Phương trình lượng giác và ứng dụng nâng cao

) Page - 1 -      22 sin x cos x 1  tanx.cotx 1  sin x tan x cos x  cos x cotx sin x  os 2 2 1 1 tan x cx  2 2 1 1 cot x sin x        sin2x 2sinx.cos x  22 22 cos x sin x cos2x 2cos x 1 1 2sin x  os 2 1 c 2x sin x 2  os os 2 1 c 2x cx 2  3 sin3x 3sinx 4sin x  3 cos3x 4 cos x 3cosx    sin a b sina.cosb cosa.sinb  osc a b cosa.cosb sina.sin b  tana tan b tan a b 1 tana.tan b  tana tan b tan a b 1 tana.tan b  π 1 tan x tan x 4 1 tan x  π 1 tan x tan x 4 1 tan x    a b a b cosa cosb 2 cos .cos 22  a b a b cosa cos b 2sin .sin 22  a b a b sina sin b 2sin .cos 22  a b a b sina sin b 2cos .sin 22  sin a b tana tan b cosa.cos b  sin a b tana tan b cosa.cos b    cos a b cos a b cosa.cosb 2  sin a b sin a b sina.cosb 2  cos a b cos a b sina.sin b 2    ππ sin x cos x 2 sin x 2 cos x 44  ππ sin x cos x 2 sin x 2 cos x 44  4 4 2 1 cos 4x cos x sin x 1 s 31 in 2x 2 4  6 6 2 3 cos 4x cos x sin x 1 s 53 in 2x 4 8 Ths.   (Nâng cao) Page - 2 -  :   sin x cos x  11 .   tan  cot     tanx  cos x 0 x k k 2 .   cotx  sin x 0 x k k .   tanx  cotx  x k. k 2 .      x      " AM   k2 n 0 0 k.360 hay a n  k ,n  n  M  ".  AM k2 3  M  3  k0 ).  AM k 6  M  6  7 6  k 0,k 1 ).  2 AM k. 43  M  11 ; 4 12  19 12 , k 0;1;2 .  k2 AM k. 4 2 4 4  M  4 , 3 4 , 5 4 ; 7 4   k 0,1,2,3 ).  xk 6  xk 3  xk 6  6  5 6  xk 3   3  4 3 .   xk 32   2 2 11 cos x cos x 22 11 sin x sin x 22    /3 5/6 4/3 /6 O ) Page - 3 - 2 2 2 2 1 cos x 2 cos x 1 0 cos2x 0 2 1 cos2x 0 2 sin x 1 0 sin x 2  2 2 sin x 1 sin x 1 cos x 1 cos x 1  22 sin x cos x 1  22 22 sin x 1 cos x 0 cos x 0 sin x 0 cos x 1 sin x 0  '' c s p''  ''''    cos cos  ''  '' sin sin , tan tan , cot tan   sin sin  ''  '' cos cos , tan tan , cot tan   0   sin cos , cos sin , tan cot , cot tan 2 2 2 2   '' ''   sinu cosv  sinu sinv  sinu cosv   sin u cos v sin u sin v 2 u v k2 u v k2 , k 22 .  2 sin x cos x 3    sin x k2 sin x cos x k2 cos x  sin x k2 sin x k cos x k2 cos x . Ths.   (Nâng cao) Page - 4 -   : u v k2 sin u sin v u v k2 : sin x 0 x k sin x 1 x k2 2 sin x 1 x k2 2  : u v k2 cos u cos v u v k2 : cos x 0 x k 2 cos x 1 x k2 cos x 1 x k2  : tan u tan v u v k Ðk : u,v k 2 : tan x 0 x k tan x 1 x k 4  : cotu cotv u v k Ðk : u,v k : cotx 0 x k 2 cotx 1 x k 4   cos3x 4cos2x 3cos x 4 0 , x 0;14  2cos x 1 2sin x cos x sin2x sin x  cos3x cos2x cosx 1 0  sin x cos x 1 sin2x cos2x 0  sin x 1 cos2x sin2x 1 cos x  1 1 7 4 sin x sin x 4 3 sin x 2  44 7 sin x cos x cot x cot x 8 3 6  44 4 sin 2x cos 2x cos 4x tan x tan x 44 9 3 x 1 3x sin sin 1 10 2 2 10 2  sin 3x sin2x sin x 1 44 ) Page - 5 - . 3 8 cos x cos 3x 1 3  3 2 sin x 2 sin x 1 4  3 sin x 2 sin x 1 4  cosx cos2x cos3x cos4x 0  2 2 2 3 sin x sin 2x sin 3x 2 .  2 2 2 sin x sin 2x sin 3x 2 .  2 2 2 2 sin x sin 3x cos 2x cos 4x  2 2 2 2 sin 3x cos 4x sin 5x cos 6x  sin 22 5x 9x cos 3x sin7x 2 2cos 4 2 2  2 2 2 sin x cos 2x cos 3x  2 2sin 2x sin7x 1 sin x .  sin x sin2x sin 3x 1 cos x cos2x  3 3 3 sin x cos3x cos x sin3x sin 4x  23 cos10x 2cos 4x 6cos3x cos x cos x 8 cos x cos 3x  3 3 2 4sin x 3cos x 3sin x sin x cosx 0 nh: 2 2sin x 1 3cos4x 2sin x 4 4cos x 3  6 6 8 8 sin x cos x 2 sin x cos x  8 8 10 10 5 sin x cos x 2 sin x cos x cos2x 4  3 3 5 5 sin x cos x 2 sin x cos x  4 2 2 4 3cos x 4cos x sin x sin x 0  33 2 3 2 cos 3x cos x sin 3x sin x 8  1 cos x cos2x cos4x cos8x 16  3 4sin 3x cos2x 1 6sin x 8sin x  1 cos x cos2x cos3x cos4x cos5x 2  sin2x 2 cos x sin x 1 0 tan x 3 Ths.   (Nâng cao) Page - 6 -  2 1 sin2x cos2x 2 sin x sin2x 1 cot x  tan x cotx 2 sin2x cos2x  2 tan x tan x tan3x 2  222 11 tan x cot x cot 2x 3  2 2 2 xx sin tan x cos 0 2 4 2  2 sin2x cotx tan2x 4 cos x  22 cot x tan x 16 1 cos4x cos2 x  1 2 tan x cot2x 2 sin2x 2sin2x  3 sin x tan x 2 1 cos x 0 tan x sin x  22 22 1 cos x 1 cos x 1 tan x sin x 1 sin x tan x 2 4 1 sin x  cos3x tan5x sin7x  11 sin2x sin x 2cotx 2sin x sin2x  44 sin x cos x 1 tan x cot2x sin2x 2  2 2 2 2 tan x.cot 2x.cot3x tan x cot 2x cot3x  x cotx sin x 1 tan x tan 4 2  Câu 1.  2sin x cosx 2cos x 3 3 sin x . Câu 2.  2tanx cosx 1 2cosx tanx . Câu 3. nh: 33 2 sin x cos x cos x sin x 8 . Câu 4.  2 2 2 cos x cos 2x cos 3x 1 . Câu 5.  22 17 sin 2x cos 8x sin 10x 2 . Câu 6.  46 cos x sin x cos2x . ) Page - 7 - Câu 7.  1 cos 4x sin 4x 0 2sin2x 1 cos4x . Câu 8.  2 21 sin x cos x cos x 2 . Câu 9.  2 x 2 3 cos x 2sin 24 1 2cos x 1 . Câu 10.  sin4x 3sin2x tanx . Câu 11.  23 cos10x 2cos 4x 6cos3x cosx cosx 8cosx cos 3x . Câu 12.  2 2 2 2cos x 2cos 2x 2cos 3x 3 cos4x 2sin2x 1 . Câu 13.  5x 7 sin 2x 3 cos x 1 2sin x , ;3 2 2 3 . Câu 14.  22 sin 4x cos 6x sin 10,5 10x , 0; 2 . Câu 15.  tan2x tan3x tan5x tan2x tan3xtan5x . Câu 16.  sin x sin2x sin 3x 3 cos x cos2x cos 3x . Câu 17.  2 1 cos x tan x 1 sin x . Câu 18.  2 4 cos x cos x 3 . Câu 19.  11 2 2 sin x 4 sin x cos x . Câu 20.  2 2 tan x cot2x 3 sin2x . Câu 21.  2 3 tan 3x cot2x 2 tan x sin4 x . Câu 22.  2 2 2 sin x sin 2x sin 3x 2 . Câu 23.  25 4x 3 sin2 x 8sin x 0 . Câu 24.  sin2x 2cos x 0 1 sin2x . Câu 25.  sin x cot5x 1 cos9x . Ths.   (Nâng cao) Page - 8 - Câu 26.  2 3 tan6x 2 tan2x cot4x sin8 x . Câu 27.  2 1 cos x tan x 1 sin x . Câu 28.  33 2 cos x cos 3x sin x sin 3 x 4 . Câu 29.  44 x x 5 sin cos 3 3 8 . Câu 30.  2 2sin 3x 1 4 sin x 1 . Câu 31.  3 3 2 cos x 4sin x 3cosx sin x sinx 0 . Câu 32.  44 xx sin cos 1 2 sin x 22 . Câu 33.  sin 3x sin2x sin x 44 . Câu 34.  2 4 4 2 sin x sin3x tan x 1 cos x . Câu 35.  2 x tan x cos x cos x sin x 1 tan tan x 2 . Câu 36.  22 x7 sin x cos4x 2 sin 2x 4 sin x , x 1 3 4 2 2 . Câu 37.  sinx sin2x sin3x 1 cosx cos2x . Câu 38.  2 2 2 2 cos x cos 2x cos 3x cos 4x 2 . Câu 39.  2 cos x cos x 1 2 1 sin x sin x cos x . Câu 40.  sinx sin2x sin3x sin4x sin5x sin6x 0 . Câu 41.  cosx cos3x 2cos5x 0 . Câu 42.  9sinx 6cos x 3sin2x cos2x 8 . Câu 43.  cosx sin x cos x sin x cos x cos2x . Câu 44.  2 2sin x 1 3cos4x 2sin x 4 4cos x 3 . Câu 45.  33 2sin x sinx 2cos x cosx cos2x . Câu 46.  2 3 4 2 3 4 sinx sin x sin x sin x cosx cos x cos x cos x . Câu 47.  2 sin x tan x 1 3sin x cosx sin x 3 . Câu 48.  2 tan2x cotx 8cos x . Câu 49.  3 cotx cos x 5 tan x sin x 2 . Câu 50.  11 2 2 sin x 4 sin x cos x . ) Page - 9 -       2 a sin x bsin x c 0 t sinx 1 t 1 2 a cos x bcosx c 0 t cosx 1 t 1 2 a tan x btanx c 0 t tanx x k , (k ) 2 2 a cot x bcotx c 0 t cotx x k , k t 2 t sin x hoc t sin x u kin là 0 t 1  cos )   2 22 1 sin2x sin x cos x 2sin x cos x sin x cosx  2 22 1 sin2x sin x cos x 2sin x cos x sin x cos x  sin2x sin x cos x 2  33 sin x cos x sin x cos x 1 sin x cos x  x 33 sin x cos x sin x cos x 1 sinx cos  22 sin x cosx sin x cos x 2 tan x cotx cos x sin x sin x cos x sin2x  22 cos x sin x cos x sin x 2cos2x cotx tan x 2cotx sin x cos x sin x cos x sin2x  4 4 2 2 1 1 1 3 1cos 4x sin x cos x 1 sin 2x cos 2x 2 2 2 4  4 4 2 2 2 2 cos x sin x sin x cos x cos x sin x cos2x  6 6 4 4 2 2 2 3 5 3cos4x sin x cos x sin x cos x sin x cos x 1 sin 2x 48  6 6 4 4 2 2 cos x sin x cos2x sin x cos x sin x cos x  x1 1 tan x tan 2 cos x  sin x cos x 2 sin x 2 cos x 44  22 cos x cos x 1 sin x 1 sin x 1 sin x cos x cos x 1 sin x cos x 1 sin x  Ths.   (Nâng cao) Page - 10 -   2 cos4x 12sin x 1 0 .  44 cos x sin x cos4x 0  cos 3x sin3x 5 sin x 3 cos2x , x 0;2 1 2sin2x  sin 3x sin5x 35  sin5x 1 5sin x  22 cos 3xcos2x cos x 0 1  44 3 cos x sin x cos x sin 3x 0 4 4 2  57 sin 2x 3 cos x 1 2 sin x; x ;2 2 2 2  2 5sin x 2 3 1 sin x tan x  2 sin2x 3 cos2x 5 cos 2x 6  66 2 cos x sin x sin x cos x 0 2 2 sin x  1 sin x cos2x sin x 4 1 cos x 1 tan x 2  11 2sin 3x 2cos 3x sin x cos x  2 cos x 2sin x 3 2 2cos x 1 1 1 sin2x  x 3x x 3x 1 cos x cos cos sin x sin sin 2 2 2 2 2  44 sin x cos x 1 1 cot2x 5sin2x 2 8sin2x  22 3cot x 2 2 sin x 2 3 2 cos x  62 3cos4x 8cos x 2cos x 3 0  2cos 4x cotx tan x sin2x  2 cotx tan x 4 sin2x sin2x  22 2sin x tan x 2 [...]... cos x 5 sin x 1 2 cot2x 1 1 sin2 x sin x sin2 x 1 1 cos2 x 2 cos x cos x cos2 x sin x 1 sin 2x tan x cot x 2 tan x cot x 2 sin 2x tan2 x cot2 x 21 1 Page - 11 - 0 3 cos x sin x 3 3 Ths uyễ Quố V ệ (Nâng cao) 4 sin5 x cos x 11 sin 2x 8 4 cos5 x sin x sin2 4x cos 2x cos 2x sin4 x cos4 x 4 cos2 x Câu 51 Câu 52 cos2 2x cos2 3x sin6 x cos2 x sin6 x cos6 x sin2 x cos6 x Câu 55 Câu 56 Câu 57 Câu 58 Câu 59... 93 sin6 x cos6 x Câu 85 Câu 86 Câu 87 cos 2x 3 cos x 3 4 cos2 sin4 2x cos2 cos 8x x 2 cos 2x cos 4x 10 cos 3 5 2 5 2 sin x 2 1 sin 2x 4 Page - 13 - 5 2 0 x 17 2 1 9 sin 2 2 x Ths uyễ Quố V ệ Câu 94 (Nâng cao) 3 sin2 2x 8 sin2 x 11 sin 2x cos 2x 3 2 Câu 95 Câu 96 Câu 97 Câu 98 sin 2x 3 2 sin2 x 1 0 1 sin x 1 1 1 cos 2x cos 2x 3 cot2x sin 4x 2 cot2x cos 2x x x sin 4 cos4 2 2 cos4 x tan x tan x 4 4 Câu... sin 2x 1 1 t tan 2 , x x 2 2 6 ; 5 7 2 3 cos x cos 2x 2 sin x cos x 2 sin x 1 y c2 3 sin 7x x sin 2 u ệ ệ cos 7x y  ệ k2 0 nên 0 k2 sin x 2 2 2 cos x 3 Page - 15 - 1 cos x 0 t x Ths uyễ Quố V ệ (Nâng cao) 3 cos x 8 sin x sin x 1 sin x cos x sin 2x 3 sin 3x 3 cos 9x 3 cos 5x 6 cos x sin 2x 2 cos2x 2 sin 2x cos2x sin 2x 1 cot2x 4 sin4 x 1 4 sin x sin3 x cos3 x sin x sin 4 x 1 4 4 cos3 x sin 3x cos... cos15x 10 cos x Câu 102 2 sin 2x sin x 1 1 3 cos x 3 sin 5x Câu 101 2 3 cos 3x cos x 2 cos2 x 139 1 3 sin 3x tan x 4 1 4 sin x 3 cos x x 2 2 sin 2x 6 Page - 17 - 2 2 9 sin 2x cos 3x 2 2 Ths uyễ Quố V ệ (Nâng cao) 21 2 Câu 104 sin Câu 105 2 sin x Câu 106 cos x 3 sin x Câu 107 sin x 2 sin 5x Câu 108 sin x Câu 109 sin 5x cos 5x 3 sin Câu 110 cos 7x sin 5x 3 cos 5x sin 7x Câu 111 sin 8x cos 6x 3 sin 6x cos... 149 cos2 x 3 sin2x sin3 x Câu 150 sin 3x 3 cos 3x sin2x 4 sin x 3 sin2x 1 3 3 cos2x 3 cos2 x 2 8 1 3 cos x sin x 2 sin x sin x sin2 x 1 3 cos2x Page - 19 - sin x 3 cos x 0 3 3 3 cos x Ths uyễ Quố V ệ (Nâng cao) –  a.sin2 X  c.cos2 X b.sin X cos X d 1 a, b, c, d cos2 X (hay sin2 X )   X 2 1  Khi X 2 cos X k , k 0 2 sin x y 1 Hay X u ệ cos X k , k k 0 2 sin x Hay X 1 ệ y k 1 cho cos2 X (hay sin2... cos4 x 8 cos3 x 3 cot2 x 5 sin 2x cos x cos3 x 3 sin4 x sin3 x 0 cos x 4 cot x 2 cos3 x 3 tan x sin3 x 3 cos3 x 2 2 sin x 3 tan2 x 6 sin x 3 sin x cos2 x 2 sin x Page - 21 - 0 3 ;2 1 Ths uyễ Quố V ệ (Nâng cao) 2 2 cos3 x 4 sin3 x 3 cos x 4 cos3 x sin x cos x 3 sin x 4 cos2 x 4 3 sin x cos x 3 sin x cos x sin4 x cos4 x 2 sin 2x cos 2x sin4 x cos4 x 3 2 sin2 x 0 1 cos2 2x 2 5 1 sin 2x cos 2x cos2 2x 4... 2 x sin 4 x 1 cos6 x cos 2x, t a sin4 x 0 b cos2x cos6 x sin 2x, t cot x b sin 2x 1 cos 2x, t a sin6 x PP : t  tan x a sin6 x PP : t b tan x 0 0 a sin4 x PP : t cot2 x sin x cos x a sin4 x PP : t (Nâng cao) 8 cos2 cos x sin4 x cos3 x cos x 2 3 1 4 x 2 cos2x cos2 x cos x 0 3 sin 2x 3 cos3 x cos4 x ) 210 2 sin x cot x 2 sin 2x 211 cos2x 5 212 cos3 x sin3 x cos2x 213 cos3 x sin3 x 1 214 2 cos2x sin2... x cos x 3 sin x cot 3x 2 2 2 2 cos2x sin 2x 0 cos2 x sin x cos x sin x sin2 x cos x tan x cos 3x cos x cos x 0 3 cos 3x 1 cos2 x sin x 2 cos x tan 2x 2 cot 3x Page - 25 - 1 sin 2x 6 Ths uyễ Quố V ệ (Nâng cao) 238 tan2 x 239 3 tan 2x 240 tan 2x 241 tan x 242 4 sin4 x 243 sin6 x cos6 x 244 tan x cot x 2 sin 2x 245 cot x tan x 2 tan 2x 246 6 tan x 5 cot 3x 247 2 tan x cot x 248 249 50 Câu 172 Câu 173 cot2... tan2 x Câu 190 1 cos x 2 9 8 0 1 cos 3x 0 0 0 F–  :  A ● A A B B u 0 B B A 0 B ● A B B 0 A B2 0  uyệ ● A B A A B B ố B ● A Page - 27 - B A A 0 B B A A 0 B A A 0 B Ths uyễ Quố V ệ 251 252 253 254 (Nâng cao) 5 cos x sin3 x cos2x cos3 x 8 sin 2x cos2 2x 1 sin 2x 1 sin x sin x 256 3 tan x sin3 x 1 2 cos x cot x 259 cos2x 1 260 1 cos 261 Gi 262 sin x cos x 4x cos2 x 3 1 tan2 x sin x sin2 2x 264 2 3 sin... sin x 1 1 cos x cos2x 1 2 sin x cos x 1 1 cos x 1 sin x 2 sin2 x 4 sin2 x sin x 2 1 sin2 x 3 1 295 2 3 cos2x sin x 296 3 4 cos2x 2 cos x 297 2 cos2x 3 sin2x 3 sin2x Page - 29 - cos2x Ths uyễ Quố V ệ (Nâng cao) 8 sin 2x.cos2 2x 298 1 299 cos2x 300 1 sin x 1 1 cos x sin x 0 sin x cos2x Câu 196 Câu 197 3 cos x 1 Câu 198 2 sin 3x Câu 199 3 1 7 sin2x 2 cot x 2 cos2x cos 4x 1 1 3 2 2 sin x 3 Câu 202 sin x