Тригонометрические уравнения. 2

. . | | MathUs.ru

. 2

. 1 . , 1 .

, ( ) 1. (, ).

, .

1. : cos x - sin x = 1.

. 2:

1

1

1

cos x - sin x = .

2

2

2

1/

2

cos

4

,

1/

2

sin

4

:

1

cos x cos - sin x sin = .

4

42

:

cos

x+

= 1 .

4

2

x + = ? + 2n 44

(n Z),

x = - ? + 2n 44

(n Z).

, ,

, -:

x = 2n,

x = - + 2n 2

(n Z).

:

2n,

- 2

+

2n,

n

Z.

2. : cos x + 3 sin x = 2.

. 2:

1

3

cos x + sin x = 1.

2

2

1

1/2

cos

3

,

3/2

sin

3

:

cos x cos + sin x sin = 1.

3

3

:

cos x - = 1.

3

: 3 + 2n, n Z.

x - = 2n 3

(n Z),

x = + 2n 3

(n Z).

3. : 2 cos x + sin x = 1.

. 2,

22 + 12 = 5:

2 cos x + 1 sin x = 1 .

5

5

5

1; (1)

, , ? :

22

12

+ = 1.

5

5

, ,

cos = 2 , sin = 1 .

(2)

5

5

(1):

1 cos x cos + sin x sin = ,

5

1 cos (x - ) = .

5

1 x - = ? arccos + 2n (n Z),

5

x = ? arccos 1 + 2n (n Z). 5

(2) , 0; ,

2 . -

, :

2

1

= arccos = arcsin .

5

5

2

 . :

x = arcsin 1 ? arccos 1 + 2n (n Z).

5

5

,

2

:

1

1

arcsin + arccos = .

5

52

:

1

1

x = + 2n, 2

x = arcsin - arccos + 2n

5

5

(n Z).

1

1

:

2

+ 2n, x = arcsin - arccos

5

5

+ 2n,

n Z.

, - , .

a cos x + b sin x = c

(3)

a, b c. a2 + b2 = 1, a2 + b2:

a

b

c

cos x +

sin x =

.

(4)

a2 + b2

a2 + b2

a2 + b2

:

a

2

b

2

+

= 1.

a2 + b2

a2 + b2

, ,

a

b

cos =

, sin =

.

a2 + b2

a2 + b2

(4), :

c

cos x cos + sin x sin =

,

a2 + b2

c

cos(x - ) =

.

a2 + b2

( ). .

4. : 3 cos x - 4 sin x = 2.

. 32 + 42 = 5:

3

4

2

cos x - sin x = .

5

5

5

3

32

42

+

= 1,

5

5

0; ,

2

3

4

cos = , sin = .

5

5

, :

2 cos x cos - sin x sin = ,

5 2 cos(x + ) = . 5

2

x = - ? arccos + 2n 5

(n Z),

3

2

x = - arccos ? arccos + 2n

5

5

(n Z).

3

2

: - arccos 5 ? arccos 5 + 2n, n Z.

. a cos x + b sin x = c .

(5) ,

3 cos x - 4 sin x = 2.

:

3

cos2 x - sin2 x

xx - 4 ? 2 sin cos = 2

cos2 x + sin2 x

,

2

2

22

2

2

5 sin2 x + 8 sin x cos x - cos2 x = 0.

2

22

2

cos2 x = 0 : 2

5 tg2 x + 8 tg x - 1 = 0.

2

2

:

-4 ? 21

x = 2 arctg 5 + 2n (n Z).

(6)

(5) (6) , . ! , , .

a cos x + b sin x = c , ? , , .

4

,

a cos x + b sin x = a2 + b2

a

b

cos x +

sin x =

a2 + b2

a2 + b2

= a2 + b2(cos cos x + sin sin x) = a2 + b2 cos(x - ),

, a cos x + b sin x x :

- a2 + b2 a cos x + b sin x a2 + b2 .

. .

. . 2 .

/ /:

+ -

sin + sin = 2 sin

cos

,

2

2

- +

sin - sin = 2 sin

cos

,

2

2

+ -

cos + cos = 2 cos

cos

,

2

2

+ -

cos - cos = 2 sin

sin

.

2

2

5. : sin x + sin 3x = 0.

. :

2 sin 2x cos x = 0

sin 2x = 0, cos x = 0

x =

x=

n ,

2

+ k

2

(n, k Z).

n

, + k . ,

2

2

n = 2k + 1,

n (2k + 1) + 2k

=

=

= + k .

2

2

2

2

( ,

/2 + k n/2.)

, n/2.

n : 2 , n Z.

.

6. : 3 sin 2x + cos 5x - cos 9x = 0.

. : 3 sin 2x + 2 sin 7x sin 2x = 0.

5

 :

sin 2x 3 + 2 sin 7x

=0

sin 2x = 0,

3

sin 7x = -

n x= , 2

x = (-1)n+1

n +

2

21 7

(n Z).

:

n ,

2

(-1)n+1 + 21

n ,

7

n Z.

7. : sin x = cos 3x.

. :

sin x - cos 3x = 0 sin x - sin - 3x = 0 2 sin 2x - cos - x = 0.

2

4

4

:

sin

cos

2x -

4 x- 4

= 0, = 0.

n x= + ,

82 3

x = + n 4

(n Z).

n 3

:

+ 8

2

,

4

+ n, n Z.

:

2 cos cos = cos( + ) + cos( - ), 2 sin sin = cos( - ) - cos( + ), 2 sin cos = sin( + ) + sin( - ).

8. : 3 + 2 sin 3x sin x = 3 cos 2x. . :

3 + cos 2x - cos 4x = 3 cos 2x cos 4x + 2 cos 2x - 3 = 0 2 cos2 2x - 1 + 2 cos 2x - 3 = 0 cos2 2x + cos 2x - 2 = 0.

, : cos 2x = 1 x = n (n Z), cos 2x = -2 .

: n, n Z. 9. : sin 5x cos 2x = sin x cos 6x. . 2:

2 sin 5x cos 2x = 2 sin x cos 6x.

6

 , :

sin 7x + sin 3x = sin 7x - sin 5x,

sin 3x + sin 5x = 0.

:

2 sin 4x cos x = 0

sin 4x = 0, cos x = 0

x =

x=

n ,

4

+ k

2

(n, k Z).

. , n = 4k + 2,

n (4k + 2) 2 + 4k

=

=

= + k.

4

4

4

2

( , /2 + k n/4.)

n : 4 , n Z.

:

sin2 = 1 - cos 2 , cos2 = 1 + cos 2 .

2

2

10. : sin2 x + sin2 2x = cos2 3x + cos2 4x.

. :

1 - cos 2x 1 - cos 4x 1 + cos 6x 1 + cos 8x

+

=

+

,

2

2

2

2

cos 2x + cos 4x + cos 6x + cos 8x = 0.

, :

2 cos 3x cos x + 2 cos 7x cos x = 0 2 cos x(cos 3x + cos 7x) = 0

cos x = 0,

x = + k, 2

4 cos x cos 2x cos 5x = 0

cos 2x = 0,

x

=

4

+

n ,

2

cos 5x = 0

n x= +

10 5

(k, n Z).

, /2 + k /10 + n/5 ( ,

n = 5k + 2 ).

n n

:

+ 4

2

,+ 10

5

, n Z.

7

: cos 2 = cos2 - sin2 .

11. : cos 2x = 2(cos x + sin x).

. :

cos 2x = cos2 x - sin2 x.

, ,

:

(cos x + sin x)(cos x - sin x) = 2(cos x + sin x),

(cos x + sin x)(cos x - sin x - 2) = 0.

:

cos x + sin x = 0,

cos x - sin x = 2.

( ). .

: - 4 + n, n Z. 12. : 2 sin 2x + 2 cos2 x + 2 sin x - 3 cos x - 2 = 0. . ( ):

(4 sin x cos x + 2 sin x) + (2 cos2 x - 3 cos x - 2) = 0.

, 2t2 - 3t - 2 :

2t2 - 3t - 2 = (t - 2)(2t + 1).

:

2 sin x(2 cos x + 1) + (cos x - 2)(2 cos x + 1) = 0,

(2 cos x + 1)(2 sin x + cos x - 2) = 0.

:

2 cos x + 1 = 0, 2 sin x + cos x - 2 = 0.

( 3).

2

2

2

: ? 3

+ 2n,

2

+

2n,

arcsin

5

-

arccos

5

+

2n,

n

Z.

8

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