Download - 200-_pt_luong_giac__1890_70616155[1]
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