bài tập lượng giác có lời giải

bài tập lượng giác có lời giải ÔN THI THPT

GV: Lấ ANH TUN Trng THPT ễNG SN õy l cõu hi luụn cú cỏc thi i hc cú im s l 1, cõu ny rt d ly im Hi vng 34 bi sau õy s giỳp cỏc bn cú ti liu ụn v t kt qu tt sin x + cos x Cõu : Gii phng trỡnh : sin x = ( tan x + cot x ) Gii : iu kin: sin x 1 sin 2 x sin 2 x sin x cos x 1 2 (1) = + = sin 2 x = sin x = ữ sin x cos x sin x sin x sin x Vy phng trỡnh ó cho vụ nghim 2 Cõu : Gii phng trỡnh: cos x + sin x sin x sin x = Gii : pt ó cho tng ng vi pt: 1 1 1 (1 + cos x ) + (cos 3x cos x) (1 cos x) = cos x cos x + cos 3x cos x + = 2 2 2 2 x= +k cos x + = 15 cos x + cos 3x = 2 x = + k cos 3x = Cõu : nh m phng trỡnh sau cú nghim 4sin3xsinx + 4cos 3x - ữcos x + ữ cos 2x + ữ+ m = 4 Gii : 4sin3xsinx = ( cos2x - cos4x ) ; +/ 4cos 3x - ữcos x + ữ = cos 2x - ữ+ cos4x = ( sin 2x + cos4x ) 4 +/ cos 2x + ữ = + cos 4x + ữữ = ( sin 4x ) 2 1 Do ú phng trỡnh ó cho tng ng: ( cos2x + sin2x ) + sin 4x + m - = (1) 2 t t = cos2x + sin2x = 2cos 2x - ữ (iu kin: t ) Khi ú sin 4x = 2sin2xcos2x = t Phng trỡnh (1) tr thnh: t + 4t + 2m = (2) vi t (2) t + 4t = 2m õy l phung trỡnh honh giao im ca ng ( D ) : y = 2m (l ng song song vi Ox v ct trc tung ti im cú tung 2m) v (P): y = t + 4t vi t Trong on 2; , hm s y = t + 4t t giỏ tr nh nht l ti t = v t giỏ tr ln nht l + ti t = Do ú yờu cu ca bi toỏn tha v ch m + 2 m 2 SU TM GV: Lấ ANH TUN Trng THPT ễNG SN 2(s inx cos x) = tanx + cot 2x cot x s inx.cos x Gii : iu kin : sinx.cosx cot x Phng trỡnh ó cho tng ng vi phng trỡnh: Cõu : Gii phng trỡnh : ( s inx cosx ) s inx cos2x cos x s inx + cos x s in2x s inx x = + k2 (k Z) Gii c cos x = x = + k2 = i chiu iu kin ta c nghim ca phng trỡnh l: x = Cõu : Gii phng trỡnh: + k2, (k Z) 5x x cos + sin 2x + 3cos x + 2 =0 2sin x sin x cos x cos Phng trỡnh ó cho tng ng vi phng trỡnh: Gii : iu kin : sin x sin 2x cos x cos 3x cos 2x + sin 2x + 3cos x + = sin 2x ( cos x + 1) ( cos 3x cos x ) ( cos 2x 1) + cos x + = sin 2x ( cos x + 1) + cos x.sin x + 2sin x + cos x + = sin 2x ( cos x + 1) + 2sin x ( cos x + 1) + ( cos x + 1) = ( cos x + 1) ( ) sin 2x + 2sin x + = ( cos x + 1) ( ) sin 2x cos 2x + = cos x = x = + 2k cos x + = ( k ) cos 2x + = x = k; x = + k sin 2x cos 2x + = ữ 3 i chiu iu kin ta c nghim ca phng trỡnh l: x = k; x = + k2; x = + k2(k Z) 3 Cõu : Gii phng trỡnh: sin x cos x + sin x = Gii : Pt tng ng: sin x cos x + sin x = (3sin x 4sin x) cos x + sin x = sin x (3 4sin x) cos x + = { [3 2(1 cos x)]2 cos x + 1} = sin x (1 + cos x) cos x + = sin x ( cos x + cos 2 x + cos x + 1) = sin x = x = k sin x ( cos x + 1) ( cos x + 1) = cos x = (k  ) x = + k cos 2 x + = (VN) 2 SU TM GV: Lấ ANH TUN Trng THPT ễNG SN 2 ( cos x sin x ) = tan x + cot x cot x cos x.sin x.sin x ( tan x + cot x ) Gii : iu kin: cot x Cõu : Gii phng trỡnh lng giỏc: Phng trỡnh tng ng sin x cos x + cos x sin x 2sin x.cos x = sin x x = + k cos x = ( k Â) x = + k = ( cos x sin x ) cos x.sin x = sin x cos x cos x sin x Giao vi iu kin, ta c h nghim ca phng trỡnh ó cho l x = Cõu : Gii phng trỡnh trờn khong (0; ) : 4sin + k ( k  ) x cos x = + cos ( x ) Gii : ( cos x ) cos 2x = + + cos 2x ữ cos x cos 2x = sin 2x cos x = cos 2x sin 2x ( Chia v cho ) cos x = cos 2x sin 2x cos 2x + ữ = cos ( x ) 2 +k a ) x = + h2 ( b ) ( 18 Do x ( 0, ) nờn h nghim (a) chn k=0, k=1, h nghim (b) chn h = Do ú pt cú ba nghim x x= 17 , x2 = , x3 = 18 18 Cõu : Gii phng trỡnh lng giỏc + sin x cos x = cos x(sin x + cos x) + tan x Gii : iu kin: cosx Bin i PT v: cos2x(1 + sin2x cos2x) = cos2x (2sinx + 2cosx)1 + sin2x cos2x = 2(sinx + cosx) thuc ( 0, ) l: x1 = (sinx + cosx)2 (cos2x sin2x) 2(sinx + cosx) = (sinx + cosx)[sinx + cosx (cosx sinx) 2] = (sinx + cosx)(2sinx 2) = sinx + cosx = hoc 2sinx = tanx = hoc sinx = (khụng tha cosx = 0) x = + k , (k ẻ Z) Cõu 10 : Gii phng trỡnh : 2.cos5 x sin( + x) = sin + x ữ.cot x SU TM ( vỡ cosx 0) GV: Lấ ANH TUN Trng THPT ễNG SN Gii : K: sin x pt 2cos5 x + sin x = cos x.cot x 2cos5 x sin x + sin x cos3 x = cos x.cos3 x 2cos5 x sin x cos5 x = cos5 x( sin x 1) = k x= + 12 (t/m k) +) sin x = x = + k +) cos5 x = x = k + (t/m k) 10 2 Cõu 11 : Gii phng trỡnh : tan x + ( + tan x ) ( 3sin x ) = Gii : iu kin cos x Phng trỡnh vit li 3sin x = sin x = ;sin x = so sỏnh /k chn sin x = tan x 3sin x = cos2 x 2sin x 3sin x + = + tan x x = + k ; x = + k ( k  ) 6 Cõu 12 : Gii phng trỡnh cos x ữ+ cos x + ữ = cos x 4 Gii : cos x.cos = ( cos x 1) 2cosx = cos x cos x cos x = (cos x 2)( cos x + )=0 cos x = x = + k 2 Cõu 13 : Gii phng trỡnh: cos x + = 2(2 cos x) sin( x ) Gii : Phng trỡnh (cosxsinx)2 4(cosxsinx) = cos x sin x = cos x sin x = (loai vi cos x sin x 2) x = + k sin x = sin x = sin (k Z ) 4 x = + k ( ) ( Cõu 14 : Gii phng trỡnh: ) cos3 x cos x = ( + sin x ) sin x + cos x Gii : K: sin x + cos x Khi ú PT ( sin x ) ( cos x 1) = ( + sin x ) ( sin x + cos x ) ( + sin x ) ( + cos x + sin x + sin x.cos x ) = ( + sin x ) ( + cos x ) ( + sin x ) = sin x = cos x = x = + k (tho iu kin) x = + m SU TM ( k , m Z) GV: Lấ ANH TUN Trng THPT ễNG SN + k v x = + m2 ( k , m Z) 4sin x + 4cos ( x ) Cõu 15 : Gii phng trỡnh (1) =2 cos2x Gii : K: cos2x x + k (k  ) 2 (1) (1 cos2x) + + cos(2x- ) ữ = 2cos2x (1 cos2x) + (1 + sin 2x) = 2cos2x 2cos2x+2sin 2x = 2cos2x 2cos2x-sin2x = 2(cos x sin x) (cosx+ s inx) = Vy phng trỡnh ó cho cú nghim l: x = x = + k cosx+sinx = (cosx+sinx)(cosx 3sinx) = (k Â) cosx 3s inx = x = arctan + k Kt hp vi iu kin phng trỡnh ó cho cú nghim l x = arctan + k (k  ) 4sin x.sin( x + ) + sin x + 3(cos x + 2) Cõu 16 : Gii phng trỡnh: =1 cos x + k PT 2.cos(2 x + ) + 5( sin x + cos x ) + = 4.sin ( x + ) + 10 sin( x + ) + = 6 sin( x + ) = 1/ x = + k (L) sin( x + ) = (VN ) x = + k Vậy S = { + k } Gii : K : x Cõu 17 : Gii phng trỡnh: Gii : K: x cos x.( cos x 1) sin x + cos x = ( + sin x ) + k PT (1 + sin x)(1 sin x)(cos x 1) = 2(1 + sin x)(sin x + cos x) + sin x = x = + k ( Tho iu kin) + sin x = sin x + cos x + sin x cos x + = ( + sin x ) ( cos x + 1) = x = + k Cõu 18 : ( sin x + cos x ) Gii phng trỡnh : 2sin x = sin x ữ sin x ữ + cot x Gii : iu kin sin x hay x k ; k Z Phng trỡnh ó cho tng ng vi SU TM GV: Lấ ANH TUN Trng THPT ễNG SN cos x ữsin x cos x ữ( sin x 1) = k x= + cos x = ữ ( k, m Z ) x = + m sin x = k + ; x = + m2 ; ( k , m Z ) So vi iu kin nghim ca phng trỡnh l x = 2 ( cos x + sin x ) sin x = 2(s inx cos x) = tanx + cot 2x cot x s inx.cos x Gii : iu kin : cot x Cõu 19 : Gii phng trỡnh : Phng trỡnh ó cho tng ng vi phng trỡnh: s inx cos2x + cos x s in2x = ( s inx cosx ) cos x s inx s inx 3 x = + k2 , x = + k2 (k Z) 4 + k2, (k Z) i chiu iu kin ta c nghim ca phng trỡnh l: x = sin x + = 2cosx Cõu 20 : Gii phng trỡnh: sin x + cos x tan x Gii : K : sin x 0, cos x 0,sin x + cos x cos x sin x cos x + cos x = Phng trỡnh ó cho tng ng : sin x sin x + cos x Gii c cos x = cos x = cos x sin( x + ) sin x ữ = sin x sin x + cos x +) cos x = x = + k , k x = + m2 x = x + + m2 t 4 m, n Z x = + , t +) sin x = sin( x + ) 4 x = + n x = x + n2 4 t , k, t i chiu iu kin ta cú nghim ca phng trỡnh l x = + k ; x = + sin( x ) + cos( x) x Cõu 21 : Gii phng trỡnh: (cos x + s inx.tan ) = cos x cos x cos x Gii : iu kin x cos cos x SU TM GV: Lấ ANH TUN Trng THPT ễNG SN 2 cos( x) + cos( x) x 3 (cos x + 2sin ) = cos x cos x cos( x) cos = s inx tan x = t anx (cos x + cos x) = cos x cos x cos x cos x x = k tan x = tan x tan x = (k Z ) x = + k tan x = x = 2l (l Z ) i chiu iu kin ta thy nghim ca phng trỡnh l x = + l cos x sin x = Cõu 22 : Gii phng trỡnh: cos 2 x + sin x sin x Gii : K: sin x + sin x + sin x cos x sin x = cos x sin x = ( sin x + cos x ) sin 2 x + sin x + cos x sin x = cos x + sin x cos x + = cos x x + = x + k x = + k x = + k x + = x + + k So li iu kin c nghim phng trỡnh ó cho x = + k x 4cos3xcosx - 2cos4x - 4cosx + tan t anx + Cõu 23 : Gii phng trỡnh: =0 2sinx - x Gii : iu kin: s inx v cos v cosx 2 cosx = Bin i pt v: 4cos x - cos x cosx + = cosx = 2 Cõu 24 : Gii phng trỡnh 2cos x + sin x cos x + = 3(sin x + cos x) Gii : Phng trỡnh cos x + sin x cos x + = 3(sin x + cos x) (sin x + cos x) 3(sin x + cos x) = sin x + cos x = sin x + cos x = (1) Phng trỡnh sin x + cos x = vụ nghim vỡ 12 + ( ) < Nờn (1) tan x = x = + k ( k  ) Vy, PT cú nghim l: x = + k ( k  ) 3 SU TM GV: Lấ ANH TUN Trng THPT ễNG SN x ữsin x = Cõu 25 : Gii phng trỡnh : 2 cos 12 Gii : sin x ữ+ sin = 12 12 sin x = = sin sin x ữ+ sin 12 12 12 = cos sin ữ = sin ữ 12 12 = ữ = sin sin 12 x = + k 2x = + k 12 12 sin x ( k Â) ữ = sin ữ 12 12 x = + k x = 13 + k 12 12 4 Cõu 26 : Gii phng trỡnh: sin x + sin x + sin x + sin x = cos x + cos x + cos x + cos x Gii : sin x cosx = (sin x cosx).[ + 2(sin x + cosx) + sin x.cosx ] = + 2(sin x + cosx) + sin x.cosx = + Vi sin x cosx = x = + k ( k Z ) + Vi + 2(sin x + cosx) + sin x.cosx = , t t = sin x + cosx (t 2; ) x = + m t = (m Z ) c pt : t2 + 4t +3 = t = -1 x = + m t = 3(loai ) Vy : x = + k , x = + m , x = + m (m Z , k Z ) Cõu 27 : Gii phng trỡnh : 2cos3x.cosx+ 3(1 + s in2x)=2 3cos (2 x + ) Gii : PT cos4x+cos2x+ 3(1 + sin x ) = + cos(4x+ ) ữ cos4x+ sin x + cos2x+ sin x = x = +k 18 sin(4 x + ) + sin(2 x + ) = 2sin(3 x + ).cosx=0 6 x= + k Vy PT cú hai nghim x = + k v x = + k 18 sin x + = 2cosx Cõu 28 : Gii phng trỡnh: sin x + cos x tan x Gii : iu kin: sin x 0, cos x 0,sin x + cos x SU TM GV: Lấ ANH TUN Trng THPT ễNG SN cos x Pt ó cho tr thnh + sin x cos x cos x = sin x + cos x sin x cos x cos x = cos x sin( x + ) sin x ữ = sin x sin x + cos x +) cos x = x = + k , k x = + m2 x = x + + m2 t 4 m, n x = + , t +) sin x = sin( x + ) 4 x = + n2 x = x + n t , k, t i chiu iu kin ta cú nghim ca pt l : x = + k ; x = + Cõu 29 : Gii phng trỡnh + sin x cos x sin 2x + cos 2x = Gii : Phng trỡnh ( sin2x) + ( sinx cosx) + ( cos2x sin2x) = ( sinx cosx).[(sinx cosx) + (sinx + cosx)] = ( sinx cosx).( 2cosx) = tan x = 1; cos x = x = + k ; x = + l ( k , l  ) ( k,l Z) 2 Cõu 30 : Gii phng trỡnh sin x cos x + cos x ( tan x 1) + sin x = Gii : iu kin cos x sin x cos x + cos x ( tan x 1) + sin x = sin x ( 2sin x ) + sin x + 2sin x = x = + k ; x = + k ; x = + k 2 6 Kt hp iu kin, phng trỡnh cú nghim S = + k ; + k 6 1 + Cõu 31 : Gii phng trỡnh: 2.cos x = (1) sin x cos x 2sin x + sin x = sin x = 1;sin x = Gii : iu kin: x k (1) 2.cos x cos x + sin x =0 (cos x sin x)(cos x + sin x)sin x (cos x + sin x) = sin x.cos x (cos x + sin x) (cos x sin x)sin x = sin x + cos x + sin x = ữ= (cos x sin x)sin x = (cos x sin x) ( (cos x sin x) ) = x = + k sin x + ữ = + k S: x = x= + k (cos x sin x)3 (cos x sin x) + = SU TM GV: Lấ ANH TUN Trng THPT ễNG SN 3x Cõu 32 : Gii phng trỡnh: 4cos4x cos2x cos4x + cos = Gii : 3x 4cos4x cos2x cos4x + cos = 3x 3x cos2x + cos (1 + cos2x)2 cos2x (2cos x 1) + cos = =2 4 cos2x = x = k ( vỡ VT vi mi x) m8 (k ; m  ) x = 8n ( n  ) 3x x = cos = Cõu 33 : Gii phng trỡnh sau : x x + cos = sin 2 Gii : 2x + cos x x = cos x + cos = sin + 2 4 2x + + cos = cos x + cos 2a = cos 3a x a = ữ + ( cos a 1) = ( cos a cos a ) + cos a + cos3 a cos a = cos a ( cos a + cos a ) = cos a = cos a = cos a = x x cos = = + k x= + k cos x = cos x = + k x = + k 3 ( loaùi ) Cõu 34 : Gii phng trỡnh : 2cos3x.cosx+ 3(1 + s in2x)=2 3cos (2 x + ) Gii : PT cos4x+cos2x+ 3(1 + sin x ) = + cos(4x+ ) ữ cos4x+ sin x + cos2x+ sin x = sin(4 x + ) + sin(2 x + ) = 6 x = +k 18 2sin(3x + ).cosx=0 x= + k Vy PT cú hai nghim x = + k v x = + k 18 SU TM GV: Lấ ANH TUN Trng THPT ễNG SN ON TAP LUONG GIAC sin x + cos x Cõu : Gii phng trỡnh : sin x = ( tan x + cot x ) 2 Cõu : Gii phng trỡnh: cos x + sin x sin x sin x = Cõu : nh m phng trỡnh sau cú nghim 4sin3xsinx + 4cos 3x - ữcos x + ữ cos 2x + ữ+ m = 4 2(s inx cos x) = tanx + cot 2x cot x 5x x sin x cos x cos cos + sin 2x + 3cos x + Cõu : Gii phng trỡnh: 2 =0 2sin x Cõu : Gii phng trỡnh: sin x cos x + sin x = Cõu : Gii phng trỡnh : Cõu : Gii phng trỡnh lng giỏc: ( cos x sin x ) = tan x + cot x cot x Cõu : Gii phng trỡnh trờn khong (0; ) : 4sin x cos x = + cos ( x ) Cõu : Gii phng trỡnh lng giỏc + sin x cos x = cos x(sin x + cos x) + tan x Cõu 10 : Gii phng trỡnh : 2.cos5 x sin( + x) = sin + x ữ.cot x 2 Cõu 11 : Gii phng trỡnh : tan x + ( + tan x ) ( 3sin x ) = Cõu 12 : Gii phng trỡnh cos x ữ+ cos x + ữ = cos x 4 Cõu 13 : Gii phng trỡnh: cos x + = 2(2 cos x) sin( x ) Cõu 14 : Gii phng trỡnh: cos3 x cos x = ( + sin x ) sin x + cos x 4sin x + 4cos ( x ) Cõu 15 : Gii phng trỡnh (1) =2 cos2x 4sin x.sin( x + ) + sin x + 3(cos x + 2) Cõu 16 : Gii phng trỡnh: =1 cos x Cõu 17 : Gii phng trỡnh: Cõu 18 : cos x.( cos x 1) sin x + cos x ( sin x + cos x ) Gii phng trỡnh : Cõu 19 : Gii phng trỡnh : = ( + sin x ) 2sin x = sin x ữ sin x ữ + cot x 2(s inx cos x) = tanx + cot 2x cot x SU TM GV: Lấ ANH TUN Trng THPT ễNG SN sin x + = 2cosx sin x + cos x tan x sin( x ) + cos( x) x Cõu 21 : Gii phng trỡnh: (cos x + s inx.tan ) = cos x cos x cos x sin x = Cõu 22 : Gii phng trỡnh: cos 2 x + sin x x 4cos3xcosx - 2cos4x - 4cosx + tan t anx + Cõu 23 : Gii phng trỡnh: =0 2sinx - Cõu 20 : Gii phng trỡnh: Cõu 24 : Gii phng trỡnh cos x + sin x cos x + = 3(sin x + cos x) x ữsin x = Cõu 25 : Gii phng trỡnh : 2 cos 12 Cõu 26 : Gii phng trỡnh: sin x + sin x + sin x + sin x = cos x + cos x + cos x + cos x 2cos3x.cosx+ 3(1 + s in2x)=2 3cos (2 x + ) Cõu 27 : Gii phng trỡnh : sin x + = 2cosx Cõu 28 : Gii phng trỡnh: sin x + cos x tan x Cõu 29 : Gii phng trỡnh + sin x cos x sin 2x + cos 2x = 2 Cõu 30 : Gii phng trỡnh sin x cos x + cos x ( tan x 1) + sin x = 1 + (1) sin x cos x 3x Cõu 32 : Gii phng trỡnh: 4cos4x cos2x cos4x + cos = Cõu 31 : Gii phng trỡnh: 2.cos x = Cõu 33 : Gii phng trỡnh sau : Cõu 34 : Gii phng trỡnh : x x + cos = sin 2 2cos3x.cosx+ 3(1 + s in2x)=2 3cos (2 x + ) SU TM GV: Lấ ANH TUN Trng THPT ễNG SN Gii : iu kin: sin x 1 sin 2 x sin 2 x sin x cos x 1 2 (1) = + = sin 2 x = sin x = ữ sin x cos x sin x sin x sin x Vy phng trỡnh ó cho vụ nghim Gii : pt ó cho tng ng vi pt: 1 1 1 (1 + cos x ) + (cos 3x cos x) (1 cos x) = cos x cos x + cos 3x cos x + = 2 2 2 2 x = + k cos x + = 15 cos x + cos 3x = x = + k cos 3x = 3 Gii : 4sin3xsinx = ( cos2x - cos4x ) ; +/ 4cos 3x - ữcos x + ữ = cos 2x - ữ+ cos4x = ( sin 2x + cos4x ) 4 +/ cos 2x + ữ = + cos 4x + ữữ = ( sin 4x ) 2 1 Do ú phng trỡnh ó cho tng ng: ( cos2x + sin2x ) + sin 4x + m - = (1) 2 t t = cos2x + sin2x = 2cos 2x - ữ (iu kin: t ) Khi ú sin 4x = 2sin2xcos2x = t Phng trỡnh (1) tr thnh: t + 4t + 2m = (2) vi t (2) t + 4t = 2m õy l phung trỡnh honh giao im ca ng ( D ) : y = 2m (l ng song song vi Ox v ct trc tung ti im cú tung 2m) v (P): y = t + 4t vi t Trong on 2; , hm s y = t + 4t t giỏ tr nh nht l ti t = v t giỏ tr ln nht l + ti t = Do ú yờu cu ca bi toỏn tha v ch m + 2 m 2 s inx.cos x Gii : iu kin : sinx.cosx cot x Phng trỡnh ó cho tng ng vi phng trỡnh: ( s inx cosx ) s inx cos2x cos x s inx + cos x s in2x s inx x = + k2 (k Z) Gii c cos x = x = + k2 = SU TM GV: Lấ ANH TUN Trng THPT ễNG SN i chiu iu kin ta c nghim ca phng trỡnh l: x = + k2, (k Z) Phng trỡnh ó cho tng ng vi phng trỡnh: Gii : iu kin : sin x sin 2x cos x cos 3x cos 2x + sin 2x + 3cos x + = sin 2x ( cos x + 1) ( cos 3x cos x ) ( cos 2x 1) + cos x + = sin 2x ( cos x + 1) + cos x.sin x + 2sin x + cos x + = sin 2x ( cos x + 1) + 2sin x ( cos x + 1) + ( cos x + 1) = ( cos x + 1) ( ) sin 2x + 2sin x + = ( cos x + 1) ( ) sin 2x cos 2x + = cos x = x= + 2k cos x + = ( k ) cos 2x + = x = k; x = + k sin 2x cos 2x + = ữ 3 i chiu iu kin ta c nghim ca phng trỡnh l: x = k; x = + k2; x = + k2(k Z) 3 Gii : Pt tng ng: sin x cos x + sin x = (3sin x 4sin x) cos x + sin x = sin x (3 4sin x) cos x + = { [3 2(1 cos x)]2 cos x + 1} = sin x (1 + cos x) cos x + = sin x ( cos x + cos 2 x + cos x + 1) = sin x = x = k sin x ( cos x + 1) ( cos x + 1) = cos x = (k  ) x = + k cos 2 x + = (VN) cos x.sin x.sin x ( tan x + cot x ) Gii : iu kin: cot x 2 ( cos x sin x ) cos x.sin x = = sin x Phng trỡnh tng ng sin x cos x cos x cos x + cos x sin x sin x 2sin x.cos x = sin x x = + k cos x = ( k  ) Giao vi iu kin, ta c h nghim ca phng trỡnh ó x = + k cho l x = + k ( k  ) Gii : ( cos x ) cos 2x = + + cos 2x ữ cos x cos 2x = sin 2x SU TM GV: Lấ ANH TUN Trng THPT ễNG SN cos x = cos 2x sin 2x ( Chia v cho ) cos x = cos 2x sin 2x cos 2x + ữ = cos ( x ) 2 +k a ) x = + h2 ( b ) ( 18 Do x ( 0, ) nờn h nghim (a) chn k=0, k=1, h nghim (b) chn h = Do ú pt cú ba nghim x x= 17 ,x = ,x = 18 18 Gii : iu kin: cosx Bin i PT v: cos2x(1 + sin2x cos2x) = cos2x (2sinx + 2cosx)1 + sin2x cos2x = 2(sinx + cosx) ( vỡ cosx 0) (sinx + cosx)2 (cos2x sin2x) 2(sinx + cosx) = (sinx + cosx)[sinx + cosx (cosx sinx) 2] = (sinx + cosx)(2sinx 2) = sinx + cosx = hoc 2sinx = tanx = hoc sinx = (khụng tha cosx = 0) x = + k , (k ẻ Z) 10 Gii : K: sin 3x pt 2cos5 x + sin x = cos x.cot x 2cos5 x sin x + sin x cos3 x = cos x.cos3 x thuc ( 0, ) l: x1 = 2cos5 x sin x cos5 x = cos5 x( sin x 1) = k x= + 12 (t/m k) +) sin x = x = + k +) cos5 x = x = k + (t/m k) 10 11 Gii : iu kin cos x Phng trỡnh vit li 3sin x = sin x = ;sin x = so sỏnh /k chn sin x = tan x 3sin x = cos2 x 2sin x 3sin x + = + tan x x = + k ; x = + k ( k  ) 6 12 Gii : = ( cos x 1) 2cosx = cos x cos x cos x = (cos x 2)( cos x + )=0 cos x = x = + k 2 cos x.cos 13 Gii : Phng trỡnh (cosxsinx)2 4(cosxsinx) = cos x sin x = cos x sin x = (loai vi cos x sin x 2) SU TM GV: Lấ ANH TUN Trng THPT ễNG SN x = + k 2 sin x = sin x = sin (k Z ) 4 x = + k ( ) ( ) 14 Gii : K: sin x + cos x Khi ú PT ( sin x ) ( cos x 1) = ( + sin x ) ( sin x + cos x ) ( + sin x ) ( + cos x + sin x + sin x.cos x ) = ( + sin x ) ( + cos x ) ( + sin x ) = x = + k ( k , m Z) (tho iu kin) x = + m2 Vy phng trỡnh ó cho cú nghim l: x = + k v x = + m ( k , m Z) 15 Gii : K: cos2x x + k (k  ) 2 (1) (1 cos2x) + + cos(2x- ) ữ = 2cos2x (1 cos2x) + (1 + sin 2x) = 2cos2x sin x = cos x = 2cos2x+2sin 2x = 2cos2x 2cos2x-sin2x = 2(cos x sin x) (cosx+ s inx) = x = + k cosx+sinx = (cosx+sinx)(cosx 3sinx) = (k Â) cosx 3s inx = x = arctan + k Kt hp vi iu kin phng trỡnh ó cho cú nghim l x = arctan + k (k  ) 16 Gii : K : x + k PT 2.cos(2 x + ) + 5( sin x + cos x) + = 4.sin ( x + ) + 10sin( x + ) + = 6 sin( x + ) = 1/ x = + k (L) sin( x + ) = (VN ) x = + k Vy S = { + k } 17 Gii : K: x + k PT (1 + sin x)(1 sin x)(cos x 1) = 2(1 + sin x)(sin x + cos x) + sin x = x = + k ( Tho iu kin) + sin x = sin x + cos x + sin x cos x + = ( + sin x ) ( cos x + 1) = x = + k 18 Gii : iu kin xỏc nh sin x hay x k ; k Z Phng trỡnh ó cho tng ng vi SU TM GV: Lấ ANH TUN Trng THPT ễNG SN cos x ữsin x cos x ữ( sin x 1) = k x= + cos x = ữ ( k, m Z ) x = + m sin x = k + ; x = + m2 ; ( k , m Z ) So vi iu kin nghim ca phng trỡnh l x = 2 s inx.cos x 19 Gii : iu kin : sinx.cosx cot x ( cos x + sin x ) sin x = ( s inx cosx ) = Phng trỡnh ó cho tng ng vi phng trỡnh: s inx cos2x cos x s inx + cos x s in2x s inx 3 x = + k2 , x = + k2 (k Z) 4 + k2, (k Z) i chiu iu kin ta c nghim ca phng trỡnh l: x = 20 Gii : K : sin x 0, cos x 0,sin x + cos x cos x sin x cos x + cos x = Phng trỡnh ó cho tng ng : sin x sin x + cos x Gii c cos x = cos x = cos x sin( x + ) sin x ữ = sin x sin x + cos x +) cos x = x = + k , k x = + m2 x = x + + m2 t 4 m, n Z x = + , t +) sin x = sin( x + ) 4 x = + n x = x + n2 4 t , k, t i chiu iu kin ta cú nghim ca phng trỡnh l x = + k ; x = + cos x 21 Gii : iu kin x cos cos( x) + cos( x) Phng trỡnh x 3 (cos x + 2sin ) = cos x cos x cos( x) cos = s inx tan x = t anx (cos x + cos x) = cos x cos x cos x cos x x = k tan x = tan x tan x = (k Z ) x = + k tan x = cos x SU TM GV: Lấ ANH TUN Trng THPT ễNG SN x = 2l (l Z ) i chiu iu kin ta thy nghim ca phng trỡnh l x = + l sin x 22 Gii : K: sin x + sin x + sin x cos x sin x = cos x sin x = ( sin x + cos x ) sin 2 x + sin x + cos x sin x = cos x + sin x cos x + = cos x x + = x + k x = + k x = + k x + = x + + k So li iu kin c nghim phng trỡnh ó cho x = + k x 23 Gii : iu kin: s inx v cos v cosx 2 cosx = Bin i pt v: 4cos3x - cos2x cosx + = cosx = 24 Gii : cos x + sin x cos x + = 3(sin x + cos x) (sin x + cos x) 3(sin x + cos x) = sin x + cos x = sin x + cos x = (1) Phng trỡnh sin x + cos x = vụ nghim vỡ 12 + ( ) < Nờn (1) tan x = x = + k ( k  ) Vy, PT cú nghim l: x = + k ( k  ) 3 25 Gii : sin x ữ+ sin = 12 12 sin x = = sin sin x = ữ+ sin ữ = sin sin 12 12 12 12 = cos sin ữ = sin ữ 12 12 x = + k 2x = + k 12 12 sin x ( k Â) ữ = sin ữ 12 12 x = + k x = 13 + k 12 12 26 Gii : sin x cosx = (sin x cosx).[ + 2(sin x + cosx) + sin x.cosx ] = + 2(sin x + cosx) + sin x.cosx = + Vi sin x cosx = x = + k ( k Z ) SU TM GV: Lấ ANH TUN Trng THPT ễNG SN (t 2; ) x = + m t = (m Z ) c pt : t + 4t +3 = t = -1 x = + m t = 3( loai ) Vy : x = + k , x = + m , x = + m (m Z , k Z ) 27 Gii : + Vi + 2(sin x + cosx) + sin x.cosx = , t t = sin x + cosx PT cos4x+cos2x+ 3(1 + sin x) = + cos(4x+ ) ữ cos4x+ sin x + cos2x+ sin x = x = 18 + k sin(4 x + ) + sin(2 x + ) = 2sin(3 x + ).cosx=0 6 x= + k Vy PT cú hai nghim x = + k v x = + k 18 sin x 0, cos x 0,sin x + cos x 28 Gii : iu kin: cos x sin x cos x + cos x = Pt ó cho tr thnh sin x sin x + cos x cos x cos x = cos x sin( x + ) sin x ữ = sin x sin x + cos x +) cos x = x = + k , k x = + m2 x = x + + m t 4 m, n x = + , t +) sin x = sin( x + ) 4 x = + n2 x = x + n 4 t , k, t i chiu iu kin ta cú nghim ca pt l : x = + k ; x = + 29.Gii : Phng trỡnh ( sin2x) + ( sinx cosx) + ( cos2x sin2x) = ( sinx cosx).[(sinx cosx) + (sinx + cosx)] = ( sinx cosx).( 2cosx) = tan x = 1; cos x = x = + k ; x = + l ( k , l  ) ( k,l Z) 30.Gii : iu kin cos x sin x cos x + cos x ( tan x 1) + sin x = sin x ( 2sin x ) + sin x + 2sin x = x = + k ; x = + k ; x = + k 2 6 Kt hp iu kin, phng trỡnh cú nghim S = + k ; + k 6 2sin x + sin x = sin x = 1;sin x = 31.Gii : iu kin: x k SU TM GV: Lấ ANH TUN (1) 2.cos x Trng THPT ễNG SN cos x + sin x =0 (cos x sin x)(cos x + sin x)sin x (cos x + sin x) = sin x.cos x (cos x + sin x) (cos x sin x)sin x = cos x + sin x = sin x + ữ = (cos x sin x )sin x = (cos x sin x) ( (cos x sin x) ) = x= + k sin x + = ữ + k S: x = x= + k (cos x sin x)3 (cos x sin x) + = 32.Gii : 3x 4cos4x cos2x cos4x + cos = 3x 3x cos2x + cos (1 + cos2x)2 cos2x (2cos x 1) + cos = =2 4 cos2x = x = k ( vỡ VT vi mi x) m8 (k ; m  ) x = 8n ( n  ) 3x x = cos = 33.Gii : 2x + cos x x = cos x + cos = sin + 2 4 2x x + + cos = cos x + cos 2a = cos 3a a = ữ 3 + ( cos a 1) = ( cos3 a cos a ) + cos a + cos3 a cos a = cos a ( cos a + cos a 3) = cos a = cos a = cos a = x x cos = = + k x= + k cos x = cos x = + k x = + k 3 ( loaùi ) 34.Gii : SU TM GV: Lấ ANH TUN Trng THPT ễNG SN PT cos4x+cos2x+ 3(1 + sin x ) = + cos(4x+ ) ữ cos4x+ sin x + cos2x+ sin x = sin(4 x + ) + sin(2 x + ) = 6 x = 18 + k 2sin(3 x + ).cosx=0 x= + k Vy PT cú hai nghim x = + k v x = + k 18 SU TM