Преобразования тригонометрических уравнений

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. 1 . 2 . , . , .

. (, . -, 2007 )

6 sin 2x - + 5 sin x - + 1 = 0.

6

3

. sin = cos

2

-

,

:

2

2

sin 2x - = cos - 2x + = cos - 2x = cos 2x - .

6

2

6

3

3

2

6 cos 2x - + 5 sin x - + 1 = 0

3

3

t = sin

x

-

3

:

t = 1,

6(1 - 2t2) + 5t + 1 = 0 12t2 - 5t - 7 = 0

7

t=- .

12

.

:

5 6

+ 2n,

3

+

(-1)n+1

arcsin

7 12

+ n,

n

Z.

. (, 2003 )

cos 3x 2 cos2 x - sin2 x cos 2x - 1 = 1. cos x

. : 2 cos2 x - sin2 x cos 2x - 1 = cos 2x - sin2 x cos 2x = cos 2x cos2 x.

cos 3x cos 2x cos x = 1,

cos 3x cos 2x cos x = 1,

(1)

cos x = 0.

x, (1), . , (1) | cos x| = 1, x1 = 2n x2 = + 2n (n Z). , x1 x2 (1).

: n, n Z.

1

. ( ! , 2014 )

cos 11x + = sin 17x + .

4

4

. cos = sin

2

-

-

:

0 = sin 17x + - cos 11x + = sin 17x + - sin - 11x = 2 sin 14x cos + 3x ,

4

4

4

4

4

14x = n,

+ 3x = + n

4

2

(n Z).

:

n 14

,

12

+

n 3

,

n Z.

. (, 1999 )

sin 3x - cos x = 1.

cos 3x - sin 5x

.

sin 3x - cos x = cos 3x - sin 5x, (2)

sin 3x - cos x = 0.

(2):

sin 3x + sin 5x = cos 3x + cos x 2 sin 4x cos x = 2 cos 2x cos x cos x(sin 4x - cos 2x) = 0 cos x cos 2x(2 sin 2x - 1) = 0.

, (2)

cos x = 0,

cos 2x = 0,

1

sin 2x = .

2

cos x

=

0,

x

=

2

+n

(n

Z).

x

sin 3x

=

?1;

(2) , x .

cos 2x

=

0,

x

=

4

+

n 2

(n

Z).

(2),

x , 2n.

11

x1 = 4 + 2n

sin 3x1

-

cos x1

=

2

-

2

=

0;

3

11

x2 = 4 + 2n

sin 3x1

- cos x1

=

2

+

2

=

0;

11

x3

=

- 4

+

2n

sin 3x1

-

cos x1

=

- 2

-

2

=

0;

3

11

x4 = - 4 + 2n

sin 3x1

-

cos x1

=

- 2

+

2

=

0.

2

,

x2

x3.

:

x

=

-

4

+

n.

,

sin 2x

=

1 2

,

x

=

(-1)n

12

+

n 2

(n

Z).

,

,

, x (2).

:

2

+ n,

-

4

+ n,

(-1)n

12

+

n 2

,

n Z.

. ( , 2010 )

sin 7x cos x - sin 5x cos 3x = 0.

cos 2x - sin 2x

.

cos 2x - sin 2x = 0

(3)

2 sin 7x cos x - 2 sin 5x cos 3x = 0 sin 8x + sin 6x - (sin 8x + sin 2x) = 0 sin 6x - sin 2x = 0 sin 2x cos 4x = 0.

sin 2x

=

0,

x

=

n 2

(n

Z).

x

cos 2x

=

?1,

(3) .

cos 4x = 0,

0 = cos2 2x - sin2 2x = (cos 2x - sin 2x)(cos 2x + sin 2x).

(3)

cos 2x + sin 2x = 0 sin 2x + = 0,

4

x

=

-

8

+

n 2

(n

Z).

:

n 2

,

-

8

+

n 2

,

nZ

. (, 2002 )

cos2 x + cos2 2x = 1 + ctg 3x.

. :

1 + cos 2x 1 + cos 4x

+

= 1 + ctg 3x cos 2x + cos 4x = 2 ctg 3x

2

2

cos 3x

1

2 cos 3x cos x = 2

cos 3x cos x -

= 0.

sin 3x

sin 3x

sin 3x = 0

(4)

cos 3x = 0, (5)

cos x sin 3x = 1.

cos 3x

=

0,

x

=

6

+

n 3

(n

Z).

sin 3x

=

?1,

(4)

.

3

 cos x sin 3x = 1. , | cos x| = 1, x = n (n Z). x sin 3x = 0. , (5) .

:

6

+

n 3

,

n

Z

. ( , 2009 )

sin 3x

= ctg2 x.

cos 3x + 2 cos x

. , :

sin 3x

3 sin x - 4 sin3 x

sin x (3 - 4(1 - cos2 x)) sin x(4 cos2 x - 1)

=

=

=

.

cos 3x + 2 cos x (4 cos3 x - 3 cos x) + 2 cos x

cos x(4 cos2 x - 1)

cos x(4 cos2 x - 1)

4 cos2 x - 1 = 0

1 cos 2x = -

(6)

2

tg x = ctg2 x tg3 x = 1 tg x = 1,

x

=

4

+

n

(n

Z).

,

x

(6).

:

4

+ n,

n

Z.

. (, 2007 ) 11 + cos 8x = -12 cos 4x - 10 ctg2 3x. sin 3x

. : 11 + cos 8x = 11 + 2 cos2 4x - 1 = 10 + 2 cos2 4x.

sin 3x = 0

(7)

5 + cos2 4x sin2 3x + 6 cos 4x sin 3x + 5 cos2 3x = 0 cos2 4x sin2 3x + 6 cos 4x sin 3x + 5 = 0.

t = cos 4x sin 3x t2 + 6t + 5 = 0 t = -1 t = -5. , cos 4x sin 3x = -5 .

cos 4x sin 3x = -1 sin x - sin 7x = 2,

(- , 1)

sin x = 1, sin 7x = -1.

x

=

2

+ 2n

(n

Z);

,

-

, (7). , x

.

:

2

+ 2n,

nZ

4

 2

x=

-

1. (, , 2012.4 )

sin 3x = 2 cos x - sin x.

+ n,

n

Z;

(-1)k

8

+

k 2

,

k

Z

2. (, , 2013.4 )

sin 5x cos 5x sin x cos x

-

=

-

.

sin x cos x sin 5x cos 5x

k 8

,

n 6

,

k

Z\4Z,

n

Z\3Z

2 3

3. ( ! , 2014 ) x

sin x = - 3 cos . 2

4. (, 2011 ) 2|x + 2| cos x = x + 2.

5. (, , 2004-03.1 ) x (-; ), sin 2x + 5 cos 2x = 3.

2 3

+ 2n,

n

-1;

-

2 3

+

2n,

n

0;

?

3

+

2n,

n

0; n Z

2 3

+

2n,

-

6

+

(-1)n

arcsin

3 5

+

n,

6. (, . -, 2007 )

5

5 cos 2x + = 4 sin - x - 1.

3

6

n

Z

7. (, 2006 )

2

3 cos 2x + sin 2x = 7 + 3 cos 2x - .

6

2;

1 2

-

7 12

+

n,

n

Z

5

 4

+

n,

arctg

3 2

+

n,

n

Z

8. (, , 2009 )

2 cos2 x - 5 sin 2x + 6t = 0,

t = arccos

cos

23 3

- arcsin

-

3 2

.

9. (, , 2010 ) 2 sin4 x + 7 cos3 x = 2.

10. (, , 2005 ) cos 4x = 4 cos x cos 2x - 1.

2

+

n,

?

3

+

2n,

n

Z

4

+

n 2

,

? arccos

1- 2

+ 2n,

nZ

3

11. ( ! , 2015 )

cos 4x - 6 cos2 2x + 8 cos2 x

= 0.

6x - x2 - 5

3

,

2 3

,

4 3

12. ( ! , 2016 ) sin10 x + cos10 x = 29 cos4 2x. 16

8

+

n 4

,

n

Z

13. ( ! , 2010 )

-

2

;

2

6 2 cos x ctg x - 2 2 ctg x - 3 cos x + 1 = 0 ?

.

;

?

arccos

1 3

14. (, 2003 ) cos 5x 1 - 2 sin2 x - sin2 x cos 2x = 1. cos x

n, n Z

6

 3

+

n,

n

Z

2

+

n,

n

Z

x=

-

28

+

k 7

,

x=

12

+

2k 3

,

15. ( , 2014 )

3 cos x

sin x

= tg 2x +

.

sin x + cos x

sin x - cos x

16. ( , 2014 )

cos x cos

x

-

4

9 cos2 x + 7 sin2 x - 8

-

8

cos x sin - 7 cos2 x

x

-

4

- 9 sin2 x

=

tg2x 2

.

17. ( , 2020, 11 )

cos 11x - cos 3x - sin 11x + sin 3x = 2 cos 14x.

x

=

5 44

+

2k 11

,

k

Z

Z,

p

Z

18. ( , 2020, 11 )

cos 8x

sin 8x

+

= 2.

cos 3x + sin 3x cos 3x - sin 3x

19. ( ! , 2017 ) ctg2 x - tg2 x = 12 . cos 2x

20. (, 1993 )

sin 6x

cos 6x

=

,

sin x - cos x cos x + sin x

-

2

;

0

.

21. (, 1994 ) 2 cos x + sin2 x = tg 2x. ctg x - sin 2x

x=

44

+

2k 11

,

k

=

11p + 4,

k

?

12

+

n 2

,

n

Z

-

20

,

-

9 20

?

arccos

1 3

+

2n,

n

Z

7

 4

+

n 2

,

n

Z

n, n Z

2

+

n,

n

Z

n 17

,

-

12

+

n 3

,

n

Z

22. (, 2002 )

3 + 4 cos 2x - 8 cos4 x

1

=

.

sin 2x - cos 2x

sin 2x

23. (, . . -, 2010 ) cos x + sin 2x = 1. cos 3x

24. (, -, 2007 ) sin 5x - sin 3x = 1. 2 sin x

25. ( ! , 2014 )

sin 14x + = cos 20x + .

4

4

26. (, -, 2006 )

cos x + sin x + cos 3x + sin 3x = - 6 cos x.

27. (, 1999 ) cos 3x + sin 5x = -1. cos x + sin 3x

28. (, 2006 ) 5 sin 3x + 16 cos x + 5 sin x = 12 cos3 x.

Z

29. ( ! , 2014 )

cos 5x + cos x 1 + cos 4x

=

.

cos 4x + cos 2x cos x

2

+ n,

8

?

5 12

+ n,

nZ

2

+ n,

4

+ n,

(-1)n+1

12

+

n 2

,

nZ

2

+

n,

-

4

+

n,

- arctg

1 4

+

n,

n

4

+

n 2

,

n

Z

8

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