PHẦN 2 LƯỢNG GIÁC

www.TOANTUYENSINH.com PHN LNG GIC 2.1 Giỏ tr lng giỏc cot + tan v 00 < < 900 Tớnh giỏ tr ca biu thc A = cot tan + Bin i c A = cos 25 + Thay cos = , ta c A = Lu ý HS cú th tớnh sin , suy tan , cot , thay vo A Cõu Bit cos = Cõu Cho l gúc m tan =2 Tớnh P = sin sin + 3cos3 tan sin cos P= = sin + 3cos3 tan + (1 + tan ) tan (1 + 2 )2 10 = = = tan + +3 11 Cõu Cho gúc thừa : < < Ta cú sin = cos = sin v cos =- Tớnh P = 3 sin + 3cos = 9 nờn sin 1 1+ = cos = 2 cos cos sin = tan .cos = 2 Do ú: A = cos 10 sin cos = ì + 10 ì ì = + = 5 Ta cú: + tan = Cõu 18 Cho tan = ( (0; )) 2 + 3cos 2+ Tớnh giỏ tr biu thc P = sin + 2cos 2 2sin tan = tan + tan = Vỡ tan = ( (0; )) nờn 2 2 tan 2 Suy tan = + hoc tan = (l ) Do tan > 2 tan + 1 + = + =2 Thay vo ta cú P = 5 tan + 2 ổ ổ pữ p ữ ỗ sin a = ; pữ sin a + ữ Cõu 19 Cho gúc a ẻ ỗ m Tớnh (1) ỗ ỗ ỗ ỗ ố2 ữ ứ ố ứ 6ữ ổ p ; pữ ữ T h thc: cos2 a + sin a = v a ẻ ỗ ỗ ữ ỗ ố2 ứ =5 ổ pử 3- a+ ữ = Thay (2) vo (1) ta c: sin ỗ ữ ỗ ữ ỗ ố 6ứ Suy ra: cos a = - 1- sin a = - 1- ổ 3p a a + cos = 2 a a 1 T sin + cos = - ị + sin a = ị sin a = 2 ; 2pữ ữ Cõu 20 Cho gúc a ẻ ỗ m sin ỗ ữ ỗ ố ứ Nguyn Vn Lc (2) Tớnh sin 2a Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com ỡù ùù cos2 a = - sin a = 1- = ù 16 16 ị Do ùớù ổ 3p ữ ùù a ẻ ỗ ;2pữ ỗ ữ ỗ ùù ữ ố2 ứ ợ Vy sin 2a = sin a.cos a = - cos a = 7 ổ 3p ổ pữ ữ ỗ tan a cos a = ữ ữ m Tớnh ỗ ữ ỗ ứ ố ứ 4ữ 41 p; Cõu 21 Cho gúc a ẻ ỗ ỗ ỗ ố ổ 3pữ 92 40 40 ữ ị sin a = cos a = =ị tan a = ữ ữ ứ 41 41 40 - ổ pử tan a 31 ữ = = = ỗa - ữ Do ú tan ỗ ữ ữ + tan a ỗ 40 49 4ứ ố 1+ p; ỗ Do a ẻ ỗ ỗ ố Ta cú: ( sin 4a + 2sin 2a ) cos a = ( cos2a + 1) 2sin 2a.cos a Cõu 22 Cho a l gúc m sin a = Tớnh ( sin 4a + 2sin 2a ) cos a = cos2 aaa sin cos2 ổ 225 ữ = 8( 1- sin aa = ) sin = 8ỗỗỗ1- 16 ữ ữ ữ ố ứ 128 Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com 2.2 Phng trỡnh lng giỏc bc nht Cõu Gii phng trỡnh: cos x + (1 + cos x)(sin x cos x) = cos x + (1 + cos x)(sin x cos x ) = sin x cos x = (sin x cos x)(sin x cos x 1) = sin x cos x = x = + k sin( x ) = x = + k ( k  ) sin( x ) = x = + k Cõu Gii phng trỡnh: sin x + = sin x + cos x sin x + = 6sin x + cos x (sin x 6sin x) + (1 cos x) = 2sin x ( cos x ) + 2sin x = 2sin x ( cos x + sin x ) = sin x = sin x + cos x = 3(Vn) x = k Vy nghim ca PT l x = k , k Z Cõu Gii phng trỡnh: sin x + 2cos x + ( sin x + cos x ) = + cos x sin x + cos x + 4( sin x + cos x ) = + cos x sin x cos x + cos x cos 2 x + 4( sin x + cos x ) = cos x( sin x + cos x ) + 2( sin x + cos x ) = cos x ( sin x cos x + sin x ) + 2( sin x + cos x ) = ( sin x + cos x )( cos x sin x + 1) = Vi sin x + cos x = x = + k , k Z Vi cos x sin x + = (1 sin x ) sin x + = ( sin x 1) ( sin x 1) = sin x = x = + 2m , m Z Cõu Gii phng trỡnh: cos2x + 2sin x 2sin x cos 2x = + PT cos2 x ( 2sin x ) ( 2sin x ) = ( cos2 x 1) ( 2sin x ) = + Khi cos2x=1 x = k , k Z Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Khi s inx = x = + k hoc x = + k , k Z 6 Cõu Gii phng trỡnh: : sin x cos x + sin x = (x R) sin x cos x + sin x = (1) (1) (sin x cos x)(1 + sin x cos x) = x = + k sin x cos x = (k Z ) + sin x cos x = x = 2k x = + 2k Cõu Gii phng trỡnh: s inx + cos x = cos2 x Ta cú: s inx + cos x = cos2 x s inx + cos x = cos x sin x (s inx + cos x) [ (cos x s inx) ] = 2cos( x ) = s inx + cos x = cos x s inx = 2cos( x + ) = x = + k x= + k cos( x ) = 2cos( x ) = 4 x + = + k x = k 4 2cos( x + ) = c os( x + ) = x = + k x + = + k 2 4 Cõu Gii phng trỡnh: lng giỏc: PT (1) sin x + cos x = 3sin x + cos x + sin x + ữ = 3sin x + cos x + (x ) 2sin x cos x 3sin x + cos x cos x = ( cos x 3) sin x + ( cos x + 1) ( cos x 3) = ( sin x + cos x + 1) ( cos x 3) = x = + k sin x + ữ = cos x = (VN )  x = + k (k ) sin x + cos x = Phng tri nh cocac nghiờm: x = Nguyn Vn Lc + k , x = + k (k  ) Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Cõu Gii phng trỡnh: : 3cos5 x sin 3x.cos2 x s inx = x= +k 18 cos5 x sin x = s inx sin x ữ = s inx PT 2 x = + k Cõu Gii phng trỡnh: + sin x = cos x x = k sin x = + sin x = cos x 2sin x cos x = 2sin x x = + k cos x = sin x Cõu 10 Gii phng trỡnh: cos x + 3sin x = cos x + 3sin x = 2sin x + 3sin x = 2sin x 3sin x + = x = + k sin x = x = + k , k  sin x = x = + k Cõu 11 Gii phng trỡnh: sin x cos2 x = 2sin x Bin i phng trỡnh v dng: 2s inx(cos x 1) + 2sin x = s inx = s inx(sin x + cos x 1) = sin x + cos x = Vi s inx = x = k x = k Vi cos2x = sin x + cos x = sin( x + ) = , k Z x = + k 2 Vy phng trỡnh cú h nghim x = k , x = Nguyn Vn Lc + k , k Z Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Cõu 12 Gii phng trỡnh: cos x ( cos x + sin x 1) = cos x ( cos x + sin x 1) = cos x = sin x + ữ = k + ( k  ) x = k (k  ) +) Vi sin x + ữ = x = + k +) Vi cos x = x = Cõu 13 Gii phng trỡnh: 2(cos x + sin x) = + sin x(1 + cos x) Phng trỡnh ó cho tng ng vi: cos x + 2sin x = + 4sin x.cos x (1 cos x)(2sin x 1) = cos x = sin x = x = + k x = + k (k Z ) 12 x = + k 12 Vy pt cú nghim l: x = + k + k ; x = + k ; x = 12 12 (k Z ) Cõu 14 Gii phng trỡnh : sin 2x ( sin x + cos x 1) ( 2sin x cos x 3) = PT ( sin x + cos x ) = ( sin x + cosx 1) ( 2sin x cos x ) ( sin x + cos x ) ( sin x + cos x ) + = ( sin x + cosx 1) ( 2sin x cos x ) x = k2 sin x + cos x = x = + k2 sin x cos x = 4(VN) Cõu 15 Gii phng trỡnh: 3sin x cos x + cos x sin x = sin x cos x + + 2sin x + 2sin x 2sin x cos x = (1+2sinx)(sinx - cosx +1) = Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Cõu 18 Gii phng trỡnh: sin x + = 8cosx + s inx s inx = (vn) Bin i phng trỡnh v dng: (s inx-4)(2 cos x 1) = cos x = Vi cosx = x = + k 2 Kl: phng trỡnh cú h nghim: x = + k , Cõu 19 Gii phng trỡnh: 2sin x = cos x sin x 2sin x = cos x 2sin x.cos x cos x = 2sin x = cos x(1 2sin x) sin x = -Vi cos x = x = + k , k  x = + k ,k  -Vi sin x = x = + k Cõu 20 Gii phng trỡnh: cos x + sin 4x cos3x = cos x + sin 4x cos3x = 2sin 2x.sin x + 2sin 2x.cos 2x = 2sin 2x(s inx + cos2x) = sin 2x( 2sin x + sin x + 1) = k x = x = + k2 sin 2x = s inx = x = + k2 s inx = + k2 x = Cõu 21 Gii phng trỡnh: sin x + cos x = + 2sin x cos x sin x + cos x = + sin x cos x sin x + cos x = sin x + sin x cos x = sin x Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com x = k sin x = 2sin x = sin x x = + k sin x = x = + k Cõu 22 Gii phng trỡnh sau: + 3cos x + cos x cos x = 4sin x.sin x Gii phng trỡnh: + 3cos x + cos x cos x = 4sin x.sin x (1) (1) + 3cos x + cos x cos ( x + x ) = sin x.sin x + 3cos x + cos x ( cos x.cos x sin x.sin x ) = 4sin x.sin x + 3cos x + cos x ( cos x.cos x + sin x.sin x ) = + 3cos x + cos x cos x = + cos x + cos x = cos x + cos x = cos x = cos x = x = + k ;k  x = + k Cõu 23 Gii phng trỡnh: sin 2x sinx = x = k s inx = x = + k Pt cosx = x = + k Cõu 24 Gii phng trỡnh: sin x cos x = 4sin x sin x cos x = 4sin x sin x cos x + cos x 4sin x = sin x cos x + 2sin x 4sin x = 2sin x Nguyn Vn Lc ( ) cos x + sin x = Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com sin x = x = k sin x = ,k  sin x + = x = + k cos x + sin x = ữ Cõu 25 Gii phng trỡnh: cos x(4 sinx + 1) = sin x + pt cos x.sin x + cos x sin x cos x = cos x.sin x sin x sin x cos x = 2sin x(2 cos x sin x cos x) = sin x = sin x = cos( x ) = cos x cos x.cos + sin x.sin = cos x 6 x = k x = + k , (k  ) x = + k 18 Cõu 26 Gii phng trỡnh: cos x 3sin x = - Ta cú phng trỡnh cos x 3sin x = 2sin x + 3sin x + = x = + k sin x = x = + k , k  sin x = x = + k - KL: Phng trỡnh cú ba h nghim Cõu 27 Gii phng trỡnh: sin x + sinx = cos x.cos 2x pt 2sin 2x cos x = cosxcos x ( ) cosx sin 2x cos 2x = cos x = x = + k Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com sin 2x cos 2x = sin 2x ữ = Pt cú nghim x = + k , x = + k Cõu 28 Gii phng trỡnh: sin x + cos x = sin 2x + sin x + cos x = sin 2x + sin x + cos x 2sin x cos x = ( ) ( 2sin x 1) cos x = * cos x = : Vụ nghim x = + k2 * 2sin x = x = + k2 Vy nghim ca phng trỡnh l x = + k2 ; , x = + k2 6 Cõu 29 Gii phng trỡnh: sin 2x + = cos x cos 2x PT sin 2x + + cos 2x cos x = sin x cos x + cos2 x cos x = cos x(sin x + cos x 2) = cos x = x = + k 2 2 sin x + cos x = (VN + < ) Vy nghim ca phng trỡnh ó cho l: x = + k Cõu 30 s inx cos x+ s inx cosx = ; TX D = Ă Phng trỡnh ó cho (2s inx 1)(cos x + 3) = sin x = cosx = 3(vô nghiệm) x = + k , vi k, l l s nguyờn Kt lun x = + l Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Cõu 31 Gii phng trỡnh: sin x Ta cú: ( 1) cos x = 2sin x (1) sin 3x cos x = sin x 2 ổ pử sin ỗ 3x - ữ ữ= sin x ỗ ỗ ố ứ 3ữ ộ ờ3 x ờ ờ3 x p = x + k 2p p = p - x + k 2p ộ p ờx = + k 2p ờ 4p k 2p + ờx = 15 (kẻ Â) p Vy nghim ca phng trỡnh ó cho l x = + k 2p, x = 4p k 2p + (kẻ Â) 15 5x 3x cos + ( 8sin x - 1) cos x = 2 ( 1) ( cos x + cos x ) + 8sin x - cos x = Cõu 32 Gii phng trỡnh: cos Ta cú: (1) cos x + 8sin x - = 4sin 2 x - 8sin x + = ã sin x = : phng trỡnh vụ nghim ộ p ờ2 x = + k 2p p ã sin x = sin x = sin ẻ 5p + k 2p ờ2 x = Vy nghim ca phng trỡnh ó cho l x = ộ p ờx = + k p 12 ờ 5p + kp ờx = 12 p 5p + k p, x = +k pẻ ( k 12 12 Cõu 33 Gii phng trỡnh: cos x.cos 3x + sin x = cos8 x Ta cú: (k Â) Â) (1) ( 1) cos8 x + cos x + sin x = cos8 x ộsin x = 2sin x - sin x - = = ờsin x = Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com ã sin x = x = p + k 2p ộ p ờx = - + k 2p ổ p ã sin x = - sin x = sin ỗ - ữ ẻ ữ ỗ ữ ỗ ố ứ 7p + k 2p ờx = (k Â) Vy nghim ca phng trỡnh ó cho l x= p p 7p + k 2p; x = - + k 2p, x = +k 2pẻ ( k 6 Â) Cõu 34 Gii phng trỡnh: ( sin x - cos x ) = - sin x Ta cú: ( 1) sin x - 2 cos x + 2sin x cos x - = ( sin x cos x + ( sin x ã sin x - ) ( - ) 2 cos x + = )( 2 cos x + ) =0 = sin x = : phng trỡnh vụ nghim ã cos x + = cos x = - 3p 3p cos x = cos x = + k 2pẻ ( k 4 Vy nghim ca phng trỡnh ó cho l x = 3p + k 2pẻ ( k Cõu 35 Gii phng trỡnh: sin x + cos x = + sin x Ta cú: (1) Â) Â) (1) ( 1) sin x + cos x - 2sin x cos x - = ( sin x - 2) ( cos x - 1) = ã sin x - = sin x = : phng trỡnh vụ nghim ã cos x - = cos x = p p cos x = cos x = + k 2pẻ ( k 3 Â) p Vy nghim ca phng trỡnh ó cho l x = + k 2pẻ ( k  ) Cõu 36 Gii phng trỡnh: sin 3x + cos x - sin x = Ta cú: ( 1) cos x sin x + cos x = = cos x ( 2sin x + 1) = cos x = x = Nguyn Vn Lc p p kp + k p x = + (1) ã (kẻ Â) Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com ộ p ờx = - + k 2p ổ pử ã 2sin x + = sin x = - sin x = sin ỗ - ữ ẻ (k Â) ữ ỗ ữ ỗ ố ứ 7p + k 2p ờx = p kp p 7p , x = - + k 2p, x = + k 2pẻ ( k Vy nghim ca phng trỡnh ó cho l x = + 6 Cõu 37 Gii phng trỡnh: cos x + sin x = sin 3x Ta cú: Â) (1) ( 1) cos x + sin x - sin x = cos x - cos x sin x = cos x ( sin x - 1) = p p kp + k p x = + (kẻ Â) p ã sin x - = sin x = x = +k 2pẻ ( k  ) p kp p , x = +k 2pẻ ( k Vy nghim ca phng trỡnh ó cho l x = + 2 ã cos x = x = Â) Cõu 38 Gii phng trỡnh: sinx+sin2x+sin3x+sin4x+sin5x+sin6x=0 (1) ( 1) ( sin x + sin x ) + ( sin x + sin x ) + ( sin x + sin x ) = 2sin x 5x x 3x 7x 3x cos + cos + cos = 4sin cos ( 2cos x + 1) = ữ 2 2 k 7x x = sin = 3x k cos = x = + ;k Z 3 2cos x + = x = + k Cõu 39 Gii phng trỡnh lng giỏc: cos(2x + ) + 4s inx.sin3x - = Gii phng trỡnh : cos(2x + ) + 4s inxsin3x = (1) 2(cos2xcos sin 2x sin ) + 4sin x sin 3x = 3 cos2x s in2x+4sin x sin 3x = 2s in x-2 sin x cos x + 4sin x sin 3x = s inx(2s in3x-sin x- cos x) = Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com s inx = s inx + cos x = 2sin 3x *s inx = x = k (k z) *s inx + cos x = 2sin 3x s inx + cos x = sin 3x 2 3x = x + + k2 x = + k sin(x + ) = sin 3x (k z) 3x = x + k2 x = + k (k z) vy phng trỡnh ó cho cú nghim x = k ; x = + k Cõu 40 Giai phng trỡnh sin(2x + 17 x ) + 16 = 3.sin x cos x + 20 sin ( + ) 2 12 *Bin i phng trỡnh ó cho tng ng vi c os2x sin 2x + 10c os(x + ) + = c os(2x + ) + 5c os(x + ) + = 2c os (x + ) + 5c os(x + ) + = 6 Gii c c os(x + ) = v c os(x + ) = (loi) 6 *Gii c os(x + ) = c nghim x = + k v x = + k 2 Cõu 41 Gii phng trỡnh sau: cos x ữ sin x + ữ= 4 x ữ sin x + ữ= cos x ữ sin x + ữ= 4 cos x + sin x sin x cos2 x = sin x (1 cos x) + cos x(1 cos x) = (sin x + cos x)(1 cos x) = tan x = x = + k cos x + sin x = (k  ) cos x = x = + k cos x = Pt ó cho cos Vy phng trỡnh ó cho cú h nghim: x = ổ p + k , x = + k , (k  ) - xữ + sin x = ữ Cõu 42 Gii phng trỡnh cos ỗ ỗ ữ ỗ ố ứ Nguyn Vn Lc (1) Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Ta cú: ( 1) sin x + 2sin x cos x = sin x ( + cos x) = ã sin x = x = k p ã + cos x = cos x = - 2p 2p cos x = cos x = + k 2p 3 Vy nghim ca phng trỡnh ó cho l x = k p , x = ổ pử x+ ữ ữ Cõu 43 Gii phng trỡnh + tan x = 2 sin ỗ ỗ ữ ỗ ố iu kin: cos x x Ta cú: ( 1) + 2p + k 2pẻ ( k Â) (1) 4ứ p + kp sin x = ( sin x + cos x ) cos x ( sin x + cos x) ( cos x - 1) = ã sin x + cos x = tan x = - x = - ã cos x - = cos x = p + k pẻ ( k Â) p p cos x = cos x = + k 2pẻ ( k 3 Â) i chiu iu kin: cỏc nghim tỡm c u tha iu kin Vy nghim ca phng trỡnh ó cho l x = - Nguyn Vn Lc p p + k p, x = + k 2pẻ ( k Ninh Kiu Cn Th Â) 0933.168.309 www.TOANTUYENSINH.com 2.3 Phng trỡnh bc hai i vi sin, cos Cõu Gii phng trỡnh: (sinx + cosx)2 = + cosx Ta cú: (s inx + cosx)2 = + cosx + sin xcosx = + cosx cosx(2 sin x-1) = x = + k cosx = x= + k2 (k Z) s inx= x = + k2 Cõu Gii phng trỡnh: 2cos 2 x + 3cos x + cos x + 3cos x = Khi ú , phng trỡnh tng ng vi : cos2 x ( cos x + 3cos x + ) = x = + k x = + k 4 x = + k cos2 x = cos x = x = k (k  ) cos2 x + 3cos x + = 2cos x + 3cos x + = cos x = x = + k 2 Vy nghim phng trỡnh l: x = + k ; x = + k 3 ( cos x + cos x ) + s inx ( cos x ) = Cõu Gii phng trỡnh Phng trỡnh ó cho tng ng vi ( 2sin x )( ) ( ) ( ) 3s inx + cos x 2sin x 3s inx + cos x = s inx + cos x = x = + k 3 s inx = 2 x = + k cos x ữ = x = + k , k  Cõu Gii phng trỡnh: sin x cos x = 3sin x cos x Phng trỡnh ó cho tng ng 2sin x 3sin x + 2sin x cos x + cos x = ( sin x + 1) ( sin x + cos x ) = sin x + cos x = : Phng trỡnh vụ nghim Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com x = + k (k  ) 2sin x + = x = + k Vy phng trỡnh ó cho cú nghim: x = + k , x = + k (k  ) 6 Cõu Gii phng trỡnh: 2sin2x + 3cosx = 2sin2x + 3cosx = (1) Pt (1) 2(1 cos2x) + 3cosx = 2cos2x 3cosx = (*) t t = cosx (t 1) t = Pt (*) tr thnh : 2t2 3t = So sỏnh iu kin t = tha t = Vi t = cosx = x = k2 (k Z) Vy nghim ca phng trỡnh l : x = k2 (k Z) Cõu Gii phng trỡnh lng giỏc: cos x + cos x + 3sin x 3sin x = 2 sin x ữ cos x + cos x + 3sin x 3sin x = cos x + ữ = cos x + cos x + 3 = sin x sin x + cos x = 2 3 sin x cos x = = + sin x 2 x = + k (1) tan x = (2) sin x ữ = sin (1) (2) x = + k2 x = + k2 Vy phng trỡnh cú hai h nghim l x = + k hay x = + k2 Cõu Gii phng trỡnh cos x + 6sin x.cos x = + Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Tp xỏc nh Ă ( *) ( + cos x ) + 3sin x = + cos x + 3sin x = 3 cos x + sin x = sin x + ữ = 2 x + = + k x = 12 + k k  x + = + k x = + k Cõu Gii phng trỡnh: 2sin x + sin x = 2sin x + sin x = sin x cos x = x = + k sin x ữ = sin 6 x = + k 1 sin x cos x = 2 ( k  ) Cõu Giai phng tri nh: cos x sin x + = Ta cú: cos x sin x + = 2sin x + sin x = (sin x 1)(2sin x+3)=0 sin x = (do 2sin x + > x Ă ) sinx = x = + k ( k  ) Vy nghim ca phng trỡnh ó cho l x = + k ( k  ) Cõu 10 Gii phng trỡnh trỡnh sau trờn s thc: ) sin2x - cos2x = vi x (o; sin2x - cos2x = cosx(sinx- 3cosx)=0 x = + k cos x = x = + k tan x = Trờn (0,3/2) ta cú nghim l: , , 3 Cõu 11 Gii phng trỡnh sin x + cos x = Nguyn Vn Lc (1) Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com ổ ( 1) - cos ỗỗỗ5 x + ố Ta cú: pử ữ ữ ữ+ cos x = 2ứ ổ pử cos ỗ 5x + ữ ữ= cos x ỗ ỗ ố ứ 2ữ ộ ờ5 x + ờ ờ5 x + ộ ờx = ờ ờx = p = x + k 2p p = - x + k 2p p k 2p + p k 2p + 14 (kẻ Â) (kẻ Â) Vy nghim ca phng trỡnh ó cho l x = - p k 2p p k 2p + , x =+ 14 Cõu 12 Gii phng trỡnh ( + 2sin x ) cos x = + sin x + cos x (kẻ Â) (1) ( 1) ( + sin x) sin x - ( + sin x) = ( + sin x ) ( 2sin x - 1) = Ta cú: sin x = - x = - p + k 2pẻ ( k Â) ộ ộ p p ờ2 x = + k 2p ờx = + k p p ờ 12 2sin x- = sin x = sin x = sin ẻờ (k Â) 5p ờ 5p + k 2p ờx = + kp ờ2 x = 12 ở p p 5p Vy nghim ca phng trỡnh ó cho l x = - + k 2p, x = + k p, x = +k pẻ ( k  ) 12 12 2 Cõu 13 Gii phng trỡnh : 2cos x ữ+ cos x = cos x (1) ( 1) + cos x ữ+ cos x = cos x sin x + cos x = ( cos x 1) sin x + cos x = cos x cos x ữ = cos x 2 x= k + k x = + ,k  12 36 Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com 2.4 Phng trỡnh cha mu Cõu Gii phng trỡnh: cos x(2 cos x + 1) s inx =1 cos x iu kin: cos x x k , k  Vi iu kin trờn phng trỡnh ó cho tng ng: cos x(2 cos x + 1) s inx = cos x 2sin x sin x = sin x = x = + k , k  ; x = + k , k  (tha iu kin) 4 Cõu Gii phng trỡnh: 3(2.cos x + cos x 2) + (3 2cos x).sin x =0 2cos x + K: Pt ó cho tng ng vi pt: Vy pt cú h nghim Cõu Gii phng trỡnh: hoc sin x + ữ + tan x + cos x = sin x cos x K : cos2x Bin i phng trỡnh ( sin x + cos x ) + sin x x + cos x.cos x = pt cos x.cos x = pt cos 2 x + cos x = cos x = (tha K), cos2x = -2 (vn) k Vy cos2x = x = + , k Z k Vy phng trỡnh cú h nghim x = + , k Z Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Cõu Gii phng trỡnh: cot x = tan x + 2cos x (1) sin x sin x k ,k Z K: cos x sin x x sin x x = l 2cos4 x cos x 2cos4 x = cos4 x = cos2 x ,l Z ( 1) cot x tan x = x = l sin x sin x sin x Kim tra iu kin ta c x = + l , l Z Cõu Gii phng trỡnh: 4cos3 x + 2cos x ( 2sin x 1) sin x ( sin x + cos x ) = (1) 2sin x k ,k Z K: 2sin x cos2 x x + ( 1) 4cos x ( sin x + cos x ) cos x ( sin x + cos x ) ( sin x + cos x ) = x = + m ( sin x + cos x ) ( cos x 1) ( cos x + 1) = x = m2 , m Z + m2 x = m2 ,m Z Kim tra iu kin ta c nghim x = Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 [...]... x + 6 = 3 + k 2 x = 12 + k k  2 x + = 2 + k 2 x = + k 6 3 4 Cõu 8 Gii phng trỡnh: 2sin 2 x + 3 sin 2 x 2 = 0 2sin 2 x + 3 sin 2 x 2 = 0 3 sin 2 x cos 2 x = 1 x = + k 6 sin 2 x ữ = sin 6 6 x = + k 2 3 1 1 sin 2 x cos 2 x = 2 2 2 ( k  ) Cõu 9 Giai phng tri nh: 2 cos 2 x sin x + 1 = 0 Ta cú: 2 cos 2 x sin x + 1 = 0 2sin 2 x + sin x 3 = 0 (sin x 1)(2sin x+3)=0 ... x.sin 2 x ) = 0 1 + 3cos x + cos 2 x 2 cos x = 0 1 + cos x + cos 2 x = 0 2 cos x + cos x = 0 2 cos x = 0 cos x = 1 2 x = 2 + k ;k  x = 2 + k 2 3 Cõu 23 Gii phng trỡnh: sin 2x 2 sinx = 0 x = k s inx = 0 x = + k 2 Pt cosx = 2 4 2 x = + k 2 4 Cõu 24 Gii phng trỡnh: 3 sin 2 x cos 2 x = 4sin x 1 3 sin 2 x cos 2 x = 4sin x 1 2 3 sin x cos x + 1 cos 2 x 4sin x = 0 2. .. p + k 2p 2 ộ p ờx = - + k 2p ổ ử 1 p ờ 6 ã sin x = - sin x = sin ỗ - ữ ẻ ờ ữ ỗ ữ ỗ ố 6 ứ ờ 7p 2 + k 2p ờx = ờ 6 ở (k Â) Vy nghim ca phng trỡnh ó cho l x= p p 7p + k 2p; x = - + k 2p, x = +k 2pẻ ( k 2 6 6 Â) Cõu 34 Gii phng trỡnh: 2 ( sin x - 2 cos x ) = 2 - sin 2 x Ta cú: ( 1) 2 sin x - 2 2 cos x + 2sin x cos x - 2 = 0 ( sin x 2 cos x + ( sin x ã sin x - ) ( 2 - ) 2 2 cos x + 2 = 0 )( 2 2 cos... 6 5 2 x = 6 + k2 Cõu 2 Gii phng trỡnh: 2cos 2 2 x + 3cos 3 x + 4 cos 2 x + 3cos x = 0 Khi ú , phng trỡnh tng ng vi : cos2 x ( cos 2 x + 3cos x + 2 ) = 0 x = + k x = + k 4 4 2 x = + k 2 cos2 x = 0 cos x = 1 x = k (k  ) 2 cos2 x + 3cos x + 2 = 0 2cos 2 x + 3cos x + 1 = 0 1 2 cos x = x = + k 2 2 3 2 Vy nghim phng trỡnh l: x = + k ; x = + k 2 4 3 3 ( 2 cos 2 x +... = 0 2 1 2sin x = 1 sin x x = + k 2 1 sin x = 6 2 5 x = + k 2 6 Cõu 22 Gii phng trỡnh sau: 1 + 3cos x + cos 2 x 2 cos 3 x = 4sin x.sin 2 x Gii phng trỡnh: 1 + 3cos x + cos 2 x 2 cos 3 x = 4sin x.sin 2 x (1) (1) 1 + 3cos x + cos 2 x 2 cos ( 2 x + x ) = 4 sin x.sin 2 x 1 + 3cos x + cos 2 x 2 ( cos x.cos 2 x sin x.sin 2 x ) = 4sin x.sin 2 x 1 + 3cos x + cos 2 x 2 ( cos x.cos 2 x +... + sin 4x cos3x = 0 2sin 2x.sin x + 2sin 2x.cos 2x = 0 2sin 2x(s inx + cos2x) = 0 sin 2x( 2sin 2 x + sin x + 1) = 0 k x = 2 x = + k2 sin 2x = 0 2 s inx = 1 x = + k2 1 6 s inx = 2 7 + k2 x = 6 Cõu 21 Gii phng trỡnh: sin 3 x + cos 2 x = 1 + 2sin x cos 2 x sin 3 x + cos 2 x = 1 + 2 sin x cos 2 x sin 3 x + cos 2 x = 1 sin x + sin 3 x cos 2 x = 1 sin x Nguyn Vn Lc Ninh Kiu Cn Th... k 2pẻ ( k 2 Â) ộ ộ p p 2 x = + k 2p ờx = + k p 1 p ờ ờ 12 6 2sin 2 x- 1 = 0 sin 2 x = sin 2 x = sin ẻờ (k Â) ờ 5p 2 6 ờ ờ 5p + k 2p ờx = + kp 2 x = ờ 12 ờ 6 ở ở p p 5p Vy nghim ca phng trỡnh ó cho l x = - + k 2p, x = + k p, x = +k pẻ ( k  ) 2 12 12 2 2 Cõu 13 Gii phng trỡnh : 2cos 2 x ữ+ 3 cos 4 x = 4 cos x 1 (1) 4 ( 1) 1 + cos 4 x ữ+ 3 cos 4 x = 4 cos 2 x 1 sin 4 x + 3 cos 4 x = 2. .. sin 2 x sin 2 x sin 2 x 3 Kim tra iu kin ta c x = + l , l Z 3 Cõu 5 Gii phng trỡnh: 4cos3 x + 2cos 2 x ( 2sin x 1) sin 2 x 2 ( sin x + cos x ) = 0 (1) 2sin 2 x 1 k ,k Z K: 2sin 2 x 1 0 cos2 x 0 x + 4 2 ( 1) 4cos 2 x ( sin x + cos x ) 2 cos x ( sin x + cos x ) 2 ( sin x + cos x ) = 0 x = 4 + m 2 ( sin x + cos x ) ( cos x 1) ( 2 cos x + 1) = 0 x = m2 , m Z 2 + m2 x = 3 m2... + k 2p ờ 3 ờ ờ 4p k 2p + ờx = ờ 5 ở 15 (kẻ Â) p 3 Vy nghim ca phng trỡnh ó cho l x = + k 2p, x = 4p k 2p + (kẻ Â) 15 5 5x 3x cos + 2 ( 8sin x - 1) cos x = 5 2 2 ( 1) 2 ( cos 4 x + cos x ) + 8sin 2 x - 2 cos x = 5 Cõu 32 Gii phng trỡnh: 4 cos Ta cú: (1) 2 cos 4 x + 8sin 2 x - 5 = 0 4sin 2 2 x - 8sin 2 x + 3 = 0 ã sin 2 x = 3 : phng trỡnh vụ nghim 2 ộ p 2 x = + k 2p 1 p ờ 6 ã sin 2 x = sin 2 x...www.TOANTUYENSINH.com 2 s inx cos x = 1 sin(x ) = 4 2 s inx = 1 1 2 s inx = 2 7 x = 6 + k 2 x = + k 2 6 x = 3 + k 2 2 x = k 2 k Cõu 16 Gii phng trỡnh: cos 2 x + 5 = 2( 2 cos x )(sin x cos x) (cos x sin x )2 4(cos x sin x) 5 = 0 x = + k 2 x = + k 2 2 Cõu 17 Gii phng trỡnh: : 3 ( cos 2 x - sin x ) + cos x ( 2sin x + 1) = 0 sin 2 x + 3 cos 2 x = 3 sin x cos x 1 3 3 1 sin 2 x + cos 2 x =