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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