1

a/kiÕn thøc cÇn nhí vµ ph©n lo¹i bµi to¸n

d¹ng 1 Ph­¬ng tr×nh bËc nhÊt vµ bËc hai , bËc cao víi 1 hµm sè l­îng gi¸c

Gi¶i ph­¬ng tr×nh:

1. 2cos2x- 4cosx=1

sinx 0

2.

1-5sinx+2cosx=0

cos 0x

3. 4sin3x+3 2 sin2x=8sinx 4. 4cosx.cos2x +1=0

5. Cho 3sin3x-3cos2x+4sinx-cos2x+2=0 (1) vµ cos2x+3cosx(sin2x-8sinx)=0 (2).

T×m n0 cña (1) ®ång thêi lµ n0 cña (2) ( nghiÖm chung sinx=1

3)

6. sin3x+2cos2x-2=0 7. a) tanx+3

cot x -2 = 0 b)

2

4

cos x+tanx=7 c* ) sin6x+cos4x=cos2x

8. sin(5

22

x

)-3cos(7

2x

)=1+2sinx 9.

2sin 2sin 2 2sin 1x x x

10. cos2x+5sinx+2=0 11. tanx+cotx=4 12. 2 4sin 2 4cos 2 1

02sin cos

x x

x x

13. sin 1 cos 0x x 14. cos2x+3cosx+2=0 15.

2 44sin 2 6sin 9 3cos 20

cos

x x x

x

16. 2cosx- sin x =1

d¹ng 2: Ph­¬ng tr×nh bËc nhÊt ®èi víi sinx vµ cosx : asinx+bcosx=c

C¸ch 1: asinx+bcosx=c

§Æt cosx=2 2

a

a b ; sinx=

2 2

b

a b

2 2 sin( )a b x c

C¸ch : 2 sin cosb

a x x ca

§Æt tan sin cos .tanb

a x x ca

sin( ) cosc

xa

C¸ch 3: §Æt tan2

xt ta cã

2

2 2

2 1sin ;cos

1 1

t tx x

t t

2( ) 2 0b c t at b c

§¨c biÖt :

1. sin 3 cos 2sin( ) 2cos( )3 6

x x x x

2. sin cos 2 sin( ) 2 cos( )4 4

x x x x

3. sin 3 cos 2sin( ) 2cos( )3 6

x x x x

§iÒu kiÖn Pt cã nghiÖm : 2 2 2a b c

1. 2sin15x+ 3 cos5x+sin5x=k víi k=0 vµ k=4 víi k=0

2. a) 1

3 sin coscos

x xx

b) 6

4sin 3cos 64sin 3cos 1

x xx x

c) 1

3 sin cos 33 sin cos 1

x xx x

3. cos7 3sin7 2 0x x (t×m nghiÖm 2 6

( ; )5 7

x

) 4. ( cos2x- 3 sin2x)- 3 sinx-cosx+4=0

5. 2

1 cos cos 2 cos3 2(3 3 sin )

2cos cos 1 3

x x xx

x x

6. 2

cos 2sin .cos3

2cos sin 1

x x x

x x

D¹ng 3 Ph­¬ng tr×nh ®¼ng cÊp (thuần nhất) ®èi víi sin x vµ cos x

2

1.a) 3sin2x- 3 sinxcosx+2cos2x cosx=2 b) 4 sin2x+3 3 sinxcosx-2cos2x=4

c) 3 sin2x+5 cos2x-2cos2x-4sin2x=0 d) 2 sin2x+6sinxcosx+2(1+ 3 )cos2x-5- 3 =0

2. sinx- 4sin3x+cosx=0 4

x k

3. tanx sin2x-2sin2x=3(cos2x+sinxcosx)

4. 3cos4x-4sin2xcos2x+sin4x=0 5. 4cos3x+2sin3x-3sinx=0

6. 2 cos3x= sin3x 7/ cos3x- sin3x= cosx+ sinx 8. sinx sin2x+ sin3x=6 cos3x 9/sin3(x- /4)= 2 sinx

Dang 4 Ph­¬ng tr×nh chứa sin x cosx và sinx.cosx

* a(sin x+cosx)+bsinxcosx=c ®Æt t= sin x+cosx 2t

at + b

2 1

2

t =c bt2+2at-2c-b=0

* a(sin x- cosx)+bsinxcosx=c ®Æt t= sin x- cosx 2t

at + b

21

2

t=c bt2 -2at+2c-b=0

1. a) 1+tanx=2sinx + 1

cos x b) sin x+cosx=

1

tan x-

1

cot x

2. sin3x+cos3x=2sinxcosx+sin x+cosx 3. 1- sin3x+cos3x= sin2x 4. 2sinx+cotx=2 sin2x+1

5. 2 sin2x(sin x+cosx)=2 6. (1+sin x)(1+cosx)=2

7. 2 (sin x+cosx)=tanx+cotx 8. 1+sin3 2x+cos32 x=3

2sin 4x

9. a*) 3(cotx-cosx)-5(tanx-sin x)=2 b*) cos4x+sin4x-2(1-sin2xcos2x) sinxcosx-(sinx+cosx)=0

10. sin cos 4sin 2 1x x x 11. cosx+1

cos x+sinx+

1

sin x=

10

3

12. sinxcosx+ sin cosx x =1

dang 5 Gi¶i ph­¬ng tr×nh b»ng ph­¬ng ph¸p h¹ bËc

C«ng thøc h¹ bËc 2

cos2x= 1 cos 2

2

x ; sin2x=

1 cos 2

2

x

C«ng thøc h¹ bËc 3

cos3x= 3cos cos3

4

x x ; sin3x=

3sin sin 3

4

x x

1. sin2 x+sin23x=cos22x+cos24x 2. cos2x+cos22x+cos23x+cos24x=3/2 3.sin2x+ sin23x-3 cos22x=0

4. cos3x+ sin7x=2sin2(5

4 2

x )-2cos2

9

2

x 5. sin24 x+ sin23x= cos22x+ cos2x víi (0; )x

6. sin24x-cos26x=sin(10,5 10x ) víi (0; )2

x

7. cos4x-5sin4x=1 8. 4sin3x-1=3- 3 cos3x

9. sin22x+ sin24x= sin26x 10/ sin2x= cos22x+ cos23x 11. (sin22x+cos42x-1): sin cosx x =0

12. 4sin3xcos3x+4cos3x sin3x+3 3 cos4x=3 ;

24 2 8 2

k kx

13. 2cos22x+ cos2x=4 sin22xcos2x 14. cos4xsinx- sin22x=4sin2(4 2

x )-7/2 víi 1x <3

§¼ng cÊp bËc 2: asin2x+bsinx.cosx+c cos2x=0 C¸ch 1: Thö víi cosx=0 Víi cosx 0 .Chia 2 vÕ cho cos2x ta ®­îc: atan2x+btanx +c=d(tan2x+1) C¸ch2: ¸p dông c«ng thøc h¹ bËc

§¼ng cÊp bËc 3: asin3x+b.cos3x+c(sinx+ cosx)=0 hoÆc asin3x+b.cos3x+csin2xcosx+dsinxcos2x=0 XÐt cos x=0 vµ cos x 0 Chia 2 vÕ cho cos3x ta ®­îc Pt bËc 3 ®èi víi tanx

3

15. 2 cos32x-4cos3xcos3x+cos6x-4sin3xsin3x=0 16. sin3xcos3x +cos3xsin3x=sin34x

17. * 8cos3(x+3

)=cos3x 18. cos10x+2cos24x+6cos3xcosx=cosx+8cosxcos23x

19. sin 5

5sin

x

x=1 20. cos7x+ sin22x= cos22x- cosx 21. sin2x+ sin22x+ sin23x=3/2 22. 3cos4x-2 cos23x=1

Dang 6 : Ph­¬ng tr×nh LG gi¶i b»ng c¸c h»ng ®¼ng thøc

* a3 b3=(a b)(a2 ab+b2) * a8+ b8=( a4+ b4)2-2 a4b4

* a4- b4=( a2+ b2) ( a2- b2) * a6 b6=( a2 b2)( a4 a 2b2+b4)

1. sin4

2

x+cos4

2

x=1-2sinx 2. cos3x-sin3x=cos2x-sin2x 3. cos3x+ sin3x= cos2x

4.

4 4sin cos 1(tan cot )

sin 2 2

x xx x

x

5. cos6x-sin6x=

13

8cos22x

6. sin4x+cos4x=7

cot( )cot( )8 3 6

x x

7. cos6x+sin6x=2(cos8x+sin8x) 8. cos3x+sin3x=cosx-sinx

9. cos6x+sin6x=cos4x 10. sinx+sin2x+sin3x+sin4x= cosx+cos2x+cos3x+cos4x

11. cos8x+sin8x= 1

8 12. (sinx+3)sin4

2

x-(sinx+3) sin2

2

x+1=0

Dang 7 : Ph­¬ng tr×nh LG biÕn ®æi vÒ phương trình tÝch

1/ cos2x- cos6x+ cos4x=1 2/sinx+2cosx+cos2x-2sinxcosx=0 3/sin2x-cos2x=3sinx+cosx-2

4/sin3 x+2cosx-2+sin2 x=0 5/ 3sinx+2cosx=2+3tanx 6/ 3

2sin2x+ 2 cos2x+ 6 cosx=0

7/ 2sin2x-cos2x=7sinx+2cosx-4 8/ sin 3 sin 5

3 5

x x 9/ 2cos2x-8cosx+7=

1

cos x

10/ cos8x+sin8x=2(cos10x+sin10x)+5

4cos2x 11/ 1+ sinx+ cos3x= cosx+ sin2x+ cos2x

12/ 1+sinx+cosx+sin2x+cos2x=0 13/ sin2 x(tanx+1)=3sinx(cosx-sinx)+3

14/ 2sin3x-1

sin x=2cos3x+

1

cos x 15/cos3x+cos2x+2sinx-2=0

16/cos2x-2cos3x+sinx=0 17/ tanx–sin2x-cos2x+2(2cosx-1

cos x)=0

18/sin2x=1+ 2 cosx+cos2x 19/1+cot2x=2

1 cos 2

sin 2

x

x

20/ 2tanx+cot2x=2sin2x+1

sin 2x 21/cosx(cos4x+2)+ cos2x-cos3x=0

22/ 1+tanx=sinx+cosx 23/ (1-tanx)(1+sin2x)=1+tanx

24/ 2 2 sin( )4

x

=1 1

sin cosx x 25/ 2tanx+cotx=

23

sin 2x

26/ cotx-tanx=cosx+sinx 27/ 9sinx+6cosx-3sin2x+cos2x=8

Dang 8 : Ph­¬ng tr×nh LG ph¶i thùc hiÖn c«ng thóc nh©n ®«i, h¹ bËc

cos2x= cos2x- sin2x =2cos2x-1=1-2sin2x sin2x=2sinxcosx

tan2x=2

2 tan

1 tan

x

x

sinx =2

2

1

t

t ; cosx=

2

2

1

1

t

t

tanx=

2

2

1

t

t

1/ sin3xcosx=1

4+ cos3xsinx 2/ cosxcos2xcos4xcos8x=1/16 3/tanx+2cot2x=sin2x

4/sin2x(cotx+tan2x)=4cos2x 5/ sin4x=tanx 6/ sin2x+2tanx=3 7/ sin2x+cos2x+tanx=2 8/tanx+2cot2x=sin2x 9/ cotx=tanx+2cot2x

4

10/a*) tan2x+sin2x=3

2cotx b*) (1+sinx)2= cosx

Dang 9 : Ph­¬ng tr×nh LG ph¶i thùc hiÖn phÐp biÕn ®æi tæng_tÝch vµ tÝch_tæng

Gi¶i ph­¬ng tr×nh 1/ sin8x+ cos4x=1+2sin2xcos6x 2/cosx+cos2x+cos3x+cos4x=0

3/sin3 sin

sin 2 cos 21 cos 2

x xx x

x

t×m 0;2x 4/ sinx+sin2x+sin3x+sin4x=0

5/ sin5x+ sinx+2sin2x=1 6/ 3 cos 2 cot 2

4sin coscot 2 cos 2 4 4

x xx x

x x

7/ tanx+ tan2x= tan3x 8/ 3cosx+cos2x- cos3x+1=2sinxsin2x

Dang 10 : Ph­¬ng tr×nh LG ph¶i ®Æt Èn phô gãc A hoÆc ®Æt hµm B

1/ sin(3

10 2

x )=

1

2sin(

3

10 2

x ) 3 4 14

2 ; 2 ; 25 15 15

x k k k

2/ sin(34

x

)=sin2x sin(4

x

) 4 2

x k

3/ (cos4x/3 – cos2x):21 tan x =0 3x k 4/ cosx-2sin(

3

2 2

x )=3 4x k

5/ cos(7

22

x

)=sin(4x+3 ) ;

6 2

kx k

6/ 3cot2x+2 2 sin2x=(2+3 2 )cosx

2 ; 23 4

x k k

7/ 2cot2x+ 2

2

cos x+5tanx+5cotx+4=0

4x k

8/ cos2x+2

1

cos x=cosx+

1

cos x x k

9/ sinx- cos2x+1

sin x+2

2

1

sin x=5 7

2 ; 2 ; 22 6 6

x k k k

11/1 sin 2

1 sin 2

x

x

+2

1 tan

1 tan

x

x

=3 ; , tan 2x k k

Dang 11 : Ph­¬ng tr×nh LG ph¶i thùc hiÖn c¸c phÐp biÕn ®æi phøc t¹p

1/ 3 4 6 (16 3 8 2)cos 4cos 3x x 2

4x k

2/ cos 23 9 16 804

x x x

=1 t×m n0

xZ 21; 3x

3/ 5cos cos2x x +2sinx=0 2

6x k

4/ 3cotx- tanx(3-8cos2x)=0 3

x k

5/ 2 sin tan

2cos 2tan sin

x xx

x x

2

23

x k

6/ sin3x+cos3x+ sin3xcotx+cos3xtanx= 2sin 2x

24

x k

7/ tan2xtan23 xtan24x= tan2x-tan23 x+tan4x 4

kx

8/ tanx+tan2x=-sin3xcos2x2

3

kx k

9/ sin3x=cosxcos2x(tan2x+tan2x) x k 10/ 2sin sin 1 sin cosx x x x 5 1

;sin2

x k x

11/cos2 2sin 2 cos4

x x

-1=tan2 2tan4

x x

2

4x k

12/ 2 32 cos 6 sin 2sin 2sin

5 12 5 12 5 3 5 6

x x x x

5 5 5

5 ; 5 ; 512 3 4

x k k k

Dang 12 : Ph­¬ng tr×nh LG kh«ng mÉu mùc (®¸nh gi¸ 2 vÕ , sum of squares, b»ng ®¹o hµm)

1/ cos3x+22 cos 3x =2(1+sin22x) x k 2/ 2cosx+ 2 sin10x=3 2 +2sinxcos28x

4x k

3/ cos24x+cos26x=sin212x+sin216x+2 víi x 0; 4/ 8cos4xcos22x+ 1 cos3x +1=0 22

3x k

5/sin

cosx

x 0x 6/ 5-4sin2x-8cos2x/2 =3k t×m k Z* ®Ó pt cã nghiÖm

7/ 1-2

2

x=cosx 8/( cos2x-cos4x)2=6+2sin3x

2x k

9/ 11 cos 1 cos cos 2 sin 4

2x x x x

24

x k