PTLG

<span class='text_page_counter'>(1)</span>I. Ph¬ng tr×nh lîng gi¸c c¬ b¶n Bµi 1. Gi¶i c¸c ph¬ng tr×nh sau.  3  1  ) sin(5 x  )  cos(3 x  )  6 2 ; 2. 3 2 2 1. ; 3.  3 3 3 cos(  4 x )  cos(2 x  40 0 )  sin(  5 x )  8 2 ; 5. 2 ; 6. 8 4. 2 3  tan(5 x  )  cot(3 x  )  3 3 3 4 7. ; 8. ; 9. tan(3 x  1) 4 Bµi 2 : Gi¶i c¸c ph¬ng tr×nh sau :    sin(3 x  )  sin 2 x 0 cos(4 x  )  cos(  3 x ) 0 3 3 4 1. ; 2.    2 sin(7 x  ) cos(3 x  ) cos2 (2 x  ) sin 2 (  x) 6 3 ; 4. 4 5 3. sin(2 x . 5.. (. sin 2 x −. π π + sin 3 x+ =0 4 3. ) (. ). ;. 6.. 7. tan(3 x  2) cot( x  3) ; 8. Bµi 7 : Gi¶i c¸c ph¬ng tr×nh sau : 4 4 1. sin x  cos x 1 ; 4 4 3. sin x  cos x sin 2 x 6 6 5. sin x  cos x cos4 x. ;. (. cos 2 x −. tan(4 x . 2 2 2 3. π π +sin x + =0 4 4. ) ( ). 9 3 )  cot(2 x  ) 0 8 4. 6 6 2. sin x  cos x 1 5 sin 6 x  cos6 x  8 4. 4 4 6. sin x  cos x cos4 x  2 cos2 ( cos2 x ) 1  cos( sin 2 x ) 2 8.. ;. cos x=√ 3 sin x 7. ; II. Phơng trình bậc hai đối với một hàm số lợng giác: Bµi 1. Gi¶i c¸c ph¬ng tr×nh sau: a) cos 2 x+ sin 2 x +2 cos x+ 1=0 d). b) 4 sin 2 x +8 cos 2 x − 9=0. √ 3 cot2 x −4 cot x+ √3=0. c) sin 2 3 x −2 sin 3 x −3=0. e) 1− 5 sin x +2 cos 2 x=0. 5 f) cos 2 x − 4 cos x+ =0 2. x g) 5 −4 sin 2 x −8 cos 2 =− 4 h) cos 2 x+5 sin x +2=0 i) 2 2 2 2 sin x −cos x − 4 sin x +2=0 2 k) 9 cos x − 5 sin 2 x −5 cos x +4=0 l) 5 sin x ( sin x − 1 ) − cos2 x=3 π π m) cos 2 3 x + − cos2 3 x −3 cos −3 x +2=0 n) 3 cos 2 x+2 ( 1+ √ 2+sin x ) sin x − ( 3+ √ 2 )=0 2 2 √3 =3 cot x+ √ 3 p) tan 2 x+ ( √ 3− 1 ) tan x − √3=0 q) sin2 x. (. ). Bµi 2. Gi¶i c¸c ph¬ng tr×nh sau: 1 a) sin4 x +cos 4 x=sin 2 x − 2 1 2 5 d) − tan 2 x − + =0 2 cos x 5 1 tan ( x −15o ) . cot ( x+ 15o )= 3. (. ). b) cos 2 x+ sin 2 x +sin x= e) 2 cos 2 x +2 tan 2 x=5. 1 c) cos 2 x+3 sin x=2 4 f).

<span class='text_page_counter'>(2)</span> III. Phơng trình bậc nhất đối với sinx và cosx: Bµi 1. Gi¶i c¸c ph¬ng tr×nh sau: a) √ 3 cos 3 x +sin 3 x=√ 2 b) sin x − cos x=√ 2 c) cos x − √ 3 sin x=2 √ 3 cos x+ sin x=1 e) 3 cos x+ 4 sin x=5 f) 3 sin 3 x − √ 3 cos 9 x=1+ 4 sin 3 3 x g) cos 7 x . cos 5 x − √ 3sin 2 x=1 −sin 7 x . sin 5 x 3. h) cos 7 x − √ 3sin 7 x=− √ 2 i) 4sin x  1 3sin x  2 2 cos x − √3 sin 2 x=1+sin x k) 4 ( sin4 x +cos 4 x ) + √3 sin 4 x =2 Bµi 2. T×m gi¸ trÞ lín nhÊt, nhá nhÊt cña biÓu thøc sau: 2+ cos x sin x +2 cos x+ 1 a) y= b) y= sin x +cos x −2 sin x +cos x +2 Bµi 3. Gi¶i c¸c ph¬ng tr×nh sau: a) 3 sin 5 x −2 cos 5 x=3 b) 2 sin x −cos x=. 2 5. 3 cos3 x. d). j). c) 3 cos2 x=sin 2 x+ sin2 x d). sin x ( 1 −sin x )=cos x ( 1− cos x ) IV. Phơng trình đẳng cấp bậc Hai đối với sinx và cosx: Bµi 1. Gi¶i c¸c ph¬ng tr×nh sau: a) sin2 x+2 sin x cos x +3 cos 2 x −3=0 b) sin 2 x −3 sin x cos x+1=0 c) 2 2 cos x − 3 sin x cos x − 2 sin x −1=0 d) cos 2 x − sin x cos x −2 sin2 x −1=0 e) 6 sin2 x+sin x cos x − cos2 x=2 π 3π f) 4 sin x cos − x + 4 sin ( π + x ) cos x +2 sin − x cos ( π + x )=1 2 2 3π π g) 2 sin x cos + x − 3 sin ( π − x ) cos x +sin + x cos x=0 2 2 h) sin2 x+sin 2 x+ 3 cos2 x=3 i) 4 cos2 x+5 sin 2 x −6 sin2 x=4 V. Phơng trình đối xứng và nửa đối xứng đối với sinx và cosx:. ( ) ( ). ( ) ( ). Bµi 1. Gi¶i c¸c ph¬ng tr×nh sau: a) 3 ( sin x+cos x ) +2 sin x cos x +3=0 b) √ 2 ( sin x +cos x ) − sin x cos x =1 c) ( 1+ √ 2 ) ( sin x +cos x ) −2 sin x cos x − ( 1+ √ 2 )=0 d) 2 sin x cos x − ( sin x+cos x )+1=0 e) sin 2 x − 4 ( sin x+ cos x )+3=0 f) sin 2 x=sin x +cos x +1 g) 2 sin2 x − ( 2− √ 2 ) ( sin x +cos x −1 )=0 Bµi 2 Gi¶i c¸c ph¬ng tr×nh sau: a) sin x − cos x − sin x cos x +1=0 b) 6 ( sin x − cos x )+ sin x cos x +6=0 c) sin 3 x+ cos3 x= √2 d) 4 − 4 ( sin x −cos x ) −sin 2 x=0 2 e) 2 sin2 x − ( √ 6+ √ 2 ) ( cos x − sin x )=2+ √ 3 3 1 f) sin4 x +cos 4 x= g) sin3 x+ cos3 x=1− sin2 x 4 2. Bµi 3. Gi¶i c¸c ph¬ng tr×nh sau: 3 b) 1+sin3 x+cos 3 x = sin 2 x. a) ( 1− sin x cos x ) ( sin x+ cos x )= √ 2. 2. 2. 2 sin2 x − 2 ( sin x +cos x ) +1=0 d) sin x cos x +2 sin x+2 cos x=2. c). e) 1+tan x=2 √ 2sin x f) sin x − cos x+7 sin2 x=1 3 1 h) sin6 2 x +cos 6 2 x= ( sin4 2 x +cos 4 2 x ) + ( sin 2 x+ cos 2 x ). g) ( sin x − cos x )2+ tan x=2 sin2 x. 2. 1 i) ( cos x − sin x )2+cos 4 x − sin4 x= sin 2 x 2. Ph¬ng tr×nh lîng gi¸c kh¸c. 2.

<span class='text_page_counter'>(3)</span> Bµi 1: Gi¶i c¸c ph¬ng tr×nh lîng gi¸c sau : 1. 2cos3x + cos2x - cosx = 0. ;. 3. cos4x – cos2x + 2sin6x = 0. 2. 4cos x - 2cos2x – cos4x = 1 2 2 2 2 4. cos x  cos 2 x  cos 3 x  cos 4 x 2. ;. cos2 x  cos2 2 x  cos2 3 x  cos2 4 x  5. 2. 2. 2. 3 2;. 2. 2 2 2 2 6. cos x  cos 3 x sin 2 x  sin 4 x. 7. sin x  sin 3 x  sin 5 x  sin 7 x 2 ; 8. sin 2 x  sin 2 3 x  sin 2 5 x  sin 2 7 x  9. 3. 2. 11. 4 cos 2 x  6sin x 3. 7 4 ;. sin 2 x  sin 2 3 x  sin 2 5 x . 3 2. 13 cos6 x  sin6 x  cos2 2 x 8 10.. cos3 x cos3 x  sin3 x sin 3 x cos3 4 x  ; 12.. 1 4. 3 3 3 3 3 13. 4sin x cos3 x  4 cos x sin3 x  3 3 cos4 x 3 ; 14. cos x sin3 x  sin x cos3 x sin 4 x.  1 sin 4 x  sin 4 ( x  )  4 4 15.. 17 sin 8 x  cos8 x  cos2 2 x 16 ; 16..   9 sin 4 x  sin 4 ( x  )  sin 4 ( x  )  4 4 8 ; 17.. cos 18.. 4x cos2 x 3. Bµi 2: Gi¶i c¸c ph¬ng tr×nh lîng gi¸c sau : 3 3 1. cos x  sin x sin x  cos x. 3 3 2. sin x  3cos x  sin x 0. ;. 3 3. 6sin x  2cox x 5sin 2 x cos x. ;. 3 4. 4 cos x  3 2 sin 2 x 8cos x. 3 3 2 5. 4sin x  3cos x  3sin x  sin x cos x 0 3 2 2 3 6. sin x  4sin x.cos x  5sin x cos x  2 cos x 0 3 7. 2 cos x sin3 x. 9.. 3 ; 8. sin x sin 2 x  sin3 x 6 cos x ;.  2 sin3 ( x  ) 2sin x 4. sin3 ( x  ; 10..  )  2 sin x 4.  8cos3 ( x  ) cos3 x 4 4 2 3 11. ; 12 . sin x  23cos x  22 cos x 0 Bµi 3: Gi¶i c¸c ph¬ng tr×nh lîng gi¸c sau : 1. sinx + sin2x + sin3x = 1 + cosx + cos2x 2. sinx + sin2x+ sin3x = cosx +cos2x + cos3x 3. 1 + cosx + cos2x+ cos3x = 0 ; 4. cos10x – cos8x – cos6x + 1 = 0 5. cosx + cos3x + 2cos5x = 0 ; 6. 1 + sinx + cos3x = cosx + sin2x + cos2x 7. ( 2sinx – 1 )( 2sin2x + 1 ) = 3 – 4cos2x ; 8. ( 2sinx + 1 )( 3cos4x + 2sinx + 1 ) + 4cos2x = 3.

<span class='text_page_counter'>(4)</span> 3 3 3 3 9. cos x  sin x sin 2 x  sin x  cos x ; 10. cos x  sin x  cos x 0 3 2 2 3 11. cos x  cos x  2sin x  2 0 ,; 12 . sin x  sin x  cos x 0 3 3 13. 2sin x  sin x 2 cos x  cos x  cos2 x 2 3 4 2 3 4 14. sin x  sin x  sin x  sin x cos x  cos x  cos x  cos x. cos4 15.. x x x x  sin 4 sin 2 x (sin x  3)sin 4  (sin x  3)sin 2  1 0 2 2 2 2 ; 16.. 17. 2sin 2 x  cos2 x 7sin x  2 cos x  4 ;. 18. sin 2 x  2 cos2 x 1  sin x  4 cos x. Bµi 4: Gi¶i c¸c ph¬ng tr×nh lîng gi¸c sau : 1. 1 + 3tanx = 2sin2x ,2. sin2x + tanx = 33. (1 – tanx ) sin2x = 2tanx ; tanx. 4. sin4x =.   cos(2 x  )  cos(2 x  )  4sin x 2  2(1  sin x ) 4 4 5. sin5 x  cos( x  6. 2 3 sin( x  7.. 5  ) sin(2 x  ) 2 2.      ).cos( x  )  2 cos2 ( x  )  3  4[sin 2 x  cos( x  )cos( x  )] 8 8 8 3 3. 0 0 0 8. tan(120  3 x )  tan(140  x ) 2sin(80  2 x ). x x  x 1  sin sin x  cos sin 2 x 2 cos2 (  ) 2 2 4 2 9. 2 4 10. cos x sin x  cos2 x 2 cos x (sin x  cos x )  1. Bµi 5: Gi¶i c¸c ph¬ng tr×nh lîng gi¸c sau :.  1 1 2 2 sin( x  )   4 sin x cos x 1.. 2sin3 x  ; 2.. 1 1 2 cos3 x  sin x cos x. cos x (cos x  2sin x )  3sin x(sin x  2) 1 sin 2 x  1 3. ; 4sin 2 2 x  6sin 2 x  9  3cos2 x 0 cos x 4. cos2 x  3cot 2 x  sin 4 x 2 cot 2 x  cos2 x 5. 1  cos 4 x sin 4 x  1  cos4 x 7. 2sin 2 x. ;. cos x (2sin x  3 2)  2 cos2 x  1 1 1  sin 2 x ; 6. 8.. 1  cos2 x 1  cot 2 x  sin 2 x.

<span class='text_page_counter'>(5)</span> cos x  2sin x cos x  3 2 2 cos x  sin x  1 9. 48  11.. 1 2  (1  cot 2 x.cot x ) 0 2 cos4 x sin 2 x ; 12. sin3 x cos x cos2 x (tan x  tan 2 x ). cos x  13.. 3(sin x  tan x )  2 cos x 0 tan x  sin x ; 10..   4cos2 x tan( x  )tan( x  )  x x 4 4 tan  cot 2 2 14.. 1 1 10  sin x   cos x sin x 3 ;. sin 2 x  2 2. 15. sin 2 x (cot x  tan 2 x ) 4 cos x. sin 2 x  4 cos2 ;. 1 2(cos x  sin x )  cot x  1 17. tan x  cot 2 x ; 1  cos x tan x  1  cos x ; 19. 2. 16.. x 2. tan 2. 1  cos3 x tan x  1  sin3 x 18. 2. 1  cos3 x tan x  1  sin 3 x 20. 2. bµi kiÓm kiÕn thøc Bµi 1 :(4 ®iÓm) Gi¶i c¸c ph¬ng tr×nh sau : 2 1. cos x  2sin x  2 0 2. sinx + cosx + sin2x + 1 = 0 Bµi 2 :(4,5 ®iÓm) Gi¶i c¸c ph¬ng tr×nh sau :. sin 2 x cos2 x  tan x  cot x sin x 1. cos x cos2 x 1 cot x  1   sin 2 x  sin 2 x 1  tan x 2 2. 2 2 3. (1  sin x ).cos x  (1  cos x )sin x 1  sin 2 x. Bµi 3 :(1,5 ®iÓm) T×m c¸c nghiÖm thuéc  0;2  ph¬ng tr×nh sau : cos3 x  sin3 x   5  sin x   cos2 x  3 1  2sin 2 x  . x 2.

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