3 4 4

Chuy?n : PHNG TR?NH LNG GI?C

3

1/ Gii phng tr?nh: 2 2 cos2x sin2x cos x 4sin x 0 .

4

4

Gii:

(sin x cos x) 4(cos x sin x) sin 2x 4 0

3

x k ; x k2 ; x k2

4

2

2/ Gii phng tr?nh: sin2 3x cos2 4x sin2 5x cos2 6x

k

Gii:

cos x(cos7x

cos11x)

0

x x

2 k

9

3/ T?m nghim tr?n khong 0; ca phng tr?nh:

2

4 sin2

x

3 sin

2x

1 2 cos2

x

3

2

2

4

5 2

Gii:

sin 2x

3

sin

2

x

x x

18 5

k 3

l2

(k Z) (l Z)

(a) (b)

6

5

V? x 0; n?n x= .

2

18

1

1

4/ Gii phng tr?nh: sin 2x sin x

2 cot 2x (1)

2sin x sin 2x

Gii :

(1)

cos2 2x cos x cos 2x 2 cos 2x

cos2x = 0

x

k

sin 2x 0

42

3sin 2x 2sin x

5/ Gii phng tr?nh:

2

(1)

sin 2x.cos x

2(1 cos x)sin x(2cos x 1) 0

Gii :

2cosx ? 1 = 0 x k 2

sin x 0, cos x 0

3

6/ Gii phng tr?nh: cos 2x 5 2(2 cos x)(sin x cos x) (1)

Gii: (cos x ? sin x)2 4(cos x ? sin x) ? 5 0

x

k2

x k2

2

7/ T?m c?c nghim thc ca phng tr?nh sau tho m?n 1 log1 x 0 :

3

sin x.tan 2x 3(sin x 3 tan 2x) 3 3 (2)

Gii: (2) (sin x 3)(tan 2x 3) 0 x k ; k Z

62

5 Kt hp vi iu kin ta c k = 1; 2 n?n x ; x

36

8/ Gii phng tr?nh: cos3x cos3 x sin 3x sin3 x 2 3 2

8

2

Gii: cos4x = x k

2

16 2

1

Chuy?n : PHNG TR?NH LNG GI?C

9/ Gii phng tr?nh: 9sinx + 6cosx ? 3sin2x + cos2x = 8

Gii: PT (1? sinx)(6cosx + 2sinx ? 7) = 0 1? sinx = 0 x k 2

2 10/ T?m nghim ca phng tr?nh: cos x cos2x sin3 x 2 tho m?n : x 1 3

Gii: PT (cos x 1)(cos x sin x sin x.cos x 2) 0 x k 2 . V? x 1 3 2 x 4

n?n nghim l?: x = 0

(sin 2x sin x 4) cos x 2

11/ Gii phng tr?nh:

0

2sin x 3

Gii:

PT

(2 cos x 1)(sin x cos x 2)

0

x

k2

2sin x 3 0

3

12/ Gii phng tr?nh: sin x cos x 4sin 2x 1 .

Gii: t

t

sin x cos x ,

t 0.

PT

4t2 t 3 0

xk

.

2

13/ Gii phng tr?nh: cos23x.cos2x ? cos2x = 0.

Gii: D?ng c?ng thc h bc. S: x k (k Z )

2

14/ Gii phng tr?nh:

4cos4x

?

cos2x

1

3x

cos 4x cos

7 =

2

42

cos 2x 1 x k

3x

Gii: PT cos2x + cos = 2 3x

m8 (k;m ) x = 8n

4

cos 4 1

x 3

cos2 x.cos x 1

15/ Gii phng tr?nh:

21 sin x

sin x cos x

Gii: PT (1 sin x)(1 sin x)(cos x 1) 2(1 sin x)(sin x cos x)

1 sin

sin x 0 x cos x

sin

x

cos

x

1

0

1 sin x 0

1 sin xcos

x

1

0

x x

2

k2 k 2

16/ Gii phng tr?nh:

1

sin

x 2

sin

x

cos

x 2

sin 2

x

2 cos2

4

x 2

Gii: PT

sin

x

sin

x

1

2 sin 2

2

x

x

2

2 sin

2

1

0

x k x k 4

x k

17/ Gii phng tr?nh:

sin3 x.sin 3x cos3 x cos3x 1

8 tan x tan x

6 3

Gii:

iu kin:

sin x

6

sin

x

3

cos x

6

cos

x

3

0

Ta

c?

tan x

6

tan

x

3

tan

x

6

cot

6

x

1

PT sin3 x.sin 3x cos3 x cos3x 1 8

1 cos 2x cos 2x cos 4x 1 cos 2x cos 2x cos 4x 1

2

2

2

2

8

2

Chuy?n : PHNG TR?NH LNG GI?C

2(cos 2x cos 2x cos 4x) 1 cos3 2x 1 cos 2x 1

x

6

k

(loa?i)

2

8

2

x

k

6

Vy phng tr?nh c? nghim x k , (k )

6

18/ Gii phng tr?nh: sin3 x.(1 cot x) cos3 x(1 tan x) 2sin 2x .

k Gii: KX: x sao cho sin 2x 0 .

2

Khi ?, VT = sin3 x cos3 x sin2 x cos x cos2 x sin x

= (sin x cos x)(sin2 x sin x cos x cos2 x) sin x cos x(sin x cos x) = sin x cos x

sin x cos x 0

PT sin x cos x

2 sin

2x

(sin

x

cos

x)2

2 sin

2x

(1)

(1) 1 sin 2x 2sin 2x sin 2x 1( 0) 2x 2k x k

2

4

tho m?n iu kin sin x cos x 0 , c?c nghim ch c? th l?: x 2k

4

19/ Gii phng tr?nh:

sin

3x

4

sin

2x

sin

x

4

.

Gii : PT sin 3x cos3x sin 2x(sin x cos x)

sin x cos x 0 tan x 1

(sinx + cosx)(sin2x 1) = 0 sin 2x 1 0

sin 2x 1

x 4

k

20/ Gii phng tr?nh: cos2x + cosx + sin3x = 0

Gii: PT cosx(1 + cosx) + 8 sin3 x cos3 x = 0 2cos2 x cos x (1 cos x)sin x 0

22

2

x cos 2 0

sin x cos x sin x.cos x 0

1 21/ Gii phng tr?nh: cos3x cos 2x cos x

2

x Gii: Nu cos 0 x k 2 , k Z , phng tr?nh v? nghim.

2

x

x

Nu cos 0 x k 2 , k Z , nh?n hai v phng tr?nh cho 2cos ta c:

2

2

x

x

x

x

2cos cos3x 2cos cos 2x 2cos cos x cos

t?chth?nh t?ng

7x cos 0

2

2

2

2

2

2 x k , k , i chiu iu kin: k 3 + 7m, mZ .

77

22/ T?m tng tt c c?c nghim x thuc [ 2; 40] ca phng tr?nh: sinx ? cos2x = 0.

Gii: Ta c?:

sinx ? cos2x = 0 2sin2x + sinx ?1 = 0

x

2 k

,k

.

63

2

3

3

V? x [ 2; 40]

n?n

2 k 6

3

40

2

2

6

k

2

40

6

0,7 k 18,8 k 1,2,3,...,18 .

3

Chuy?n : PHNG TR?NH LNG GI?C

2 Gi S l? tng c?c nghim tho YCBT: S = 18. (1 2 3 ... 18) 117 .

63

23/ Gii phng tr?nh:

tan x

6

tan

x

3

.sin 3x

sin

x sin 2x

Gii:

iu kin:

cos

x

6

. cos

x

3

0

PT

sin

x

6

sin

x

3

sin 3x

sin

x

sin

2x

? sin3x = sinx + sin2x

cos

x

6

cos

x

3

sin2x(2cosx

+

1)

=

0

sin 2x 0

cos x

1 2

x x

k

2 2 3

k 2

k

Kt hp iu kin, nghim ca phng tr?nh

x l?:

2

x 2 2k

3

24/ Gii phng tr?nh:

2 cos

x

1

cos2

x

3

8

sin

2( x

)

3cos

x

21

1

s in2x

.

3

3

2 3

1 sin x 0 Gii: PT (1 sin x)(6 cos x sin x 8) 0 6 cos x sin x 8 0 1 sin x 0

1

1

25/ Gii phng tr?nh: sin 2x sin x

2cot 2x

2sin x sin 2x

Gii: PT cos22x cosxcos2x = 2cos2x v? sin2x 0

cos 2x 0

2 cos2

x cos x 1 0(VN )

cos2x

=

0

2x

k

x

k

2

42

26/ Gii phng tr?nh:

2 sin x

4 (1 sin 2x) 1 tan x cos x

Gii: iu kin cos x 0 x k , k .

2

Ta c? PT cos x sin x cos x sin x2 cos x sin x (cos x sin x)(cos 2x 1) 0

cos x

cos x

cos cos

x sin x 2x 1 0

0

x x

4

m

m

,

m

.

27/ Gii phng tr?nh: tan2 x tan2 x.sin3 x cos3 x 1 0

Gii: K: x k . PT tan2 x(1 sin3 x) (1 cos3 x) 0 2

(1 cos x)(1 sin x)(sin x cos x)(sin x cos x sin x cos x) 0

x k2 ; x k ; x k2 ; x k2

4

4

4

4

Chuy?n : PHNG TR?NH LNG GI?C

28/ Gii phng tr?nh: 2cos3x + 3 sinx + cosx = 0

Gii:

PT

cos

x

3

cos 3x

cos

x

3

cos(

3x)

x

3

k

2

sin6 x cos6 x 1

29/ Gii phng tr?nh:

tan 2x

cos2 x sin2 x 4

k Gii: iu kin: cos2x 0 x (k )

42

PT

1

3 sin2 2x

1 sin 2x

3sin22x

+

sin2x ? 4

=

0

4

4

sin2x = 1 x k ( kh?ng tho). Vy phng tr?nh v? nghim.

4

30/ Gii phng tr?nh: cos3 x cos 3x sin 3 x sin 3 x 2 4

1

Gii: PT cos 2x= x= k (k )

2

8

31/ Gii phng tr?nh: cot x 3 tan x 2cot 2x 3 .

Gii: iu kin: sin x cos x 0 x k .

2

cos 2x cos2 x sin2 x

Ta c?: 2cot 2x 2

2

cot x tan x .

sin 2x 2sin x cos x

cot x 3

PT

3

cot

x

3 cot

x

cot

2

x

7 cot

x

6

0

cot

x

1

x

4

k ,k

32/

Gii phng tr?nh:

2

cos2

3x 4 cos 4x 15sin 2x 21

4

Gii: PT

sin3 2x 2 sin2 2x 3sin 2x 6

0

sin 2x

1

x

k

4

33/ Gii phng tr?nh: (1 4 sin2 x)sin 3x 1 2

Gii: Nhn x?t: cosx = 0 kh?ng phi l? nghim ca PT. Nh?n 2 v ca PT vi cosx, ta c:

PT 2 sin 3x(4 cos3 x 3 cos x) cos x 2 sin 3x.cos3x cos x

sin 6x sin x

2

k2

k2

x

x

14 7

10 5

34/

Gii phng tr?nh:

1 sin x sin 2x

1 cos x cos2

x

2

Gii: PT (sin x 1)(sin x cos x 2) 0 sin x 1 x k2 .

2

5

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