Преобразования тригонометрических уравнений
. . | | MathUs.ru
. 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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