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1

ÑÖÔØNG TROØN LÖÔÏNG GIAÙC

Baøi 1: Bieåu dieãn caùc cung (hoaëc goùc) sau treân ñöôøng troøn löôïng giaùc:

1) x 0 0 0 0 0 0 030 ;45 ;60 ;90 ;120 ;150 ;180

2) x 2 5 3

; ; ; ; ; ;6 4 3 2 3 6 2

Baøi 2: Bieåu dieãn caùc cung (hoaëc goùc) sau treân ñöôøng troøn löôïng giaùc vaø vieát laïi döôùi

nhieàu daïng coâng thöùc khaùc nhau:

1) x =

4

+ k2 2) x =

4

+ k 3) x =

4

+

2

3

k

4) x =

4

+

2

k 5) x =

4

+

3

k 6) x =

4

+

4

k

7) x =

6

+ k2 8) x =

6

+ k 9) x =

6

+

2

3

k

10) x =

6

+

2

k 11) x =

6

+

3

k

HAØM y = sin(x)

Baøi 3: Duøng ñöôøng troøn löôïng giaùc tính giaù trò haøm soá löôïng giaùc y= sin(x) cuûa caùc

goùc sau :

1) 4950 ; 1830

0 ;

6

43 ; -

17

3

2) (3

2x

) ; (

5

2x

) ; (

11

2x

) ; (

7

2x

) ; (

13

2x

) vôùi x =

3

; x =

6

Baøi 4: Söû duïng ñöôøng troøn löôïng giaùc, giaûi phöông trình löôïng giaùc caên baûn

1) sin 3

(2 )6 2

x

sin 2

(4 )5 2

x

2)

2

1

35sin

x 3) sin2

(x+

4

) =

3

4

4) sin2

(x +

4

) =

1

2 5) sin

2(2x +

2

) = 1 6) sin

2 (2x +

3

) =

1

2

Baøi 5:

1) sin (5 ) sin 2 03

x x

2) sin3

(7 ) cos 5 04 6

x x

3)sin (5 ) sin 2 03

x x

4) sin2

(3 ) sin 4 03 6

x x

5) sin4 5

(2 ) cos 3 03 4

x x

6) sin 5

(7 ) cos 3 06 3

x x

7) sin3

(7 ) cos 5 04 6

x x


2

HAØM y = cos (x)

Baøi 6: Duøng ñöôøng troøn löôïng giaùc tính giaù trò haøm soá löôïng giaùc y= cos(x) cuûa caùc

goùc sau :

1) 4950 ; 1830

0 ;

6

43 ; -

17

3

2) (3

2x

) ; (

5

2x

) ; (

11

2x

) ; (

7

2x

) ; (

13

2x

) vôùi x =

3

; x =

6

Baøi 7:

1)

2

3x4

8cos

2)cos

3( 4 )8 2

x 3) cos

3( 6 )3 2

x

4) cos 2 5 1

(3 )10 4

x

5) cos2(2x +

4

) =

1

4 6) cos

2(2x +

6

) =

3

4

7) cos2(2x +

4

) =

1

2

8) cos2

(2x +

3

) = 1

Baøi 8:

1)cos

(4 ) cos 3 03 4

x x

2) cos3 3

(4 ) cos 6 05 2

x x

3) cos (4 ) cos 3 03 4

x x

4) cos3 3

(4 ) cos 6 05 2

x x

HAØM y = tg (x)

Baøi 9: Giaûi phöông trình :

1) tg (5 ) 36

x

2) 6tg 5 2 33

x

3) cotg2 1

(7 )5 3

x

4) cotg 2

2 15 5

x

5) 3tg2

2 15

x

6) cotg2 3 3

4x

Baøi 10:

1) tg 9 3

(4 ) cot 2 08 4

x g x

2) tg5

(6 ) cot 4 06 3

x g x

3) tg4 5

(5 ) cot 3 07 14

x g x

Baøi 11: Ruùt goïn caùc bieåu thöùc sau:

1) A = sin( + x) – cos(

2

– x) + cotg( 2 – x) + tg(

3

2

– x)

2) B = cos( – x) + sin(x – 3

2

) – tg(

2

+ x) cotg(

3

2

– x)

3) C = cos(270o

– x) – 2sin(x - 450o

) + cos(x + 900o

) + 2sin(720o

– x) + cos(540o

– x)


3

GIAÛI PHÖÔNG TRÌNH LÖÔÏNG GIAÙC SÖÛ DUÏNG COÂNG THÖÙC

CĂN BẢN - NHAÂN ÑOÂI VAØ HAÏ BAÄC 2

Baøi 12: Giaûi caùc phöông trình sau: (Söû duïng coâng thöùc: sin2x + cos

2x = 1)

a/ 3sin2

2x + 7cos 2x – 3 = 0

b/ 2cos 2x + 5sin x – 4 = 0

c/ 2sin2x – cos

2 x – 4sin x + 2 = 0

d/ 9cos2x – 5sin

2 x – 5cos x + 4 = 0

e/ 5sinx(sinx – 1) – cos 2x = 3

f/ cos2( 3x +

2

) – cos

2 3x – 3cos ( 3 ) 2 0

2x

g/ 1 – (2 +2

2 22)sin 0

1 cotx

g x

Baøi 13: Giaûi caùc phöông trình sau: (Söû duïng coâng thöùc: cos2x = 2cos2 x – 1 = 1 –

2sin2x)

1)

a/ cos 2x + 3sinx = 2

b/ 1

cos cos 02 43

x x

c/ cos 8 cos 04 8

x x

d/ cos 25

4cos3 6 2

x x

e/ 1 – cos 3

sin 02

xx

2)

a/ 2sin3

x – cos2x – sin x = 0

b/ 2cos2x – 8cosx + 7 = 1

cos x

Baøi 14: Giaûi caùc phöông trình sau: (Phoái hôïp baøi 1 vaø baøi 2)

1) cos2x + sin2x + sin x =

1

4

2) cos2x + sin2 x + 2cosx +1 = 0

3) 6sin2x + 2 sin

22x = 5

4) 3cos2x + 2(1 + 2 + sinx)sinx – (3 + 2 ) = 0

5)

2 24sin 2 6sin 9 3cos 20

cos

x x x

x

6)

cos (cos 2sin ) 3sin (sin 2)1

sin 2 1

x x x x x

x

7) 4 sin2

x + 3tg2

x =1

Baøi 15: Giaûi caùc phöông trình sau:

1)

a/ sin4x + cos

4 x =

5

8

b/ 4 4sin s

3 3

x xco

= 5

8


4

c/ 4 4sin s 1 2sin

2 2

x xco x

d/ sin4x + cos

4x = cos2x

e/ sin4x + cos

4x = cos

2 2x +

1

4

f/ sin4x + cos

4x = sin 2x –

1

2

g/ sin42x + cos

4 2x = sin2x.cos2x

2)

a/ 5(1 + cosx) = 2 + sin4x – cos

4x

b/ 3(1 – sinx) + sin4x = 1 + cos

4x

c/ 2 2

1 1 8

cos 3 sin 3 3x x

3)

a/ 4 4 1

sin 3 sin 34 4

x x

b/ 4 4 1

sin cos4 4

x x

c/ sin4x + sin

4

4x

+ sin

4 9

4 8x

4)

a/ sin6 x + cos

6x =

7

16

b/ sin6x + cos

6x =

1

4sin

22x

c/ sin6x + cos

6x = cos4x

d/ 16(sin6 x + cos

6x – 1) + 3sin6x = 0

e/ 6 6 213cos sin cos 2

8x x x

f/ 6 6

2

4 4

1 sin cos2cos 3

1 sin cos

x xx

x x

Baøi 16:

1)

a/ 2sin4x +

5

4sin

22x – cos

2x = cos 2x

b/ (sin x + cos x) 4

+ (sin x – cos x) 4

= 3 – sin 4x

2)

a/ cos4 x + sin

6x = cos 2x

b/ cos 6

x + sin 6

x – cos 2

2x – 1

6 = 0

c/ 2cos 62x – cos

42x +

3

2sin

24x – 3sin

22x = 0

d/ 2(sin6 x + cos

6x) – 3(sin

4 x + cos

4x) = cos 2x

e/ sin 4x (3sin 4x – 2cos 4x) = sin2 2x – 16 sin

2x cos

2 x cos

2 2x + cos

2 2x


5

SÖÛ DUÏNG COÂNG THÖÙC NHAÂN BA VAØ HAÏ BAÄC

Baøi 17: Duøng coâng thöùc nhaân ba caên baûn vaø kyû thuaät bieán ñoåi ñoàng cung ñeå giaûi toaùn

a/ sin3xcos3x = sin2x

b/ 1 + sin 2x = (cos3x + sin3x)2

c/ cos 6x = 2sin3

22

x

d/ 2cos4

3

x+ (sinx + cos x)

2 = 0

e/ cos2x = cos

4

3

x

f/ 1 + 2cos2 3

5

x= 3cos

4

5

x

g/ 2cos2

2

x – 1 = sin3x

h/ cos23x-sin

23x + 3cosx.cos3x +

3sinx.sin3x =

4

2

i/ cos32x=cos

24x

Baøi 18: Duøng coâng thöùc nhaân ba ñeå haï baäc haøm baäc ba vaø kyû thuaät ñoàng cung

1)

a/ cos3xcos3x + sin

3x sin 3x =

2

4

b/ cos3xcos3x + sin

3x sin 3x = cos

24x

c/ sin3x sin 3x + cos

3xcos3x = cos

34x

d/ cos3xcos3x – sin

3x sin 3x = cos

34x +

1

4

2)

a/ cos3xsin 3x + sin

3x cos3x = sin

34x

b/ 4sin3x cos3x + 4 cos

3x sin 3x = 3sin 2x

3)

a/ 4 sin3x sin 3x + 4 sin

3x cos3x + 3 3 cos4x = 3

Baøi 19: Bieán ñoåi ñoàng cung baèng phöông phaùp ñaët aån phuï cuûa cung

a/ cos 25

4cos3 6 2

x x

b/ Sin )3

x2(

=5sin )6

x(

+cos 3x

c/ 32 cos 6

4x

- sin 6x =1

d/ 38cos cos3

3x x

e/ 2cos sin 3 cos36

x x x

f/ cos 9x + 2 cos( 6x+2

) 2 03

g/ sin 3x = 2 cos

6x

h/ cos 3x = 2 sin( x+5

6

)

i/ Sin 3x = 2 cos

6x

j/ Cos 3x = 2 Sin5

6x

k/ sin3 1 3

sin10 2 2 10 2

x x

l/ sin3 3

3sin2 10 10 2

x x

m/

3sin 3sin

2 4 4 2

x x


6

PHÖÔNG TRÌNH ÑOÁI XÖÙNG VAØ NÖÛA ÑOÁI XÖÙNG VÔÙI SINX, COSX


1) Phöông trình ñoái xöùng:

a/ 2sin 2x – 2(sin x + cos x) + 1 = 0

b/ sin xcos x + 2sin x + 2cos x = 2

c/ 1 + sin2x = sinx + cos x

d/ sin 2x + 5(sinx + cos x) + 1 = 0

e/ 5sin 2x – 11(sin x + cos x) + 7 = 0

2) Phöông trình nöûa ñoái xöùng:

a/ 5(1 – sin 2x) – 16 (sin x – cos x) + 3 = 0

b/ 4 – 4( cos x – sin x) – sin 2x = 0

c/ sin 2x + ( sin x – cos x) + 1

2 = 0

d/ 1 – sin 2x = cos x – sin x

e/ sinx – cosx + 7 sin 2x = 1

f/ ( 1 + 2 ) ( sin x – cos x) + 2 sinx cosx = 1 + 2

3) Daïng phoái hôïp: (Ñöa veà phöông trình ñaïi soá baäc 3)

a/ ( sin x + cos x) ( 2 sin 2x – 1) = 1

b/ (1 – sinx cosx ) ( sinx + cos x) = 2

2

c/ (sinx – cos x + 1) 1 1

sin 22 2

x

d/ (sinx – cos x – 1) 3 3

sin 22 2

x

e/ sin x + cos x = 2 3

1 sin cos3

x x

Baøi 2: (Moät soá daïng ñoái xöùng theo sinx, cosx thöôøng gaëp, caàn nhôù)

1)

a/ sin 2x + 2 sin4

x

= 1

b/ 1

2sin 2x – 2 sin

4x

= –1

2)

a/ (1 – cosx)(1 – sinx) = 2

b/ (1 + cosx)(1 + sinx) = 2

3)

a/ sin cos 4sin 2 1x x x


7

b/ sin cos sin 2 2 1x x x

4)

a/ sin3 x + cos

3 x =

2

2

b/ sin3

x – cos 3

x = 3 6

8

5)

a/ 1 1

cos sinx x = 2 2

b/ 1 1

cos sinx x = 2 6

6)

a/ 1 + sin3 x + cos

3 x =

3sin 2

2x

b/ 2(sin3x + cos

3x) – (sinx + cos x) + sin 2x = 0

c/ (sin x + cos x – 1)3 32(sin cos 1) 2sin 2x x x

d/ 4(sinx cos 5x + cos x sin

5x) + sin

32x = 1

e/ cos x + 1 1 10

sincos s 3

xx inx

f/ cos x+1 1 1

sincos sin 10

xx x

g/ 1 1 1

5sin cos sin cosx x x x

h/ cos -1

2x + sin-1

2x + cos -1

2x sin-1

2x -5 = 0

i/ ( sin 2x – sin-1

2x)2 + (cos

-1 2x – cos 2x )

2 = 1

7)

a/ 1 + tg x = 2 2 sinx

b/ 1 – tgx = 2 6 sinx

8)

a/ 1 + cotgx = 2 6 cos x

b/ 1 – cotgx = 2 2 sinx

9)

a/ 1 – tgx = 2 1

sincos1 2

xx

b/ 1 + tgx = 1

2sin2cos

xx

10)

a/ 2 (sin x + cosx) = tgx + cotg x

b/ 2(1 – sinx – cosx) + tgx + cotgx = 0


8

GIAÛI PHÖÔNG TRÌNH LÖÔÏNG GIAÙC BAÈNG PHÖÔNG PHAÙP BIEÁN ÑOÅI COÙ

HAØM TANG VAØ COTANG

Baøi 1: Giaûi caùc phöông trình sau: (coâng thöcù2 2

2 2

1 11 &1 cot

cos sintg x g x

x x )

1)

a/ 2 57 0

costg x

x

b/ 2 39

costg x

x

c/ 2

33cot 3

singx

x

2) Sử dụng công thức vạn năng ñöa veà haøm ñoàng cung

a/ 2cos2x + 2tg2

x = 5

b/ 1 – cos 6x = tg3x

c/ 2 12

xtg

= cosx

d/ cosx +

2

xtg = 1

e/ 1 + 3tgx = 2sin 2x

f/ 2sin2x + 3tgx = 5

g/ 2 + sinx = 3

2

xtg

h/ (1 – tgx ) (1 + sin2x) = 1 + tgx

i/ 1 1

sin 23 3

xtgx tgx

j/ 2(sin 2x + cos 2x) = 1 + tgx

k/ tgx + 2 cotg 2x = sin2x

Baøi 2: Giaûi caùc phöông trình sau: (Söû duïng coâng thöùc: cotgx + tgx = 2

sin 2x)

1)

a/ 3 ( tgx + cotg x) = 4

b/ 3 + sin2x = tgx + cotgx

c/ tgx + cotg x= 2cos-1

4x

d/ tgx+cotgx = 3- cos4x

e/ 3(tgx + cotgx) = 2(2+sin 2x)

f/ tgx +cotgx =2(sin 2x +cos 2x)

g/ 7 + 4sinxcosx +3

2(tgx + cotgx) = 0

h/

4 4sin cos 1cot

sin 2 2

x xtgx gx

x

i/ 2 (sin x+ cosx) = tgx +cotg x

j/ 1 2(cos sin )

cot 2 cot 1

x x

tgx g x gx

k/ 2tgx + cotg x =2

3sin 2x

2)

a/ cotg x – tgx = sinx +cosx

b/ cotg x = tgx +2tg2x

c/ tgx = cotg x + 2cotg3

2x


9

d/ 2

1cot 4 0

sin 2tgx gx

x

e/ cotgx- tgx = sin x+cos x

f/ 2( 1-sinx –cosx) +tgx +cotgx = 0

g/ cotgx – tgx = cos sin

sin cos

x x

x x

h/ 2 2

1 13 12 2 3( cot )

sin costgx gx

x x

3)

a/ tgx + tg 2x –tg3x =0

b/ tgx – tg2x = sinx

c/ tg 3x –tgx – 4sinx =0

d/ tg2x + cotg x = 8 cos 2

x

e/ 6tgx + 5cotg 3x = tg 2x

f/ tgx +tg4x = 2tg 3x

g/ tg2x – cotg 3x + coïtgx =0

h/ 2(cotg2x – cotg 3x)= tg2x +cotg 3x

i/ 3tg3x + cotg2 x =2tgx+2

sin 4x

j/ 2 tgx + cotg 2x = 2 sin 2x + 1

sin 2x

k/ 3tg3x + cotg2 x =2tgx+2

sin 4x

l/ 2

2

1 5cot ( cot ) 2

cos 2g x tgx gx

x

Baøi 3: Ñöa veà daïng chaën sinax.sinbx = 1

1) cotg 2x + cotg 3x + 1

0sin sin 2 sin3x x x

Baøi 4:

1)

a/ tg2

x – tgxtg3x =2

b/ 3 tg2x – 4tg3x = tg2

3x. tg2x

2)

a/ tg2x – tg3x – tg5x = tg2x . tg3x. tg5x

3)

a/ tg2

2x.tg2

3x .tg5x= tg2

2x -tg2

3x +tg5x

b/ tg2

x.cotg2

2x .cotg3x= tg2

x-cotg2

2x +cotg3x

c/ tg2

x.tg2

3x .cotg4x= tg2

x-tg2

3x +tg4x


10

SÖÛ DUÏNG COÂNG THÖÙC BIEÁN ÑOÅI TOÅNG THAØNH TÍCH

Baøi 20: Luyeän kyõ naêng bieán ñoåi thaønh tích:

1)

a/ 1 + cos x + cos 2x + cos 3x = 0

b/ cos x + cos 2x + cos 3x + cos 4x = 0

c/ cos 9x – cos 7x + cos 3x – cos x = 0

d/ cos 7x + sin 8x = cos 3x – sin 2x

e/ cosx + cos 3x + 2 cos 5x = 0

f/ cos 5x + cos 7x = cos ( + 6x)

g/ sinx – sin 2x + sin 5x + sin 8x = 0

h/ cos 10 x – cos 8x – cos 6x + 1 = 0

2)

a/ sin x + sin 2x + sin 3x = 1 + cos x + cos 2x

b/ sinx + sin 2x + sin 3x = cosx + cos 2x + cos 3x

c/ sinx + sin 2x + sin 3x + sin4x + sin 5x + sin 6x = 0

d/ 1+ sin x + cos 3x = cos x + sin2x + cos 2x

e/ sin 6x + sin 7x + sin 8x = cos 6x + cos 7x + cos 8x

f/ sin x + sin 7x – cos 5x + cos ( 3x – 2 ) = 0

g/ sin 2x – sin 3x + sin 8x = cos 3

72

x

3)

a/ sin x + sin 2 x + sin

3x + sin

4x = cos x + cos

2x + cos

3x + cos

4x

4)

a/ cos2x + cos

2 2x + cos

2 3x =

3

2

b/ cos2x + cos

2 2x + cos

2 3x = 1

c/ sin2x + sin

2 2x + sin

2 3x =

3

2

d/ cos2

3x + cos24x + cos

25x =

3

2

e/ sin 2 3x – sin

2 2x – sin

2x = 0

5)

a/ cos2x + cos

2 2x + cos

2 3x + cos

24x = 2

b/ cos2x + cos

2 2x + cos

2 3x + cos

24x =

3

2

c/ sin22x + sin

23x + sin

24x + sin

25x = 2


11

6)

a/ sin2x = cos

2 2x + cos

2 3x

b/ cos2

x + cos2 2x – cos

2 3x – cos

24x = 0

c/ sin23x + sin

24x = sin

25x + sin

26 x

d/ sin 2 3x – cos

2 4 x = sin

2 5x – cos

2 6x

e/ sin2x + sin

23x = cos

22x + cos

24x

7)

a/ 2 sin 2x + cos 5x – cos 9x = 0

b/ 1+ sin x – cos 6x – sin 7x = 0

c/ sin 3x – sin 7x = 3sin 2x

d/ cos 2x + cos 6x + 2 sin 2x = 1

e/ 2cos 2x + 2 cos

2 2x + 2 cos

2 3x – 3 = cos4x(2sin 2x + 1)

Baøi 21: Bieán ñoåi thaønh tích

1)

a/ sin22x – cos

28x = sin

1710

2x

b/ sin24x – cos

26x = sin(10,5 + 10 x)

c/ cos3x + sin7x = 2sin2

25 92cos

4 2 2

x x

d/ 2 2sinx cos 4x + 2sin 2x = 1- 4 sin4 2

x

e/ 2 2sin sin sin8 8

x x x

f/ sin 3x + sin 5x = 2 (cos2

2x – sin2 3x)

g/ 2 21cos5 cos7 cos 2 sin 3 0

2x x x x

BIEÁN ÑOÅI THAØNH TÍCH

Baøi 5:

1)

a/ tg2

x = 1 cos

1 sin

x

x

b/ cotg2

x =1 sin

1 cos

x

x

c/ tg2

x =

3

3

1 cos

1 sin

x

x

d/ tg2

x =

3

3

1 cos

1 sin

x

x

e/ cotg2

x =

3

3

1 cos

1 sin

x

x


12

2)

a/ cos2x + sin

3x + cosx = 0

b/ sin2x + cos

3x + sinx = 0

c/ cos2x + cos

3x + 2sinx -2 = 0

d/ sin4x – sin

2x + 4(sin x + 1) = 0

e/ sin3x – sinx + sin2x = 0

f/ cos3x + sin

3x = sinx – cos x

3)

a/ (2sinx – 1) (2sin 2x + 1) = 3 – 4cos 2x

b/ (2sinx + 1) (3cos 4x + 2sinx – 4) + 4 cos2x = 3

c/ 2 sin x + cotg x = 2 sin 2x +1

4)

a/ sin2x – 3 sinx cosx + cos2x + cosx = 0 (cos 2x = cos

2x –sin

2x)

b/ 4 sin 2x – 3cos 2x = 3( 4sinx-1)

Baøi 6:

1)

a/ 2cos3x + cos2x + sinx = 0

b/ 2sin3x – cos2x + cosx = 0

c/ 2sin3x + cos2x - sinx = 0

2)

a/ 2sin2x (4sin

4x – 1) = cos 2x(7cos

22x + 3 cos2x – 4)

b/ 21 1

2 sin4 sin cos

xx x

c/ 2 sin x + 23 1

3 coscos sin

xx x

d/ sin 3x + cos

3 x = 2(sin

5x + cos

5 x)

e/ sin 8x + cos

8 x = 2(sin

10x + cos

10 x) +

5

4cos 2x

Baøi 7: Giaûi phöông trình

1)

a/ (cosx – sin x) cosx sinx = cosx cos 2x

b/ cos3x – sin

3x = sinx – cos x

c/ sin3x + cos

3x = cos 2x

d/ cos3x – sin

3x = cos2x

e/ 3 3sin cos

cos 22cos sin

x xx

x x

f/ 1 + sinx + cosx + sin2x + 2cos2x = 0

g/ (1+sin2x)cosx+(1+cos2x)sinx = 1+sin2x.

h/ 2 + cos2x +2sinx(sinx + cosx) + 2sin2x + 2sin2x(sinx + cosx) – 3(sinx + cosx) = 0

2)

a/ (cotg x – 1)(1 + sin 2x) = 1 + cotgx

b/ sin3 x(1 – cotg x) + cos

3x(1 – tgx) =

6

2cos 2x

c/ 3sin 3

2(1 cos ) 0sin

x tgxx

x tgx


13

Baøi 8: Kyõ thuaät bieán ñoåi cung nhaân ba :

1)

a/ cos3 sin 3

5 sin cos 2 31 2sin 2

x xx x

x

b/ 2(sin3x – cos3x) = 1

sin x +

1

cos x

2)

a/ 2( tgx – sin x)+ 3( cotgx – cos x) +5 = 0

b/ 3(cotgx – cos x) – 5 ( tgx- sinx) =2

3)

a/ (2cos2x + 5) cos4x – (2cos2x + 5) sin

4x = 3

b/ 1+ cotg 2x = 2

1 cos 2

sin 2

x

x

4)

a/ 4 sin 5x cos 5x (cos 4x – sin

4x) = sin 4x

b/ cos 4

2

x – sin

4

2

x= sin 2x

Baøi 9: Nhoùm töø töø 2 giai ñoaïn ôû leân :

1)

a/ cos3x + sin

3x = sin 2x + sinx + cos x

b/ 2sin2x – cos 2x = 7 sinx + 2cosx – 4

c/ 9sinx + 6 cosx – 3sin2x + cos2x = 8

d/ cos5x + sin

7x + ½ (cos

3x + sin

5x) sin2x = cosx + sinx

e/ cos10x + 2cos24x + 6cos3xcosx = cosx + 8cosxcos

33x

Baøi 10: GPT BAÈNG PHÖÔNG PHAÙP SÖÛ DUÏNG COÂNG THÖÙC BIEÁN ÑOÅ TÍCH THAØNH TOÅNG

1)

a/ sin 2x sin 6x = cosx . cos 3x

b/ cos 3x. cos 6x = cos 4x. cos 7x

c/ Cos 11 x. cos 3x = cos 17x . cos 9x

d/ Sin 18. cos 13x = sin 9x . cos 4x

e/ sin xsin 3x + sin 4x sin 8x = 0

f/ cosx – sin x = 4 cos x sin2

x

g/ Sin 2

x + sin 2x sin 4x + sin 3x. sin 9x + sin 4x. sin 16x =1

2)

a/ Sinx.sin(600

-x),sin(600

+ x) = 1

8

b/ 8cosx.cos(600

-x).cos(600

+ x) + 1 = 0

1) Sin3x + cos3x = 4cos3x – 3cosx + 3sinx – 4sin

3x = 4 (cos

3x –sin

3x) – 3(cosx – sinx )

= (cosx – sinx)[4(1 + sinx.cosx)-3) = (cosx –sinx)(1 + 4sinx.cosx)

Cos3x –sin3x = 4cos3x – 3 cosx -3sinx +4sin

3x = 4(sin

3x +cos

3x) -3(sinx + cosx) =

= (sinx + cosx)[4(1 – sinx.cosx)-3] =(sinx + cosx)(1 -4sinx.cosx)


14

c/ Sinx.cos 2x + cosx sin 4x = sin 2 sin 34 4

x x

d/ 4sin x sin 2 4

sin 4 3 cos cos cos 23 3 3 3

x x x x x

e/ 4 cos x . sin sin cos 26 6

x x x

3)

a/ 8 sin x = 3 1

cos sinx x

b/ sin 3x -2

3sin

2

x = 2 sinx cos 2x

c/ cos 3x . tg 5x = sin 7x

4)

a/ cos 2x + cos 4x + cos 6x = cos x .cos 2x .cos 3x+2

b/ cosx. cos

2

xcos

3 3 1sin sin sin

2 2 2 2

x x xx

c/ 2 sin23 1 8sin 2 cos 2

4x x x

d/ 2( cos 4x – sin xcos 3x) = sin 4x + cos 2x

5)

a/ cosx cos 2x sin 3x 1

sin124

x

b/ sin 2x sin 6x = cosx . cos 3x

c/ cos 3x. cos 6x = cos 4x. cos 7x

d/ sin 2x sin 6x = cosx . cos 3x

e/ cos 3x. cos 6x = cos 4x. cos 7x

PHÖÔNG TRÌNH COÅ ÑIEÅN

A - PHÖÔNG TRÌNH THUAÀN NHAÁT BAÄC NHAÁT (ÑAÚNG CAÁP BAÄC NHAÁT,

COÅ ÑIEÅN) VÔÙI SINX, COSX

Baøi 3: Giaûi caùc phöông trình sau: (Thöû ñieàu kieän PTVN)

1) sin2x – cos2x = 2

2) sin3x + 2 cos3x = –5

2

3) 2 2 (sinx + cosx)cosx = 3 + cos2x

Baøi 4: Giaûi caùc phöông trình sau: (Daïng , 1, 3A B chia 2 veá cho 2 2A + B

1) (Söû duïng coâng thöùc: sinx cosx)

a/ sin 3x – cos 3x =

3

2

b/ sin x(1 – sin x) = cos x( cos x –1)


15

c/ 1 + 3

8sin4x = cos

6x + sin

6x

d/ 1 + 2(cos 2x tg x – sin 2x) cos2 x = cos 2x

2)

a/ 3 cos3 sin3 2x x

b/ 3sin cos 1x x

c/ 3 cos sin 2x x

d/ 2sinx(cosx – 1) = 3

cos2x

3)

a/ cos 2x –

23s 2 1 sinin x x

b/ cos 7x. cos 5x – 3sin 2 1 sin 7 .sin5x x x

c/ 4(sin4 x + cos

4 x) +

3sin 4 2x

d/ 4sin 3x – 1 = 3sin x –

3 cos3x

e/ 3sin3x –

33 cos9 1 4sin 3x x

f/ 2 (sinx + 3 cosx) = 3 cos2x – sin2x

g/ 4sin 3 xcos 3x + 4 cos

3x sin 3x + 3 3 cos 4x = 3

Baøi 5:

1) Giaûi caùc phöông trình sau: (Daïng naâng cao, bieán ñoåi ñöa veà

, 1, 3A B chia 2 veá cho A2 + B

2 = 2)

a/ sin2x + 3 cos2x + 2sin(2x –

6

) = 2 2

b/ (sin2x + 3 cos2x)2 = 2 – 2cos(

2

3

– x)

2) Giaûi caùc phöông trình sau: (Daïng B + C = 0 PT coù nghieäm cosx = 0)

a/ 2sin3x – 3 cos3x = 3

b/ sin2x – 3cos2x = 3

3) Giaûi caùc phöông trình sau: (Daïng B + C 0 ñaët t = 2

xtg )

a/ 4sinx + cosx = 4

b/ 3 sin 2x + 2 cos 2x =3

c/ 4 sin 3x + 1

cos3 33

x

d/ sin 3x + 3 2 cos3 1x

e/ 1 3 sin 1 3 cos 2x x


16

f/ 2

54sin sin

6 62 0

cos

x x

tgxx

g/ 2 sin 4x + 3 cos 2x + 16 sin3x cosx – 5 = 0

B - PHÖÔNG TRÌNH ÑAÚNG CAÁP BAÄC 2 VÔÙI SIN X, COS X


(Söû duïng coâng thöùc: a + d = 0 PT coù nghieäm cosx =0)

1) sin2x – 2sinx cosx + 3cos

2x – 1 = 0

2) 3 sin2

x + 3

2sin2x 3 = 0

3) 2sin2x –

3

2sin2x + 3cos

2x = 2

4) 3sin2x – sinx cosx – 4 cos

2x + 3cos2x = 0

Baøi 7: Giaûi caùc phöông trình sau: (cosx 0 chia 2 veá PT cho cos2x)

1) sin2x – 3sin xcosx + 1 = 0

2) sin2x + 2sinx cosx + 3cos

2x – 3 = 0

3) 4cos 2 x + 5sin 2x – 6sin

2 x = 4

4) cos 2x – 3sin xosx – 2sin

2 x – 1 = 0

5)

2 2 54 3sin cos 4cos 2sin

2x x x x


1)

13sin cos

cosx x

x

2) cos-1

3x – 6cos 3x = 4sin 3x

3) tgx + cotgx = 2(sin 2x + cos 2x)

4) cot

6cos 2 4sin 2cot

tgx gxx x

gx tgx


1) 2 25 3

3 sin 3 2sin cos 5sin 02 2 2

x x x x

2) 4 sinx cos

3

4sin cos 2sin cos 12 2

x x x x x

3) 2 sinx cos

3

3sin cos sin cos 02 2

x x x x x

4) sin

13 7 3 3cos cos

2 2 2

x x x


17

PHÖÔNG TRÌNH ÑAÚNG CAÁP BAÄC 3 VÔÙI SIN X, COS X


1)

a/ sin3 x + sinx sin 2x – 3cos

3 x = 0

b/ 2sin3x + 2sin

2x cos x – sinx cos

2 x – cos

3 x = 0

c/ 3sin3x + 5sin

2x cos x + 2sinx cos

2 x = 0

d/ sin3

2 2 3sin cos 3sin cos 3cos 03 3 3 3 3 3

x x x x x x

e/ sin3x – 5sin

2x cos x – 3sinx cos

2 x + 3cos

3 x = 0

2)

a/ cos3x – 4sin

3x – 3 cosxsin

2 x + sin x = 0

b/ 4sin3x + 3cos

3x – 3sin x – sin

2x cosx = 0

c/ sinxsin 2x + sin 3x = 6cos 3x

d/ cos 3x – sin

3x = sinx – cosx

e/ sin2 x( tgx + 1) = 3sin x(cos x – sin x) + 3

Baøi 11:

1) 2sinx cos 2

2 23cos cos 5cos sin 02 2 2

x x x x x

2) sin2

2 33 32 cos 2 3sin 2 sin 2 2cos 2 0

2 2x x x x x

3) 3

sin 3 2 cos 2 sin 2cos 3 04 2 2 4

x x x x

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