phương trình lượng giác THPT

phương trình lượng giác THPT

PHNG TRèNH LNG GIC I) KIN THC C BN 1) Bng giỏ tr lng giỏc rad - x -180o -90o -60o -45o -30o 30o 45o 60o 90o 120o 2 2 - sin -1 cos -1 2 tan || - -1 - cot || - -1 - 2 2 3 2 2 - 3 || || 1 - 135o 150o 180o 2 2 - -1 - -1 - 3 -1 - || - 2) Giỏ tr lng giỏc ca cỏc gúc cú liờn quan c bit Gúc i cos( ) cos sin( ) sin tan( ) tan cot( ) cot Gúc bự sin( ) sin Gúc ph sin cos Gúc hn kộm Gúc hn kộm sin( ) sin sin cos cos( ) cos cos sin cos( ) cos cos sin tan( ) tan tan cot tan( ) tan tan cot cot( ) cot cot tan Nguyn Hoi Nam 0979160543 cot( ) cot cot tan Dy kốm hc sinh t L6 L12 3) Cụng thc lng giỏc 1) Cụng thc cng: 5) Cụng thc tớch thnh tng cos(a + b) = cosa.cosb sina.sinb cos(a - b) = cosa.cosb + sina.sinb tana - tanb tan(a - b) = + tana.tanb sin(a - b) = sina.cosb - cosa.sinb tana + tanb tan(a + b) = - tana.tanb sin(a + b) = sina.cosb + cosa.sinb 2) Cụng thc nhõn ụi : cosxcosy= sin2x = 2sinxcosx = (sinx+cox)2 - cos2x = cos2x sin2x = 2cos2x - = 2sin2x x y x y sinx + siny = 2sin cos x y x y sinx siny = 2cos sin 2tanx tan x cot x cot2x = 2cotx tan2x = 3) Cụng thc nhõn 3: cos3x = 4cos3 x - 3cos x sin3x = 3sin x - 4sin3 x 3tan x - tan x tan 3x = 1- 3tan x 3cot x - cot x cot 3x = 1- 3cot x 4) Cụng thc h bc: cos x c os2 x sin x cos2 tan2 cos2 cos2 cot cos2 cos x cos( x y ) cos( x y ) sinxcosy= Sin ( x y ) Sin ( x y ) sinxsiny= cos( x y ) cos( x y ) cos x.sin y = [sin( x + y ) - sin( x - y )] 6) Cụng thc tng(hiu) thnh tớch: x y x y cos x y x y cosxcosy = 2sin sin sin( x y ) tanx + tany = cos xcosy cosx + cosy = cos sin( x y ) cos xcosy sin( x y ) cotx + coty = sin xsiny sin( y x) cotx coty = sin xsiny tanx tany = (3sin x - sin x) cos3 x = (cos3 x + 3cos x) sin x = Nguyn Hoi Nam 0979160543 Dy kốm hc sinh t L6 L12 tanx= sinx ,(x k) cosx cosx ,(x k) sinx 2 sin x cos x cotx= cos x tan x,(x k) cot x,(x k) sin x tanx.cotx=1,(x k ) 3 sin x cos x (sinx cos x)(1 sinx.cos x) 3 sin x cos x (sinx cos x)(1 sinx.cos x) 1cos x sin x cos4 x sin 2 x cos x sin x cos6 x sin 2 x sin x sin x cos x sin x cos x sin x 2cos x 4 sin x cos x 2sin x 2cos x 4 4) Phng trỡnh lng giỏc a) Phng trỡnh lng giỏc c bn ộx = a + k2p Dng: sin x = sin a ờờ ờởx = p - a + k2p ộx = a + k2p Dng: cos x = cos a ờờ ờx = - a + k2p t an x = t an a x = a + k p Dng: p ék : x, a + kp cot x = cot a x = a + kp Dng: ék : x, a kp x arcsin a +k ,k +) sin x a x arc sin a + k x arc cosa +k ,k +) cosx a x arccosa +k Nguyn Hoi Nam 0979160543 ỡù ùù sin x = ị x = k p ùù ù p c biờt: ùớ sin x = ị x = + k2p ùù ùù p ùù sin x = - ị x = - + k2p ùợ ỡù ùù cos x = ị x = p + kp ùù c biờt: ùớ cos x = ị x = k2p ùù ùù cos x = - ị x = p + k2p ùù ợ ỡù t an x = x = k p ùù ùù t an x = x = p + kp c biờt: ùợ ỡù ùù cot x = x = p + k p c biờt: ùớ ùù p ùù cot x = x = + k p ùợ +) tanx a x arc tana +k , k +) cotx a x arccot a+k , k Dy kốm hc sinh t L6 L12 Bi 1: Gii cỏc phng trỡnh sau: a) 2cos x b) e) s in(x-600 ) = 3= tan x - = c) s inx+ = d) s in2x = - 2 i) sin(3x 1) sin(x- 2) n) sin2x cot x = o) sin 3x + sin x = f) cos(2x+500 ) = k)cos3x sin 2x g) tan(2 x - 1) = l) (1- 2cox)(4 - cos x) = p) tanxtan2x= -1 x x m) (cot - 1)(cot + 1) = 2 r) cos( x - x) = q) sin x = h) cot(2 x - p )= b) Phng trỡnh bc nht, bc hai i vi mt hm s lng giỏc Phng trỡnh bc nht i vi mt hm s lng giỏc: gii cỏc phng trỡnh ny ta dựng cỏc cụng thc lng giỏc a phng trỡnh v phng trỡnh lng giỏc c bn Phng trỡnh bc hai i vi mt hm s lng giỏc: l nhng phng trỡnh cú dng a.sin2x+b.sinx+c=0 (hoc a.cos2x+b.cosx+c=0, a.tan2x+b.tanx+c=0, a.cot2x+b.cotx+c=0) gii cỏc phng trỡnh ny ta t t bng hm s lng giỏc.(Chỳ ý iu kiờn ca t t t=sinx hoc t=cosx) Dng t õn phu iu kiờn a sin x + b sin x + c = t = sin x - 1Ê t Ê a cos2 x + b cos x + c = t = cos x - 1Ê t Ê a tan x + b tan x + c = t = tan x a cot x + b cot x + c = t = cot x xạ p + kp , (k ẻ Â ) x kp, (k ẻ Â ) Nờu t t = sin x hoc t = sin x thỡ iu kiờn l Ê t Ê Mt s hng ng thc lng giỏc va mi liờn h + sin 2x = sin x + cos2 x + sin x cos x = (sin x + cos x ) - sin 2x = sin x + cos2 x - sin x cos x = (sin x - cos x ) sin x cos x = sin 2x sin x + cos3 x = (sin x + cos x )(1 - sin x cos x ) sin x - cos3 x = (sin x - cos x )(1 + sin x cos x ) Nguyn Hoi Nam 0979160543 Dy kốm hc sinh t L6 L12 t an x + cot x = sin x cos x sin x + cos2 x + = = cos x sin x sin x cos x sin 2x cos x sin x cos2 x - sin x cos2x cot x - t an x = = = = cot x sin x cos x sin x cos x sin 2x 1 + 1cos 4x sin x + cos4 x = - sin 2x = + cos2 2x = 2 ( )( ) cos4 x - sin x = sin x + cos2 x cos2 x - sin x = cos 2x sin x + cos6 x = sin x + cos4 x - sin x cos2 x = - + cos 4x sin 2x = ( cos6 x - sin x = cos 2x sin x + cos x + sin x cos2 x x = cos x ổ pử ữ sin x cos x = sin ỗỗỗx ữ = ữ ữ 4ứ ố ) + t an x t an ổ pử ữ cos ỗỗỗx m ữ ữ ữ 4ứ ố cos x cos2 x - sin x + sin x = = = (mi liờn hờ gia sinx v cosx) - sin x cos x cos x (1 - sin x ) cos x (1 - sin x ) Bi 2: Gii cỏc phng trỡnh sau: ổ pử ổ pử a) 2cos2 x - 3cos x + = k) cos ỗỗỗ2x + ữữữ+ cos ỗỗỗ2x - ữữữ+ sin x = + (1 - sin x ) ứữ ứữ ố ố b) 2cos2 2x + 3sin x = l) sin x cos x - cos5 x sin x = sin 4x c) 3cos2 x - 2sin x + = m) cos x + 3cos + = d) 5sin x + 3cos x + = n) cot 2x+3cot2x+2=0 e) 2sin x + 3sinx-5 = o) 2cos 2x f) tanx+cotx=2 2 p) 3cot x 2 sin x cosx g) 3sin 2x 4cos 2x q) tan x h) sin x 2sin 2x r) 2cos 2x t anx i) cos x x t) ( cos2x tan x cos2x ) cos6 x + sin x - sin x cos x = - sin x ổ j) (tanx+cotx) -(tanx+cotx)=2 Nguyn Hoi Nam 0979160543 s) (1 + sin x + cos 2x )sin ỗỗỗốx + + t an x pữ ữ ữ 4ữ ứ = cos x Dy kốm hc sinh t L6 L12 c) Phng trỡnh bc nht i vi sinx v cosx: Dng: asinx+bcosx=c iu kiờn phng trỡnh cú nghiờm l a b c Cỏch 1: asinx+bcosx=c b a t: cos ; sin a b2 sin( x ) c 2 2 a b a b b Cỏch 2: a sin x cos x c a b c t: tan a sin x cos x.tan c sin( x ) cos a a 2t t2 x ;cos x Cỏch 3: t: t tan ta cú: sin x (b c)t 2at b c 2 t t Bi 3: Gii cỏc phng trỡnh sau: a) sinx - cos x = b) 2sin 3x + cos3 x = - c) sin 3x cos3x h) sin 2x cos2x i) sin8x cos6x sin 6x cos8x j) sin x sin 2x 3cos x p p )= d) 3sin5x 2cos5x k) 2sin( x + ) + sin( x - e) 4sin x cos x l) 4sin x + 3cos x = 4(1 + tan x) - f) sin 2x cos2x ổ x ử2 x ữ m) ỗỗỗsin + cos ữữ = ữ 2ứ ố g) sin x sin x cos x cos x n) cos 7x - cos x cos x = sin 7x = - ổ2p 6p ữ , " x ẻ ỗỗỗ ; ữ ữ ố ứữ d) Phng trỡnh lng giỏc ng cp Dng: a sin X + b sin x cos x + c cos2 x = d (1) " a, b, c, d ẻ Ă Cỏch 1: ỡù cos x = ù (Hay x = kp ) cú phi l nghiờm ca ùù sin x = ùợ phng trỡnh (1) hay khụng ? Nờu phi thỡ nhn nghiờm ny p Bc Kim tra xem x = + kp, (k ẻ Â ) p + kp, (k ẻ Â ) Bc Khi x ỡù cos x ù (Hay x kp ) Chia hai vờ ca (1) cho cos2 x ùù sin x ùợ (hay sin x ), ta c: Nguyn Hoi Nam 0979160543 Dy kốm hc sinh t L6 L12 (1) a sin x sin x cos x cos2 x d + b + c = 2 cos x cos x cos x cos2 x ( a t an x + b t an x + c = d + t an x ) (a - d )t an x + b t an x + c - d = Bc 3: t t = tan x a v phng trỡnh bc hai ó biờt cỏch gii Cỏch 2: S dung cụng thc h bc v nhõn ụi - cos 2x sin 2x + cos 2x ; cos2 x = v sin x cos x = vo (1) v rỳt gn li, 2 ta c: b sin 2x + (c - a )cos 2x = 2d - a - c (*) Bc 1: Thờ sin x = Bc 2: Gii phng trỡnh (*) , tỡm nghiờm õy l phng trỡnh bc nht i vi sin 2x v cos 2x m ó biờt cỏch gii ộa sin x + b sin x cos x + c sin x cos2 x + d cos x = (2) Dng: 2 ờởa sin x + b sin x cos x + c sin x cos x + d sin x cos x + e cos x = (3) Cỏch gii: Chia hai vờ ca (2) cho cos3 X (hay sin X ) hoc chia hai vờ ca (3) cho cos4 X (hay sin X ) v gii tng t nh trờn Bi 4: Gii phng trỡnh lng giỏc: a) cos2 x - sin 2x = + sin x b) sin x + (1 - 3) sin x cos x + (1 - 3) cos2 x = c) sin x - 5sin x cos x - 6cos2 x = d) sin x - sin x cos x + 2cos2 x = e) 2sin x + 3 sin x cos x - cos2 x = f) sin x + sin 2x + (8 - 9) cos2 x = g) sin x + sin x cos x - cos2 x = h) - sin x + sin x cos x + cos2 x = i) 2sin x - cos2 x - sin x cos x = j) sin x + cos2 x - sin x cos x = k) sin x + cos2 x - sin x cos x = l) sin x + cos2 x + sin 2x = Nguyn Hoi Nam 0979160543 Dy kốm hc sinh t L6 L12 e) Phng trỡnh lng giỏc i xng Dng Dng Dng a (sin x + cos x ) + b sin x cos x + c = ị P P : t = sin x + cos x, t Ê t2 - ị sin x cos x = a (sin x - cos x ) + b sin x cos x + c = ị P P : t = sin x - cos x, t Ê ị sin x cos x = ( 1- t2 ) a t an x + cot x + b (t an x + cot x ) + c = ỡù sin x kp éK : ùớ sin 2x x , (k ẻ Â ) ùù cos x ợ ị P P : t = t an x + cot x , t ị t an x + cot x = t - Dng ( ) a t an x + cot x + b (t an x - cot x ) + c = ỡù sin x kp éK : ùớ sin 2x x , (k ẻ Â ) ùù cos x ợ ị P P : t = t an x - cot x , t ị t an x + cot x = t + Dng ( ) a sin x + cos4 x + b sin 2x + c = ị P P : t = sin 2x, t Ê ị sin x + cos x = - Dng ( ) a sin x + cos4 x + b cos 2x + c = ị P P : t = cos 2x, t Ê ị sin x + cos x = - Dng ( ) ( 3 sin 2x = - t 4 ) a sin x + cos6 x + b cos 2x + c = ị P P : t = cos 2x, t Ê ị sin x + cos6 x = - Dng 1 1 sin 2x = + cos2 2x = + t 2 2 2 a sin x + cos6 x + b sin 2x + c = ị P P : t = sin 2x, t Ê ị sin x + cos x = - Dng sin 2x = - t 2 3 sin 2x = + cos2 2x = + t 4 4 a sin x + b cos4 x + c cos2x + d = ỡù ỡù ùù 1- t) ( cos 2x t ùù sin x = ù sin x = = 2 ị ùùớ ị P P : t = cos 2x, t Ê ị ùớ ùù ù + cos 2x + t + t ù ( ) cos x = = ùù ù 2 ùợ ùùợù cos x = Nguyn Hoi Nam 0979160543 Dy kốm hc sinh t L6 L12 Bi 5: Gii cỏc phng trỡnh sau: a) sin x cos x 6sin x cos x k) sin x cos x 4sin x cos x b) sin x cos x sin x cos x l) sin x cos x sin x cos x c) sin x cos x sin xcos x m) 2 sin x cos x 3sin 2x d) 2sin 2x 3 sin x cos x n) sin x 2sin 2x cos x e) sin x + cos3 x - = sin 2x f) (sin x + cos x ) = tan x + cot x 3 g) + cos x - sin x = sin x h) cot x - tan x = sin x + cos x i) + tan x = sin x + cos x ổ j) sin 2x + sin ỗỗỗx ỗố p ửữ ữ= ữ ứữ Nguyn Hoi Nam 0979160543 o) p) q) cos3 x + sin x = cos 2x cos3 x - sin x = sin 2x - 12 (sin x - cos x ) + 12 = sin x + cos x = r) sin 2x + sin x cos x + 2sin x + 2cos x = s) sin x + cos6 x = sin 2x t) Dy kốm hc sinh t L6 L12 f) Mt s dng phng trỡnh khỏc PHNG TRèNH LNG GIAC CHA CN VA CHA TRI TUYấT ễI Phng phỏp: Phng trỡnh cha cn thc: Ap dung cụng thc A= ỡù B A = B ùớ ùù A = B2 ùợ ỡù A ỡù B B ùớ ùớ ùù A = B ùù A = B ợ ợ Lu ý: Khi gii B , ta ỏp dung phng phỏp th li Phng trỡnh cha giỏ tr tuyt i Cỏch M giỏ tri tuyờt i da vo inh nghia Cỏch Ap dung cụng thc ộỡù A ờù ỡù B ờớù A = ùù ù A = B ộờA = B ờùợỡ ùù ờùù A < ùù ờA = - B ờớ ợở ờùù A = ởợ ộA = B A = B ờờ ờởA = - B B - B PHNG TRèNH LNG GIAC KHễNG MU MC Loi Tng hai s khụng õm: ỡù A ùù ùớ B ị ùù ùù A + B = ợ ỡù A = ù ùù B = ợ Loi Phng phỏp i lp dng 1: ỡù A Ê M ùù ỡ ùớ B M ị ớùù A = M ùù ùù B = M ợ ùù A = B ợ Loi Phng phỏp i lp dng 2: ỡù ỡù A Ê M ùù ù ù ớù B Ê N ị ùợ ùù ùù A + B = M + N ợ ỡù A = M ù ùù B = N ợ ỡù sin u = c biờt sin u sin v = ùớ ỡù sin u = - sin u + sin v = - ùớ ỡù cos u = cos u cos v = ùớ ỡù cos u = - cos u + cos v = - ùớ ùù sin v = ợ ùù cos v = ợ Nguyn Hoi Nam 0979160543 ùù sin v = - ợ ùù cos v = - ợ Dy kốm hc sinh t L6 L12 ộỡù sin u ờù ờớù sin v sin u sin v = ờùợỡ ờùù sin u ờớ ờùù sin v ởợ = = = - = - ộỡù cos u = ờù ờớù cos v = cos u cos v = ờùợỡ ờùù cos u = ờớ ờùù cos v = ởợ Nguyn Hoi Nam 0979160543 1 - - ộỡù sin u ờù ờớù sin v sin u sin v = - ờùợỡ ờùù sin u ờớ ờùù sin v ởợ ộỡù cos u ờù ờớù cos v cos u cos v = - ờùợỡ ờùù cos u ờớ ờùù cos v ởợ = - = = = - = - = = = - Dy kốm hc sinh t L6 L12