong mi tam gic ABC, chng minh

/ s i n A + %/s inB + / s c

J A B , r 7 c3 cos + 3 COS + ?/COS~V 2 V 2 V 2

< 1.

HI): Ta chng minh mi X, y > 0 t h i x + y < ^/4(x3 + y3) .

' A B C 3V3B 176: Chng minh vi ABC l tam gic bt k: COS COS COS < 2 2 2 8

HI): Dng bt ng thc Csi.

A B C ^/3Bi 177: Chng minh vi ABC tam gic bt k: tan tantan < .2 2 2 9

i n * A B * c _ c + AHD: tan---tan + tan-tan + tan tan = 1.2 2 2 2 2 2

Bi 78:Chng minh vi ABC l tam gic bt k:

X2

1 + > c o s B + (co s A + c o s C ) x ,V x 2HI): Dng tam thc bc hai.

179: Chng minh vi ABC 1

HD: Dng bin i tng ng

cBi 179: Chng minh vi ABC l tam gic bt k: sinA. sinB < COS2 .

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Bi 180: Chng minh vi ABG l tam gic bt k:

(1 cosA) (1 cosB) (1 - cosC) .8

HD: Dng bt ng thc Csi.Bi 181:Chng minh vi ABC l tam gic bt k:

1 1 1 1 1 - f - 1 ^ --

sin A sin B sinC A BCOS - cos - cos2 2 2

HD: Ghp cp ri nh gi.Bi 182: Chng minh vi ABC l tam gic nhn:

Vtan A + VtanB + VtanC s Jc ot +./cot + JcotV 2 V 2 V 2

HD: Dng bt ng thc Csi.Bi 183: Chng minh vi ABC l tam gic bt k:

1 1 + -------7 > 1 2 .

sin2 t f J sin [ f j sin

TITT* 2 A _ 1 A\ - 1 /1 b 2 + c 2 - a 2 .HD: sin ~ = (1 - cosA) ( 1 ------------------)2 2 2 2bc

Bi 184:Chng minh vi ABC l tmgic bt k:

a2(l V3cotA) + b2(l y cotB) + c2( l y/s cotC) > 0

HD: Dng bt ng thc Csi,

Bi 185:Chng minh vi ABC l tam gic bt k: ma mb mc___ 3 2 , 1-2 , 2HD: Chng minh a.ma -----* -----

Bi 186: Chng minh vi ABCl tam gic bt k: + 2 + 2a b c 4r

HD: Dng so snh.Bi 187: Chng minh vi ABC l tam gic bt k:

3 (a + b + c ) < m a + m b + m c < a + b + c4

HD: Dng hnh bnh hnh AB CD tm M.Bi 188: Chng minh vi ABC l tam gic bt k:

a2(p - b)(p - c) + b2(p - e)(p a) + c2(p - a)(p - b) < p^ 2

HD: Dng tam thc bc hai theo X = p a.

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Bi 189: Chng minh vi ABC l tam gic bt k:y cosA + 2cosB + 2 /3cosC < 4 .

HD: Dng bt ng thc tng ng

(sinB *!?> sinC)2 + (cosB + V3 cosC 2)2 > 02 2

Bi 190:Chiig minh vi ABC l tam gie bt k: < +- k+ - < 22 b + c c + a a + b

HD: Dng so snh v bt ng thc Csi.Bi 191:Chng minh vi ABC l tam gic nhn:

tan A + tannB + tannC > 3^/(3V3)nHD: Dng bt ng thc Csi.

Bi 192: Chng minh v i ABC l tam gic bt k:Nu c sin A + sin 2B + sin 2C th anA.tanB < 1HD: sin 2A + s in 2B + s n 2C2 +2cosAcosB cosC.

Bi 193: Hai gc A, B ca tam gic ABC tha mn iu kin tan + tan = .2 2

3 c Chng minh rang < tan < 1.4 2

HD: Dng bt ng thc Csi.B i 194: Chng minh vi ABC l tam gic nhn:

2 _ 1 _ _ (sinA + sinB + sinC) + (tanA + tanB + anC) > 7r3 3

HD: Xt hm s f(x) = sinx + tanx X vi 0 < X <

w 3 3 2Bi 195: Chng minh vi ABC l tam gc khng c gc t:sin A + sin B + sinC _ \2-----------------------------> 1 + cosA + cosB + cosC 2

c2 COS + sinC

0D : VT > --------------------- v dng o hm.2sin + COSc

2Bi 196: Trong cc tam gic AB C, tm gi tr ln nht ca biu thc

E = cosA cosB cosC.

BS: max E = .8

Bi 197: Trong cc tam gic ABC, tm gi tr b nht ca biu thc:

+ ~ X l + ~ h sn sin sn-r2 2 2

S: min N = 27.

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Bi 198: Trong cc tam gic ABC, tim gi tr b nht ca biu thc

R = tan2 + tan2 + tan22 2 2

HX>: Dng bt ng thc Csi.Bi 199:Trong cc tam gic ABC, tm gi tr ln nht ca biu thc

M = 3cosA + 2 (cosB + cosC).

BS: max M = ~3

Bi 200: Trong tt c tam gic ABC ni tip ng trn (0;R) cho tnrc,tm tam gic c din tch ln nht.

S: tam gic ABC u .

-DNG3.-.---'. : , :D^ G'.TAM:G 1C ________________

1. L THUYT V PHNG PHP GII TOM ________v':'i;Mt;ani gide thm cc iu kin: v cnir hoc v gQ th c dng cbf:;/tm"gic'-vungi'tm'-gic cn, tam gic.u, tam gic nhn,...hoc c gc3 0,6Q ,2Q , ..7 ^

X c p h m r a g p h p I s |b i i : .

Bin i lng gic trc tip a v phng trnh lng gic c bn^ Lp hiu so 2 v ri bin i iih gi bt ng th ci s v ng gic G ithitcho ubng ca bt ng thc

. Ch : ^ _ '1) Phi iifp cc bin iliicmg gic v cc nh l trong tam gic.

.' 2) Cc bi ton nh dng th phi gii theo bin i tang ucmg._______

2. CC BI TON IN HNH

Bi 201: Cho tam gic ABC tha mn = 2cos A.sinB

Chng minh tam gic ABC cn. Gii:

Ta c sin ^ = 2cos A nn sinC = 2cosAsinB sinB

=> sin(A + B) = 2sinB cosA=> sinA cosB + sinB cosA = 2 sinB cosA=> sinA cosB sinB cosA = 0 => sin(A B) = 0=> A B kt. V A, B l cc gc tam gic nn k = 0 o A = B.Vy tam gic tam gic ABC cn ti c .

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Bi 202: Chng t rng nu trong tam gic ABC ta c:c

tanA + tanB = 2cot th tam gic ABC cn.2

Gii:

ng thc cho tong ng vi sin~^ = 2tan~ cos A cos B 2

A + BA + B A + B _ rts 2 A T ,

2 s i n ----- c o s - = 2 -------- T m-n c o s A c o s B2 2 __A + B

cos - 2

CC)S2 f ^ | = cosAcosB

-- 1 + cos(A + B)] = [cos(A - B) + cos(A + B)] cos(A - B) = 1.i' 2

T d suy ra tam gic cho l tam gic cn.

Bi 203: Chng minh rng tam gic ABC tha mn iu kin:cos2A + cos2B + cos2C + 1 = 0

th tam gic l tam gic vung.Gii:

Ta chng minh ng thc:cos2A + cos2B + cos2C = 14cosA cosB cosC

Vy ang thc bi tng ng vicosA oosB cosC = 0o cosA = 0hoc cosB = 0 hoc cosC = 0.Vy lim gic cho l tam gic vung.

b2 + c2 < a2Bi 204: Xt dng tam gic ABC tho mn: 0 nn (b + c)2< 2(b2+ c2) < 2a2

=> b + c < a /2 = > a + b + c < a y/2 + a = a (y/2 + 1)=> sinA + sinB + sinC < (V2 + l)snA < V2 + 1 .

Du "=="xy ra: ----- v c o s 3B c o s 2B > --------------------------------- .3 12 3 12

=> (cos3A cos2A) + (cossB cos2B) > - 3 3 6

Do ng thc xy ra nn cosA = v cosB = o A = B = 60.2 2

Vy tam gic ABC u.Bi 208: Cho tam gic ABC khng t, tha mn iu kin

COS2A +2\2COSB+2\2COSc = 3. Tnh ba gc ca tam gic ABC.

( H khi A nm 204)Gii

t gi thit tam gic AB C khng t th c. _ A. _ B C , ______2 A A

s i n - r - > 0,COS - l , c o s A < COSA nn2 2

c o s 2 A + 2^/2 c o s B + 2-J2COSc - 3

= 2 cos2A 1 + 2V2 .2 cos- .cos-B - 32 2

^ 2 cos2A 1 + 2%/22c o s - 32

< 2cos A 1 + 2-42.2sin - 3 = -2{s2.sin - l)2 02 2

Do du "" xy ra nn:

B - C ,COS- = 2

COS = 1 . Vy A = 90, B = c = 45.

\Z2.sin = 12 / .

Bi 209: Cc gc A, B, c ca tam gic ABC tha.mnA B c

sinA + sinB + sinC - 2sin sin =2 sin .2 2 2

Chng minh rng c = 120.

Gii

Ta chng minh h thc sinA + sinB -- sinC = 4cos ^ COS COS nn t2 2 2

g i t h i t t h c :

A _ B c , A , B _ A + B4COS- co s COS 2 si n -^ si n . = 2 COS------r------

2 2 2 2 2 2

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A B c A . B A B A . c x4c os -r cos co s- - 2 s m - s i n ------2(cos-C O S-------s i n - f s i n - O

2 2 2 2 2 2 2 2 S:

li n c o s COS ( 2 c o s - 1) - 0 .Zi

Vi cos cos > 0 nn COS = => = 60 c ~ 120.2 2 2 2 2

Bi 210: Tnh gc c ca tam gic ABC bit rng: (1 + cotA)(l + cotB) = 2.

Ta chng minh h thc cotAcotB + cotB cote + cotC cotA 1

nn t gi thit

cotA + cotB + cotAcotB = 1cotA + cotB + cotAcotB = cotAcotB -- cotBcotC + cotCcotA

nn cotA + cotB = cotB cotC + cotCcotA = cotC (cotA + cotB)Gi s cotA + cotB = 0=> cotA = cot(B) =^A + B = 7T:V

nn cote = 1 => c = 45.

(A > B > c

Bi 211: Tnh cc gc ca tam gic ABC, bit rng

T iu kin th nht suy ra A > ; > B > 0; > c > 0

3 2 3_ 3tc ^ 3A _ TE 3jc 3B ^ A n . 3C ^ . . 3B ^ A . 3C . n=> > > 41; 0; -Z >> 0nn s in - > 0; sin > 0.2 2 2 4 2 2 2 2 2

, A v chon A = .

2 3 3. , , 5^ 5g 5(3 /

T iu kin th ba bin i thnh COS COS COS = 02 2 2

V cos 0 v do B > c nn > c > 0 CGS-^ > 0.2 6 2

do cos = 0 => B = +v chon B= . T c = .2 5 5 5 15

Bi 212; Cho tam gic ABC tho-^- = Tnh gc A. K b c

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T- 1 _ 1 1 ~ 1 (1 . 1 1 _ 1 1Ta CO - = + --------*A b e 2 c o s ' c ' b c

2

cos = + k2r A = 4- k47t2 2 2 3 3

27TVi A l gc mt tam gic nn A = .

33. BI LUYN TPBi 213: Cho tam gic ABC tha:

cosC(sinA + sinB) = sinC.cos(A B). Hy tnh cosA + cosB.S: cosA + cosB = 1 .

Bi 214: Xt dng tam gic ABC tha mn iu kin:A 3 B . B 3 A

s i n c o s = s i n COS z~ .2 2 2 2

SrTam gic ABC cn ti c.

Bi 215: Cho tam gic ABC tha mn co t = + c .2 bChng minh rng tam gic ABC l tam gic vung.

S: Tam gic ABC vung ti A hay c .

Bi 216: Cho tam gic ABC tho mn sin(B + C) + sin(C + A) + cos(A + B) = .

Chng minh tam gic cn.S: tam gic ABC cn ti c.

Bi 217: Xt dng ca tam gic ABC tha mn iu kinsin A + sinB + s in C ___. A

= c o t - c o t .sin A + s inB - s inC 2 2

S: tam gic ABC cn.Bi 218: Chng minh nu tam gic ABC tho mn iu kin

QCOS A cos B = sin 2 th tam gic cn.

2HD: Dng cng thc h bc.

Bi 219: Xt dng ca tam gic ABC tha mn iu kin0

a 2 s i n 2 B + b 2 s i n 2 A = c 2 c o t 2

BS: tam gic ABC cnB 220:C hng minh rng tam gic AB C vung hoc cn khi v ch khi:

a cosB b cosA = a sinA b sinB.HD: Dng nh l sin

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j___, ____., .r,/-,*, - cosA cosB cosC 5Bi 221: Xt dng tam gic ABC thoa- + I- - - = 3 4 5 1 2

HD: Dng tng bnh phcmg hoc vct.

Bi 222: Xc inh dang ca tam gic ABC nu s = (a + b - c)(a - b + c).4

S: Tam gic ABC vung ti A.

Bi 223: Chng minh rng nu ong tam gic ABC c_ (a j- b)(b + c - a)(a + c - b) +U

HD: Dng nh l cosin

______ _ D^D + c - a x a + c - d; . , . ,cosB = ------------------------------th tam gic l vung.

2abc

Bi 224: Xc inh dang ca tam gic ABC nu 7.cosB cosC sin B sin c

S: tam gic ABC vung

Bi 225: X t dng tam gic ABC tho sin = J ------ .2 V 2cL

S: Tam gic ABC vung.

Bi 226: Chng minh rng, nu ta c be V3 = R(2(b +c) - a) th tam gic l u.

H: snB[>/3 sinC + cosC 2] + siC[n/3 sinB + cosB 2] = 0

B i 227:Tam gic ABC tha mn 3S = 2R2(sin3A + sin3B + sin 3C).

Chng minh rng ABC l tam gic u.

HD: Dng bt ng thc Csi.___ , 9R

Bi 228: Chng minh nu tam gic ABC tho m a + mb + mc = th tam

gic ABC u.

HD: Dng nh l trung tuyn.

Bi 229: Xt dng tam gic ABC tho ab sin + be s in + ca s in^- = 2S V3 .2 2 2

1 2SHD: s = bc sinA Sy ra bc = : r~ -

2 sin A

Bi 230: Xt dng tam gic ABC tho

A . B . B . c . c . A 5 _rs i n s i n + s i n s i n + s i n s m - +

2 2 2 2 2 2 8 4R

S: tam gic u

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Bi 231: C tn ti hay khng tam gic ABC tho mn iu kin

s = a2 - (b - c)2

sin2 A + sin2 B + sin 2 c = cot A + cotB + cotC

BS: khng tn ti

Bi 232: Cc cnh tam gic ABC tho mn a4 = b4 4- c4.Chng t rng tam gi c ABC nhn v 2s in2A - tanB tanC.HX>: So snh cnh gc trong tam gic.

I Bi 233: Tnh 3 gc ca tam gic ABC tho A B C

cosA + cosB + cosC = sin + sin + sin 2 2 2

BS: A = B = c = 60.

Bi 234: Gho tam gic ABCc 3 cnh: a = x2 + x + l ; b = 2 x + l ; c X2 1.

Tnh gc ln nht. , .1 BS: A = 120.

Bi 235: Tnh 3 gc catam gic ABCtho mn: S = (a + b + c)2

B S: A = B = c = 60.

Bi 236: Tnh 3 gc ca tam gic ABC tho: b + c = +

HD: Dng nh l sin v din tch.Bi 237: Tnh cc gc ca tam gic ABC, bit chng tha mn

3 .sin(B A) sinC + sinA + cosB =2

S A = 30; B = 60; c = 90.'Tr,

1 + cosC 2a + b

Bi 238: Tnh 3 gc ca tam gic ABC tho mn: snC ^4a2 ~ b2

a 2(b + c a) = b3 + c3 a 3

BS: A = B = c = 60

Bi 239: Cho tam gic ABC tho s in ^ - - CQ- anA. Tnh gc A.sin B + cos ABS: A = 90.

Bi 240: Cho tam gic ABC c gc A, B nhn : sin2A + sin2B = %/sin c .

Tnh gc c.

BS: c = 90.

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53/341

Phng php: V- \ ''Xt cosx .= 0, xt cosx 7* Or chia 2 ye chp ,cos"x a v phng trinhtheo t = tanx. :; ^ ;V.;>. V ;' ^-:;Ncu chia sinnx th v phng teirii Aeo"t3= cofcc. : " . : - Ch : Bc tng /gim ixrTig i cua lng g ic . ; . ;sin2x = 2sinxcosx; 1 = s in 2x .+ cos2x; cos2x;= cos^x-- sin 2 :..

c = C(sin2x + cos2x) = C(sin2x + cosSc)"* I:;vPhcrag trnh bc nh t theo siiij: S (G in):Phcrag trnh bc nht theo siii es (G in): ;Dng: a. sinu + b: cosu:= c . : J /',iu kin c nghim: 2 + b 2 > c2 : ;

Phng php: Chia 2 v ch Va2 + b2 th c: . : V

a ' : ' .. b i : .' ' V C V= - ; s i n a + ; ; _. .COS a 7r;

.2 . 1.2 - / 7. -1 5? . ~ . . 9. ,

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2 _g q b i t o n i n h n h{Bai 24 l Gii phng trnh: 2cos2x c o s x - 1 = 0

-----^ Gii:

2 c o s 2x c o sx 1 = 0 c o s x 1 hay c o s x = 2

o x - k2n hay X = + k271X - z .3 3B i 242: Gii phng nh: c o s 23 x c o s 2 x c o s 2x = 0

Gii:(H khi A nm 2003)

PT (1 + cos6x)cos2x (1 + cos2x) = 0 cos6xcos2x 1 = 0cos8x + cos4x 2 = 0 2cos24x + cos4x 3 = 0

cos4x = l 3 cos4x = 1 X = , k z .

cos4x = - (lo i) 22

/ Bi 243: Gii phng trnh: 5snx 2 = 3(1 sinx)tan2x(H khi B nm 2004)G:

iu kin: cosx 0 X = + k7, k

55/341

Gii:

Ta c: sin2x + sin2x 2cos2x = *

o s i n 2 x + s in 2 x 2 c o s 2X = ( s in 2x + c o s 2x )2

sin2x + 4s inxcosx 5cos2x = 0Xt cosx = 0 phng trinh: sinx = 0 =>sin ^ + cos^ = 0 (loi)Xt cosx 5* 0. Chia 2 v cho coszx: ta n2x + 4tan x 5 = 0V a + b + c = 0 nn tanx = 1 hay tanx = 5 = ta na

V y n g h i m X = + k t, X = a + kt, k s Z .4

Bi 246: Gii phng trnh: s in3x ~ 3 sin 2xcosx H- 4cos2x . sin x = 0Gii:

Phng trnh: sin3x 3sin2xcosx -- 4cos2x . sinx = 0Xt cosx = 0 th sin 3x = 0 => sinx 0=> sin2x + coszx = 0: V l.Xt csx ^ 0. Chi 2 v cho cos3x:t a n 3x 3 t a n 2x + 4 t a n x = 0 o t a n x ( t a n 2x ~ 3 t a n x + 4 ) = 0

t a n X = 0

tan X 3t an x + 4 = 0 (v nghim v A < 0)

tanx = 0.Vy nghim X = k7,k Z .

Bi 247; Gii phng trnh: 6s in x -+- 8cosx = 5Gii:

Chia 2 v cho V36 + 6410.PT: . sin X + 4- COS X =5 5. 2

V ' l Y T i T - 1 - _ 3 -_ 4V + = 1 nn tCOS cp= , sin (p= \ 5 J W ' 5 5

X 1 7TPT: COSC) . sinx +sincp. cosx = sin x + cp) = = sin 2 2 6

X = tp + k2tVy: 6

SuX P -cp + k2E, k e z

.. . 6

Bi 248: Gii phong trinh: c 0 sx : 2 s ln ^c0 sx = s

Bi 245: Gii phng trnh: sinzx + sin 2x - 2cos X =

2 cos X + sin X 1

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G:>iu kin 2cos2x + sinx i 5*0cosx-2sinxcosx _ ^ COS X - sin 2x ^

2 cos2 X + sin X - 1 cos 2x + sin X

"J3cos2x + sin2x = cosx \3 .sinx COS 2x I= coss X K e ) { s )

x = + k 2 7 t ( l o a i )

X = k - , k


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Phng trnh tr thnh: 6u +l - u ' + 6 0

12u + 1 u2 + 12 = 0 u2 12u 13 0

y/2 c o s X + = -1 cos X + ] - l=r{ 4J l 4 j 4

u = - l

U = 13>V2

n _ 37 1 o _X + = 4 - k 2 7 T4 4t _ 371 *

X + = + k2z4 4

Vy nghim:X = +k2t , rr

2 k e Z

X = 7+ k2jt

Bi 252: Gii phng trnh: 2sinx(l + cos2x) + sin2x = 1 + 2cosx(H khi D nm. 2008)

Gii: .Phng trnh tng ng: 2s inx.2cos2X + sin2x = 1 + 2cosx 2sin 2xcosx + sin2x = 1 + 2cosxs in2x (2cosx + 1) = + 2cosx (2cosx + l) (s in 2x 1) = 0

c o s x = -----

2s i n 2 x = 1

x = - ^ + k27t3

X = + k i , k e Z 4

Bi 253: Gii phng trnh: cos3x + cos2x cosx 1 = 0(Hkh D nm 2006)

G:

PT: cos3x cosx + cos2x 1 = 0 2sinJ|x.siii2x 2sin 2x = 0 s in x ( s n 2 x + s in x ) = 0 s i n 2x (2 c o s x + 1 ) = 0

osin X = 0

1 ^ cosx = 2

X = k !

x = +k2t, k e Z3

Bi 254: Gii phng trnh: (2cosx l)(2 sin x + cosx) sn2x sinx(H khi D nm 2004)

Gi:PT (2cosx l)(s inx + cosx) = 0

To V _ 1_ n 1 X = + k2I COS X - 1 = 0 cos X = 3s i n X 4- c o s x = 0 . _ _ , n , _ _

t a n X = -1 x = - + K7T, k e Z L 4

Bi 255: Gii phng trnh: 2sir?t+ sin7x 1 = sinx(H Kh i B nm 2007)

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PT: 2snz2x 1 + sin7x sinx = 0 o cos4x 2cos4xsin3x = 0Gii:

co s4 x( l 2sin 3x) = 0 cos4x = 0

1 sin 3x - 2

71 - nX = + k

8 47t T 2 n

X " -r k 18 3

.251X " + k1, kZ18 3'Bai 256: Gii phng trnh: (1 + sin2x)cosx + (1 + cos2x)sinx = 1 + sin2x

v (H kh i A nm 2007)Gii:

PT: sinx + cosx + sinxcosx(sinx + cosx) = (sinx + cosx)2(sin x + cosx)(l + sinxcosx sinx cosx) = 0 (sinx + cosx)(l sinx)'(l cosx) = 0

s i n X + cosX = 0 s i n X = 1

COS X = 1

X = 4- k t

4X = + k2r

2X = k27t, k e Z

Bi 258: Gii phng trnh: sin X + - + COS X + *s i n X

3. BI LUYN TP

Bi 257: Gii phng trnh: sin23x cos24x = sin25x cos26x

HD: Cng thc h bc

_____

10cos X 3

HD: t t = sinx + cosx,

Bi 259: Gii phcmg trnh: 2sin 2x + 3cos2x = 5sin 2x + 6 HD: PT: 7 sin2x + lp sinxcosx + 3cos2x = 0

B i 260: Gii phng trnh: sin x + cosx + 5sin2x = 1

H: t y = sinx + cosx \/2s in (x + ), ly < V24

/2Bi 261: Gii phng trnh: s in (2 x -1 5 ) = vi 120 < X < 90

S:-105, 30,75

Bi 262: Phng trnh: tan(3x + 2) =yfz vi < X < c bao nhiu nghim.2 2

S: 3 nghim.

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S:18+-k72Bi 264: Gii phng trnh: sin22x + cos23x =1

HD: S dng cng thc h bc.Bi 265: Gii phng trnh: tan 2x tan 3x tanx = tan2x.tan3x.tan5x

HD: a v tan 5x = tan X. , , sin x + s in 2 x + sin3x nr

Bi 266: Gii phng trnh: ------------- ----- ------- = V3cos X + cos 2x + cos 3x

HD: a v tan 2x = V3Bi 267: Gii phng nh: cos2xcosx + sinxcos3x = sin2xsinx sin3xcos X

KD: a v COS 3x = sin 4x

Bi 268: Gii phng trnh: s in 4X - s in 4 (x + ) = 4sin . COS .cosx

Bi 263: Gii phng trnh: sin3x = cos2x

7r , 7T

+ k-r8 2Bi 269: Gii phng trnh: 3sinx + 2cosx = 2 + 3tan X

HD: a v (cosx l)(3tanx + 2) = 0.

. cot(x + )Bi 270: Gii phng trnh: x _

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1234 BÀI TẬP TỰ LUẬN ĐIỂN HÌNH HÌNH HỌC-LƯỢNG GIÁC - LÊ HOÀNH PHÒ