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?p ?n chuy?n : Phng tr?nh lng gi?c - Gii t?ch 11

Vn 1. Gii c?c phng tr?nh lng gi?c c bn

B?i 1 :

1.

Phng

tr?nh

sin

2x

+

3

=

sin

-

6

2x 2x

+ +

3 3

= =

-

6

+

+

6

k2 + k2

x x

= =

- + k 4

5 + k 12

,

k

2. Phng tr?nh cos(3x + 150 ) = cos 300

3x + 150 = 300 + k.3600 x = 50 + k.1200

, k

3x + 150 = -300 + k.3600 x = -150 + k.1200

3.

Phng

tr?nh

4x

+

1 2

=

arcsin

1 3

+

k2

4x

+

1 2

=

-

arcsin

1 3

+

k2

x

=

-

1 8

-

1 4

arcsin

1 3

+

k

2

, k

x

=

4

-

1 8

-

1 4

arcsin

1 3

+

k

2

4. Phng tr?nh sin(2x + 1) = sin( - 2 + x) 2

2x + 1 =

2

-

2

+

x

+

k2

x

=

2

-

3

+

k2

,

k

.

2x

+

1=

2

+

2

-

x

+

k2

x

=

6

+

1 3

+

k2 3

5. Phng tr?nh cos x = 2 = cos x = + k2, (k )

2

4

4

6. Phng tr?nh cot 2x = 3 2x = arc cot 3 + k

323

2

x = 3 arc cot 3 + 3 k (k ) .

2

22

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B?i 2 Bn c t l?m B?i 3

1.

Phng

tr?nh

tan

3x

-

3

=

tan

-

3

3x - = - + k x = k,k 33

2. Phng tr?nh cot(4x - 200 ) = cot 600

4x - 200 = 600 + k.1800 x = 200 + k.450 , k

3. Phng tr?nh sin 2x = 2 cos 2x tan 2x = 2

2x = arctan 2 + k x = 1 arctan 2 + k , k

2

2

2x = x + k x = k

4.

Phng

tr?nh

x

2

+

k

x

2

+

k

x = k, k

.

x

4

+

k

2

x

4

+

k

2

B?i 4

1. Phng tr?nh tan 2x = 3 = tan 2x = + k2

3

3

x = + k (k ) . 6

2. Phng tr?nh cos2 x - 2sin xcos x = 0

cos x(cos x - 2 sin x)

=0

cos x = 0 2 sin x = cos x

cos x = 0

tan

x

=

1 2

x x

= + k 2

= arctan 1 2

+

k

.

3.

Phng tr?nh

cos(3x

-

1)

=

sin(-2x

-

1)

=

cos

2

+

2x

+

1

3x 3x

- -

1 1

= =

2

-

+

2

2x + 1 - 2x -

+ k2 1 + k2

x x

= =

+ 2 + k2 2 - + k 2

10 5

4.

Phng

tr?nh

sin

4x

-

4

=

sin

3

-

2x

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4x

-

4

=

3

-

2x

+

k2

x

=

7 72

+

k 3

4x

-

4

=

2 3

+

2x

+

k2

x

=

11 24

+

k

5.

Phng tr?nh

cos

7

x

=

sin

5

-

2x

=

cos

3 10

+

2x

7x

=

3 10

+

2x

+

k2

x

=

3 50

+

k2 5

7x

=

-

3 10

-

2x

+

k2

x

=

-

20

+

k 5

6.

Phng

tr?nh

1 - cos 4x

=

1+

cos

2x

-

2

cos 4x

=

sin(-2x)

2

2

cos 4x

=

cos

2

+

2x

x x

= =

+ k 4 + k 12 3

7.

Phng tr?nh

x = cos 8x = cos 2x

k 3

x

=

k 5

8. Phng tr?nh

sin 2x(1 + 6cos2x) = 0

x x

= =

k 2 1 2

arccos

-

1 6

+

k

9.

Phng tr?nh sin 4x(3 + 5cos4x) = 0

x x

= =

k 4 1 4

arccos

-

3 5

+

k 2

.

B?i 5

1. iu kin: sin 2x 1 x + k 4

Phng tr?nh cos 2x = 0 x = + k 42

Kt hp iu kin ta c?: x = 3 + k l? nghim ca phng tr?nh 4

2. iu kin: sin 2x 0 x k 2

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Phng

tr?nh

cot 2x sin 3x

=0 =0

x x

= =

+ 4 k

3

k

2

Kt hp iu kin ta c? nghim ca phng tr?nh l?

x = + k ,x = m vi m 3n

42

3

3.

iu

kin:

cos 3x cos 4x

0 0

x x

6 8

+ +

k k

3 4

Phng tr?nh 4x = 3x + m x = m

Kt hp iu kin ta c? nghim ca phng tr?nh x = m .

4.

iu

kin:

sin 5x sin 8x

0 0

x x

k 5 k 8

Phng

tr?nh

cot 8x

=

tan 5x

=

cot

2

-

5x

x

=

26

+

m 13

Kt hp iu kin ta c? nghim ca phng tr?nh

x = + m , m 13n + 6 . 26 13

5. iu kin: -2 x 2

Phng

tr?nh

x = 2 sin 2x = 0

x = x =

2 k 2

Kt hp iu kin ta c? nghim ca phng tr?nh: x = 2,x = ,x = 0 . 2

6. iu kin: -1 x 1

Phng tr?nh cos x = 0 x = + k 2

Kt hp iu kin ta thy phng tr?nh v? nghim.

7. iu kin: sin 2x 0 x k 2

Ta

c?:

tan2

x + cot2

x

2

1 + cos2

3x

+

4

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N?n

phng

tr?nh

tan2 x = cot2 x

sin

3x

+

4

=

0

x x

= =

+ k 4

- +m 12

3

x = + k l? nghim ca phng tr?nh ? cho. 4

B?i 5

1. Phng tr?nh 2 sin x - 2 = k2 sin x = 1 + 3k

3

3

Do -1 sin x 1 k = 0 x = + k2 2

2. Phng tr?nh (cos x - 1) = - + k

4

4

cos x = 4k k = 0 cos x = 0 x = + k . 2

B?i 6

1

Phng

tr?nh

sin

2x

-

6

=

-

1 2

x x

= =

k 2 3

+

k

2. Phng tr?nh cos(4x - ) = 1 x = 1 arccos 1 + k

5

44

52

Trong

?

0;

2

: sin

=

3 5

,cos

=

4 5

.

3.

Phng

tr?nh

sin

3x

-

3

=

sin

2

-

5x

3x

-

3

=

2

-

5x

+

k2

x

=

5 48

+

k 4

3x

-

3

=

2

+

5x

+

k2

x

=

-

5 12

-

k

4. Phng tr?nh 1 - cos 2x + sin 2x = 2 2sin 2x - cos 2x = 3 2

Phng tr?nh v? nghim.

5. Phng tr?nh 3 sin 2x + cos 2x = sin7x - 3 cos7x

sin

2x

+

6

=

sin

7x

-

3

x x

= =

10 7 54

+ k 2 5

+ k 2 9

Group:

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