Bài tập công thức lượng giác

10   2   Page - 1 - Chương  NG   1/ 2 2 2 cos x sin x 1 2sin x- = - . 2/ 22 2cos x 1 1 2sin x- = - . 3/ 22 3 4sin x 4cos x 1- = - . 4/ sinxcot x cosxtanx sinx cosx+ = + . 5/ 4 4 2 2 sin x cos x 1 2sin xcos x+ = - . 6/ 4 4 2 2 cos x sin x cos x sin x- = - . 7/ ( )( ) 2 4cos x 3 1 2sin x 1 2sinx- = - + . 8/ ( ) ( ) 2 2 2 1 cosx sin x cosx cos x sin x+ - + = . 9/ 4 4 2 2 sin x cos x 1 2cos x 2sin x 1- = - = - . 10/ 33 sin xcosx sinxcos x sinxcosx+= . 11/ 2 2 2 2 tan x sin x tan xsin x-= . 12/ 2 2 2 2 cot x cos x cot xcos x-= .   1/ 1 tan x cot x sinxcosx += . 2/ 1 cosx sin x sin x 1 cosx - = + . 3/ 11 1 1 tan x 1 cot x += ++ . 4/ 2 11 1 1 tan x 0 cosx cosx æ öæ ö ÷÷ çç ÷÷ - + + = çç ÷÷ çç ÷÷ çç è øè ø . 5/ 2 2 2 1 sin x 1 2tan x 1 sin x + =+ - . 6/ tan x tan y tan xtan y cot x cot y + = + . 7/ 4 24 21 1 cot x sin x sin x - = - . 8/ cosx 1 tan x 1 sin x cosx += + . 9/ ( ) ( ) 2 1 1 cosx 1 cot x 1 cosx - + = + . 10/ 1 cosx 1 cosx 4cot x 1 cosx 1 cosx sin x +- -= -+ . 11/ sin x 1 cosx 2 1 cosx sin x sin x + += + . 12/ sin x cosx 1 cosx sin x cosx 1 1 sin x +- = - + + . 13/ 2 2 sin x 2cosx 1 cosx 1 cosx 2 cosx cos x +- = + +- . 14/ sin x cosx 1 2cosx 1 cosx sin x cosx 1 +- = - - + .  5    .  .  .   .  .  5.  Page - 2 - "  "   1/ 6 6 2 2 sin x cos x 1 3sin xcos x+ = - . 2/ ( )( ) 6 6 2 2 2 2 sin x cos x sin x cos x 1 sin xcos x- = - - . 3/ ( ) 2 8 8 2 2 4 4 sin x cos x 1 2sin xcos x 2sin xcos x+ = - - . 4/ ( )( ) 8 8 2 2 2 2 sin x cos x sin x cos x 1 2sin xcos x- = - - .   1/ ( )( ) 1 sin x cosx tan x 1 cosx 1 tanx+ + + = + + . 2/ ( )( ) 1 tan x 1 cot x sin xcosx 1 2sin xcosx+ + = + . 3/ ( ) ( ) ( ) 2 22 1 tan x cos x 1 cot x sin x sin x cosx+ + + = + . 4/ 22 sin xtanx cos xcot x 2sinxcosx tanx cot x+ + = + . 5/ 2 2 2 2 2 sin xtan x 4sin x tan x 3cos x 3+ - + = .   1/ 22 sin cos tan 1 1 2sin cos tan 1 a - a a - = + a a a + . 2/ 22 6 22 tan sin tan cot cos a - a =a a - a . 3/ 22 cos sin 1 sin cos 1 tan 1 cot aa + = - a a - a - a . 4/ ( ) 3 tan sin 1 sin cos 1 cos a - a = a a + a . 5/ 22 22 1 tan cot 2 sin cos = a + a + aa . 6/ 2 2 22 1 3tan tan 1 cos cos a - a = + aa . 7/ 2 2 2 2 2 2 2 2 tan tan sin sin tan tan sin sin a - b a - b = a b a b . 8/ ( ) 2 2 1 cos 1 cos 1 2cot sin sin éù +a êú -a êú - = a êú a a êú ëû . 9/ ( ) ( ) 2 1 1 cos 1 cot 1 cos - a + a = +a . 10/ 23 3 sin cos 1 tan tan tan cos a + a + a + a + a = a . 11/ 2 2 sin cos 1 cot sin cos cos sin 1 cot a a + a -= a + a a - a -a . 12/ 2 2 4 2 2 2 2 tan 1 cot 1 tan . 1 tan cot tan cot a + a + a = + a a a + a . 13/ 2 2 1 sin 1 sin 4tan 1 sin 1 sin æö ÷ + a - a ç ÷ ç - = a ÷ ç ÷ ç - a + a ÷ ç èø . 14/ 2 2 1 cos 1 cos 4cot 1 cos 1 cos æö ÷ + a - a ç ÷ ç - = a ÷ ç ÷ ç - a + a ÷ ç èø .   1/ 4 2 2 1 P sin sin cos= a + a a . 2/ 4 4 2 2 P sin cos cos= a - a + a . 3/ 2 2 2 3 P sin sin cot= a + a a . 4/ 2 2 2 4 P cos cos cot= a + a a . 5/ ( ) 2 2 2 5 P 1 sin cot 1 cot= - a a + - a . 6/ 22 6 P sin tan cos cot 2sin cos= a a + a a + a a . 10   2   Page - 3 - 7/ 2 7 2cos 1 P sin cos a- = a + a . 8/ 8 2 1 cos 1 P 1 cos sin -a =- +a a . 9/ 9 cot cot P tan tan a + b = a + b . 10/ 10 cos P tan 1 sin a = a + +a . 11/ 11 sinx tan x P sinxcot x tan x + =- . 12/ 12 2 cosxtan x P cot xcosx sin x =- . 13/ 22 13 22 cos x cot x P sin x tan x - = - . 14/ 22 2 14 2 1 sin x cos x P cos x cos x - =- . 15/ 32 15 sin x sin xcos x cosx P 1 2sin xcosx +- = - . 16/ ( ) 2 16 sin x cosx 1 P tan x sin xcosx +- = - . 17/ 2 2 2 17 2 2 2 sin x sin xtan x P cos x cos x tan x + = + . 18/ 22 2 18 2 cot x cos x P 1 cos x cot x - = + - . 19/ 22 22 19 22 cos x sin y P cot xcot y sin xsin y - =- . 20/ 20 1 sin x cosxcot x P sin x 1 cot x 1 tan x = - - ++ . 21/ 21 1 cosx 1 sinx P tan x cot x sinx cosx = + + + . 22/ 22 11 P 1 cot x 1 cot x sin x sin x æ öæ ö ÷÷ çç ÷÷ = + + + - çç ÷÷ çç ÷÷ çç è øè ø . 23/ ( ) ( ) 22 23 P 1 tanx cos x 1 cot x sin x= + + + . 24/ 24 22 1 P , 0 x 2 sinx cot x cos x æö p ÷ ç ÷ = < < ç ÷ ç ÷ ç èø .   1/ 2 A 2cos x 1=- . 2/ 2 B 3 4sin x=- . 3/ 2 C sin xcosx cos x 1= + - . 4/ 2 D sin x sin xcosx 1= + - . 5/ E 1 sinx cosx tan x= + + + . 6/ F tan x cot x sinx cosx= - + + . 7/ ( ) 2 G cosxtan x 1 cosx= - + . 8/ ( ) ( ) 2 H 3 4cos x sinx 2sinx 1= - - + . 9/ 22 I sin x 3cos x 6cosx 2sinx= - + - . 10/ 33 J cos x sin x sinx cosx= - + + . 11/ 32 K cos x cos x 2sin x 2= + + - . 12/ 23 L cos x sin x cosx= + + . 13/ ( ) 2 M 1 cosx cos x sinx 1 cosx= + + - + .  5.  Page - 4 - "  " 14/ 32 N 2cos x 2cos x sin x 1= + + - . 15/ 3 3 2 O cos x sin x 2sin x 1= + + - . 16/ ( )( ) 2 P 2sinx 1 2cosx 2sinx 1 3 4cos x= - + + - + . 17/ ( )( ) Q 2cosx 1 sinx cosx 1= - + - . 18/ 3 3 2 R 4sin x 3cos x 3sinx sin xcosx= + - - . 19/ ( ) ( ) 2 S 1 sin x tan x 1 cosx= + - + . 20/ ( ) 2 T 2 5sinx 3 1 sinx tan x= - + - . 21/ 2 U 2sin xcosx 2sin x 3sin x cosx 1= - + - - . 22/ ( ) V tan x 3cot x 4 sinx 3 cosx= - - + . 23/ X 3sinx 2cosx 3tan x 2= + - - . 24/ ( ) ( ) Y 2 tan x sinx 3 cot x cosx 5= - + - + . 25/ ( ) ( ) Z 3 cot x cosx 5 tan x sin x 2= - - - - .   1/ 4 4 2 A cos x sin x 2sin x= - + . 2/ 4 2 2 2 B sin x sin xcos x cos x= + + . 3/ 4 2 2 2 C cos x sin xcos x sin x= + + . 4/ ( ) ( ) 4 2 4 2 D cos x 2cos x 3 sin x 2sin x 3= - + - . 5/ 6 6 4 4 2 E sin x cos x 2sin x cos x sin x= + - - + . 6/ 11 F sin x. , 0 x 1 cosx 1 cosx 4 æö p ÷ ç ÷ = + < < ç ÷ ç ÷ ç +- èø . 7/ 4 2 4 2 G sin x 4cos x cos x 4sin x= + + + . 8/ 2 2 2 2 2 H cos xcot x 5cos x cot x 4sin x= + - + . 9/ ( ) ( ) 33 I 1 cot x sin x 1 tan x cos x sinx cosx= + + + - - . 10/ ( )( ) 4 4 2 2 J sin x cos x 1 tan x cot x 2= + - + + . 11/ ( ) ( ) 8 8 6 6 4 K 3 sin x cos x 4 cos x 2sin x 6sin x= - + - + . 12/ ( ) ( ) 4 2 4 2 2 2 L sin x 1 sin x cos x 1 cos x 5sin x cos x 1= + + + + + . 13/ ( ) ( ) 2 4 4 2 2 8 8 M 2 sin x cos x sin x cos x sin x cos x= + + - + . 10   2   Page - 5 -   1/ 2 cot x 1 A tan x 1 cot x 1 + =+ . 2/ ( ) 2 2 2 2 2 1 tan x 1 B 4tan x 4sin xcos x - =- . 3/ ( )( ) 2 22 1 tan x C 1 tan x 1 cot x tan x - = - + + . 4/ ( ) 2 2 2 2 2 1 cot x 1 D cot x sin x cos x - =- . 5/ 62 62 1 sin x 3tan x E cos x cos x - =- . 6/ 2 2 2 2 22 tan x cos x cot x sin x F sin x cos x =+ . 7/ 22 2 cot x cos x sin xcosx G cot x cot x - =+ . 8/ 44 66 sin x cos x 1 H sin x cos +- = + .  Cho ( ) ( ) f x 2 1 cosx ,=- ( ) g x 2 cosx 3.sinx,= + - ( ) h x 2 cosx 3.sinx= + +   ( ) ( ) ( ) 2 2 2 A f x g x h x ,= + + ( ) ( ) ( ) 4 4 4 B f x g x h x ,= + + ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 C f x .g x g x .h x h x .f x= + + .  Cho a sinxsiny bcosxcosy 0-=  2 2 2 2 11 Q a sin x bcos x a sin y bcos y =+ ++   Cho ( ) 6 6 4 4 Q sin x cos x m sin x cos x= + - +  Q   Q     Cho 44 sin x cos x 1 a b a b += +  1/ ( ) 88 3 3 3 sin x cos x 1 ab ab += +  1) 2/ ( ) 10 10 4 4 4 sin x cos x 1 ab ab += +    5.  Page - 6 - "  "  B    10   2   Page - 7 - NG    2 3 5 7 13 3 15 , , , , , , , 2, , 3 4 9 5 3 3 4 7 15 p p p 9p p p p p p - .   ( ) rad : 0 0 0 0 0 0 0 0 0 90 , 36 , 15 , 72 , 270 , 240 , 540 , 750 , 21 .   0 0 0 21 30', 75 54', 12 36' 0 0 0 , 15 30', 105 45'30'', 27 38'49''- .   ( ) 2 rad p  2,5p .   0 0 0 15 0 30 0 45 0 60 0 75 0 90 0 120 0 150 0 180 16 p 8 p 6 p 5 12 p 2 p 2 3 p 3 4 p 5 6 p p 4 3 p Radian 7 4 p 12 p 5 6 p - 2 p 2 3 p 3 4 p  0 135 0 50- 0 810    ( ) 10 cm  ( ) 00 2 3 7 13 3 3 , , , , , 2 , , 45 , rad , 75 3 18 5 3 6 4 2 p p p p p p -p .   ( ) 7 cm  ( ) 10 cm   30 t.   ( ) 55 cm  ( ) /40 km h     ( ) 1,75 m  ( ) 1,26 m  15     5.  Page - 8 - "  "   0 0 0 0 0 30 , 45 , 120 , 120 , 330 , 0 0 0 0 0 7 4 15 2010 630 , 750 , 1250 , , , 750 , 1125 , , 4 3 2 4 p p p p - - - .   ( ) k2 , k , k , k , k , k 2 3 4 ppp p p Î ¢ .   k Î ¢ . 1/ xk=p . 2/ x k2 2 p = + p . 3/ x 4k=p . 4/ xk 3 p = + p . 5/ x k2 6 p = - + p . 6/ xk 4 p = - + p . 7/ k x 3 p = . 8/ k2 x 33 pp =+ . 9/ k3 x 44 pp =+ . 10/ k x 63 pp =+ .   ¼ đs AM 6 p =  ¼ đ k s AN 798 p =  k Î ¢  k Î ¥  1 2   35 3 p  m 5 p      1/ sinx, cosx, tan x, cot x  3 x 2 p p < < . 2/ 3 sin x , cos x , tan x , cot x 4 2 2 2 æ ö æ ö æ ö æ ö p p p p ÷ ÷ ÷ ÷ ç ç ç ç ÷ ÷ ÷ ÷ + - - + ç ç ç ç ÷ ÷ ÷ ÷ ç ç ç ç ÷ ÷ ÷ ÷ ç ç ç ç è ø è ø è ø è ø  0x 2 p << . 3/ ( ) 00 A sin 40 .cos 290=- . 4/ ( ) 00 B sin 25 .cos170=- . 5/ ( ) 0 0 0 C sin225 .tan130 .cot 175=- . 6/ ( ) 0 0 0 D cos195 .tan269 .cot 98=- . 7/ ( ) 00 E sin50 .cos 300=- . 8/ 0 21 F sin215 .tan 7 p = . 9/ 32 G cot .sin 53 æö pp ÷ ç ÷ =- ç ÷ ç èø . 10/ os 4 4 9 H c .sin .tan .cot 5 3 3 5 p p p p = .   1/ 1 sin x 2 =  00 90 x 180<< . 2/ 4 sin x 5 =-  00 270 x 360<< . 10   2   Page - 9 - 3/ 3 sin x 5 =-  3 x 2 p p < < . 4/ 1 cosx 4 =  0x 2 p << . 5/ 3 cosx 5 =  0 0 x 90<< . 6/ 5 cosx 13 =-  00 180 x 270<< . 7/ 2 cosx 5 =  x0 2 p - < < . 8/ 4 cosx 5 =  00 270 x 360<< . 9/ 5 sin x 13 =  x 2 p < < p . 10/ 1 sin x 3 =-  00 180 x 270<< . 11/ tanx 3=  3 x 2 p p < < . 12/ tanx 2=-  x 2 p < < p . 13/ 1 tan x 2 =-  x 2 p < < p . 14/ cot x 3=  3 x 2 p p < < . 15/ 3 tan x 4 =  3 x 2 p p < < . 16/ tanx 2=-  x 2 p < < p . 17/ 2 cot x 3 =  0x 2 p << . 18/ cot x 3=-  x 2 p < < p . 19/ cot x 3=-  3 x2 2 p < < p . 20/ 0 cot15 2 3=+   1/ Cho tanx 2=-  12 5cot x 4tan x 2sinx cosx A , A 5cot x 4tan x cosx 3sinx ++ == . 2/ Cho cot x 2=  12 3sin x cosx sin x 3cosx B , B sin x cosx sin x 3cosx == ++ . 3/ Cho cot x 2=  12 2 2sin x 3cosx 2 C , C 3sin x 2cosx cos x sinxcosx + == - - . 4/ Cho tanx 2=  1 2sinx 3cosx D, 4sinx 5cosx + = - 2 33 3sin x 2cosx D, 5sin x 4cos x - = + 33 3 2 sin x cos x D, sin x sin xcosx - = + 4 33 sin x 5cosx D, sin x 2cos x + = - 33 5 3 8cos x 2sin x cosx D, 2cosx sin x -+ = - 33 6 32 sin x 2cos x 2sin x 3cosx D sin x cos x 5sinx - + + = -+ . 5/ Cho 3 sinx , 0 x 52 p = < <  cot x tan x E cot x tanx + = - . 6/ Cho 00 1 sinx , 90 x 180 3 = < <  2 8tan x 3cot x 1 F tan x cot x +- = + . 7/ Cho 2 cosx 3 =-  cot x 3tan x G 2cot x tan x + = + .  5.  Page - 10 - "  " 8/ Cho 0 2 sinx , 0 x 90 3 = < <  12 2 tan x cosx tan x cosx H , H cosxcot x cot x sin x - = = - . 9/ Cho 4 cosx , x 52 p = - < < p  12 cot x tan x sin x I , I cot x cot x tan x 1 cosx + = = + -+ .  Cho 5 sinx cosx 4 +=  1/ A sinx.cosx= . : 9 A 32 = . 2/ B sinx cosx=- . : 7 B 4 =± . 3/ 33 C sin x cos x=- . : 41 7 C 128 =± .  Cho tanx cot x 3-=  1/ 22 A tan x cot x=+ . : A 11= . 2/ B tanx cot x=+ . : B 13=± . 3/ 44 C tan x cot x=- . : C 33 13=± .  Cho sinx cosx m+=  1/ A sinxcosx= . 2/ 33 B sin x cos x=+ . 3/ 44 C sin x cos x=+ . 4/ D sin x cosx=- . 5/ 22 E tan x cot x=+ . 6/ 66 F sin x cos x=+ .   sinx, cosx, tan x, cot x  1/ sin x cosx 2+= . 2/ sinx cosx 2-= . 3/ 1 sinx cosx 2 += .   4/ 1 sinx cosx 5 += . : 4 3 4 3 ; ; ; 5 5 3 4 - - - . 5/ tanx cot x 4+= . : 1 2 3 ; ; 2 3; 2 3 2 2 2 3 - +- - .  Cho tanx 2cot x 1- = -  1/ 22 A tan x cot x=- . 2/ 33 B tan x cot x=+ . 3/ 44 C tan x 2cot x=+ . 4/ 55 D tan x 3cot x=- .   1/ Cho 44 3 3sin x cos x 4 +=  44 A sin x 3cos x=+ . : A 7 4 = . [...]... bc + ca à A B B C C A tan tan + tan tan + tan tan = 1 2 2 2 2 2 2 5/ r 2 æ A ö çtan + tan B + tan C ÷ à s d ÷ ç ÷ ç è 2 2 2ø 3/ Page - 21 - à s Lê Vă 5 à – D – CÔNG THỨC NHÂN  Công thức nhân đôi     Công thức hạ bậc Công thức nhân ba ở r       NG R T G N VÀ T NH GIÁ TR C A BI U THỨC r s 1/ A = cos2 3/ 5/ 7/ p p - sin2 8 8 B = 3cos100 - 4cos3 100 C = sin 200 1... 4cos x - sin x - 1 7/ G = (2cos x - 1)(sin x + cos x)- 1 8/ H = 2sin 2x - cos2x - 7 sin x - 2cos x + 4 E – CÔNG THỨC BIẾN Đ I Page - 29 - à s Lê Vă 5 à –   Công thức biến đổi tổng thành tích ● ● ● ● ● ● ● ●  ● ●  Công thức biến đổi tích thành tổng ● ● ● NG CÔNG THỨC BIẾN Đ I T NG THÀNH T CH r 1/ A= cos2x - cos 4x khi x = 200 sin 4x - sin 2x 2/ B= cos x.cos13x... = æ 1 1 1 pö ÷ 1 + sin x + 1 - sin x, ç0 < x < ÷ 16/ P = + + + cot 8x ç ÷ ç 4÷ sin 2x sin 4x sin 8x è ø 17/ O = æ p pö ÷ 1 - sin 2x + 1 + sin 2x, ç- < x < ÷ ç ÷ ç 4 ÷ 4ø è Tính giá tr c a bi u th ng giác, khi bi t 3 p , < x< p 5 2 1/ sin 2x, cos2x khi sin x = 2/ sin 2x, cos2x khi sin x + cos x = 2 Page - 24 - "A e f wer f rr w re e see s f d y " 10 – 2 s Lê Vă 5 3p ,p< x< 13 2 3/ cos2x, sin 2x,... sin 2x, tan 2x khi tan x = 2 5/ sin x, cos x khi sin 2x = - 6/ cos2x, sin 2x, tan 2x khi tan x = 7/ sin 2x, cos2x, tan 2x khi tan x = 2 - Tính giá tr c a bi u th 4 p 3p , < x< 5 2 2 7 8 3, 0 < x < ng giác, khi bi t p 16 1/ A = sin5 x cos x - cos5 x sin x 2/ B = sin 4 x + sin 3 x cos x - cos3 x sin x + cos4 x 3/ C = cos4x 4/ D = cos 2700 + 4x 5/ E = sin 4 x + cos4 x 6/ F= sin 180 = x= p 48 ( 2/ )... 14 - - 3A + B + C = - sin 2A 2 A + B + 3C = cosC 2 10/ tan A + B - 2C 3C = cot 2 2 A + 3B + C = - sin B 2 12/ cot A - 2B + C 3B = tan 2 2 "A e f wer f rr w re e see s f d y " 10 – 2 s Lê Vă à C – CÔNG THỨC C NG        :   NG d y y r 1/ A = sin120.cos 480 + cos120.sin 480 2/ B = cos380.cos220 - sin 380.sin 220 3/ C = sin100.cos550 - cos100.sin 550 4/ D = sin 360.cos60 - sin1260.cos840... sin 2C = 4sin Asin Bsin C A B C sin sin 2 2 2 5/ 1 + cos2A + cos2B + cos2C = - 4cos Acos BcosC 6/ sin2 A + sin2 B + sin2 C = 2(1 + cos Acos BcosC) 7/ cos2 A + cos2 B + cos2 C = 1 - 2cos A cos B cosC CÔNG THỨC BIẾN Đ I T CH THÀNH T NG à 1/ p 2p A = sin sin 5 5 2/ B = sin 5x cos3x 3p p cos 4 6 4/ D = sin E = sin (x + y)cos (x - y) 6/ F = sin x + 300 cos x - 300 8/ H = 8cosx sin2x sin 3x 3/ C