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# Погребная Марина Юрьевна. Методика дифференцированного обучения решению тригонометрических уравнений в 10-11 классах средней школы

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80
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............................................... ...................................................16
................20
............................................... ..........................................................34
............................................... ...................34
2.2.
...........................................................39
..............................................................45
2.4.
.........68
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.....................................87
,
-
-
.
,
-
.
-
1.
2.
3.
.
6
[1]
-
-
:
7
:
:
-
-
8
1
1.1
cos 2 x cos x sin 2x sin x 1
cos x 1
9
.
10
1.2
-
11
1.
2.
-
3.
.
12
-
13
-
1.
2.
3.
14
1.
2.
-
-
3.
1.
15
2.
-
1.3
-
10-
16
-11 [5]
.
.
-11 [51]
17
.
:
.
.
[45]
18
,
.
[14]
-
19
-
-
-
f1 ( x) a1 f 2 ( x) a2 ... f n ( x) a n
f i (x) -
ai
0
R.
1.4
,
20
1.4.1
sin x
a , cos x a , tgx a , ctgx a .
cos x a .
a 1
x
a 1
x
cos x 1 ;
cos x
cos x
x
1;
0;
x
2
x
Q
arccos a
Z(
,
2
2
,
,
2
,
Z
1, )
Z(
Z(
1, )
1, )
21
a
arccos( a )
arccos a ;
cos(arccos a)
[ 1;1]
0 arccos a
;
a.
sin x a .
a 1,
x
Q
a 1
x
( 1) n arccos a
sin x 1 ;
sin x
x
1;
sin x 0 ;
2 n, n
2
x
2
2
x
Z(
,
Z(
n, n
Z
1, )
Z(
,
1,
)
1, )
a
[ 1;1]
a
[ 1;1]
2
arccos a
2
;
arcsin a ;
arcsin( a )
arccos a arcsin a
2
.
tgx a .
x
arctga
tg ( arctga)
n, n
Z
2
arctga
2
; arctg ( a )
arctga ;
a
ctgx a .
x
arctga
n, n
Z
a
ctg ( arcctga)
[ 1;1] 0 arctga
a ; arctga arcctga
; arcctg ( a )
arcctga
2
sin f ( x)
a,
cos f ( x) a , tgf ( x) a , ctgf ( x) a .
f(x)
22
23
24
3
25
1. sin
x
2
1
;
2
x
2
( 1) n
x
( 1) n
n;
6
2 n, n
3
( 1) n
x
Z
2 n, n
3
Z.
2
cos( x
x
x
3
2 k, k
3
3
3. tg (2 x
2x
) 1;
Z;
2 k, k
4
)
4 6
5
2x
n;
12
5
n
x
,n
24 2
Z.
3
;
3
n;
Z
26
1.4
:
sin f ( x )
f ( x)
( x) 2 k , k
f ( x)
( x) 2 n.n
sin ( x )
Z
Z
cos f ( x)
f ( x)
f ( x)
cos ( x)
( x) 2 k , k Z
( x) 2 n, n Z
tgf ( x)
f ( x)
( x)
( x)
tg ( x )
k;
n
2
1.
sin( x
x
x
x
x
3
2x
3
12
2
sin( 2 x
4
,x
2
4
,
);
_x
3
2x
4
2 m;
2 m
;
3
5
36
(1 24 ),
Z;
(5 24 m), m
Z.
12
6
)
27
sin 4 x
cos x;
3
sin 4 x sin(
x);
2
3
2
4x x
n; _
2 3
x
x
6
(3 4n), n
2
2
;
Z;
Z.
(1 4 ),
10
_ 5x
1.4.3
2
t
cos 2
x
2
cos
t1
t2
cos
x
2
x
2
t , (t 1),
2
( )
( )
0
( )
t
0;
1
2
3
28
t,
cos
x
x
2
4
cos
1;
x
2
Z. x
,
2
;
3
2 arccos(
2
) 4 n, n
3
Z.
1.4.4
U
U2
V
3U + 2V
5U 3
3UV V2
U 2V
UV 2
V3
U
2( x 2
x 3) 2
3( x 2
x2
3UV
U2
U.
0;
U=V
2
2
1, 2
3
2
1
4
0,
0.
x 3 V,
x 7
2V 2
x 3)( x 7) ( x 7) 2
V = 0,5 U
7, 2
2
2
6
2
2
3
1 0,
3
17
17
3, 4
4
7,
.
U
V
1.
V.
2 cos x 3sin x
cos x
.
sin x
0.
29
sx
sx = 0
0
2 cos x0
0,
cos x 0
3 sin x0
0
sin x 0
0,
0 .
cos 2 x sin 2 x
1.
.
s x,
tgx
x
2
;
3
arctg
2.
2
3
n, n
Z.
sin 2 x 5 sin x cos x 4 cos 2 x
0.
cos 2 x
.
.
30
1.4.5
cos x(3tgx 5)
1.
cos x
cos x
0
0
3tgx
x
5,
arctg
5
3
m, m
Z.
4 cos x sin x 2 cos x 2 sin x 1 0.
2.
( 2 cos x 1)( 2 sin x 1)
2
x
2 n, n Z ,
3
x
0.
( 1) k
1
6
k, k
0,
Z.
.
1.4
a cos x b sin x
c(a b c
0)
31
a2
a cos x b sin x
b 2 cos( x
),
a
cos
a 2 b2
,
b
sin
a2
b2 .
3 cos x sin x
1.
a
2
3;
b 1;
a2
b2
( 3) 2
12
4
2
3
2.
cos
6
2
(x
cos( x
x
x
6
6
6
6
)
2;
) 1;
2 n, n
Z
2 n, n
Z
5
2.
a 5
b 12
1
2
sin
a2
b2
12 sin x 13.
5 2 13 2
169
13
32
arctg
12
5
cos( x arctg
x
arctg
cos
12
5
sin
5
13
12
13
12
) 1
5
2 n, n
Z.
.
a cos x b sin x .
a cos x b sin x
a 2 b 2 cos( x
),
tg
b
.
a
33
2.1
34
35
(
(-
(6)
7
3 4
,
;
-
;
;
36
-
-
;
-
[28]
37
;
,
[6]
38
2.2
.
:
(
)
-
39
1
,
40
2
,
3
.
sin 0,5x 1 .
1)
4 k, k
2)
1
Z
k
4 k, k
2)
2 k, k
Z
2 cos 2x 0 .
k, k
2
4
4)
Z
4.
1)
4 k, k Z
3)
Z
3)
k, k Z
2
1)
4 k, k
7
2)
1
Z
1
7
k
k, k
4)
cos 7 x
2
Z
Z
Z
1.
1
7
3)
4 k, k
7
k, k
2
4)
2
7
4 k, k Z
7
4 k, k
7
Z
cos 2 x 1 .
k
1)
1
2)
4 k, k
k, k Z
2
Z
3)
4)
4 k, k Z
k, k
Z
41
sin1,5 x 1.
1)
4 k, k
3
2)
1
Z
k
3
3)
4 k, k
3
Z
4 k, k
3
3
4)
2 k, k
3
3)
7
8
Z
Z
tg2 x 1 .
7
1)
2)
7
8
7k , k Z
7
8
7k , k Z
7 k, k
2
7
8
4)
Z
7 k, k Z
2
.
.
sin 2 x
3
1.
2
.
1)
2)
1 12k, k Z
1
k 1
1
4
3 k, k
2
Z
cos
5
3)
1
4
4)
1
3k , k
k 1
Z
6k , k
Z
3
.
2
x
.
1)
5
6
1
5
2k , k
Z
3)
5
6
1
5
k, k
Z
42
5
6
2)
1
5
k, k
Z
4)
5
6
1
5
2k , k
Z
sin 2 x cos 2 x 0,5 0 .
.
1)
k, k
6
2
3
2)
Z
3)
2 k, k Z
4)
tg3 x
4
2
3
4 k, k
Z
2 k, k Z
6
3.
1)
4
9
8 k, k
3
Z
3)
4
9
8 k, k
3
2)
4
9
4 k, k
3
Z
4)
4
9
4 k, k Z
3
tg
4
x
Z
1.
.
1)
2)
3)
2 k, k Z
4)
k, k
2
2.
2
Z
3)
1
k
k, k Z
4)
1
k
2 k, k Z
2
k, k
2
Z
2cos x sin x
1)
1
k
2)
1
k
8
8
k, k
2
Z
2
12
2 k, k
Z
k, k
Z
2
43
.
1
3
sin 2 x
2sin 2 x
1.
.
2
4 sin x 5 tg x
2.
2
cos x 1 0 .
.
2
3.
2
2 sin x 3cos x .
cos x
.
3 tg x sin 2 x
2
4.
2 cos 2 x 8sin x 5 .
.
5.
sin 1 x
2
2 sin 1 x
2
2
5 ctg x
1
3.
.
6.
4 cos x
2
sin x
1 0.
.
1)
2)
3)
44
2.3
:
.
1.
2.
.
.
sinx = a, c sx = a, tgx = a, tgx = a.
.
.
c sx=a
c sx
,
45
sinx = a
sinx
.
1)
2)
tgx = a .
.
46
tgx = a
II
.
.
, 2)
5)
4)
,
-2,4,
;
1 =0,
,
IV
V
1 = 0;
=- ;
.
.
47
1
s x = -1
2
2
2 n, n
,
3
3
4
2 n, n
sx=1
3
2 n, n
s x =1,5
3
1
2 s x -1= 0
2 s 2x =
2
2 n,
n
3
,
3
4
2 n, n
3
2
3
5
12
2 n, n
s x =1,8
3
1
3
5
s x=
6
2
s( - x) = 1
sx= 5
2 n, n
2 n, n
2
,
3
n, n
2 n, n
3
4
2 n, n
5
3
12
n, n
5
2 n, n
3
2 n, n
48
II
.
-
-
.
-
a
49
.
sx =0
asin 2x + bsinx
sx +
s 2x =0
2
.
s 2x + b sx = ,
atg 2x + btg x =
( sx, tgx).
.
sx =
1.
5sinx + 2 = 0.
tg
x
x
+ 3 tg = 4.
2
2
tg
5t + 2 = 0.
2.
2sinx
x
= z,
2
z+
3
= 4.
z
:
s5x
s5x = 0;
s5x (2sinx
1) = 0.
s5x = 0,
2sinx
3.
50
a sinx + b
sx +
sx
= 0.
b sinx
sx +
sx =
4sinx + 3 sx = 5.
sinx = (2tg x/2) / (1 +tg2x/2);
sx = (1- tg2x/2) / (1+tg2x/2);
4sinx+3 sx =5
/5 sx =1
(4/5)2 +(3/5)2
3/5=
sx = 1
s(xx-
s3/5,
6sin2x + 2sin22x = 5
a) 2 sin4x s2x = 4 s32x
3 s2x
s6x + s2x =
s6x
51
III
1)
2)
3)
4)
5)
4
I
sinx
sx
x
/2 +
sx +
2 sinx
1
2
3
4
5
1
2
3
4
5
sx
1=0
sx = 1
s3x = 0
s5x
s5x = 0
5
2sinx sx
sinx = 0
3
s2x = 1
sx = 1
sin3x = sin17x
52
1
1
2
2
3
4
5
+
1
2
3
1
+
+
4
5
+
3
+
+
4
5
+
2
3
4
+
+
5
+
.
6
x = (-1)nar
2.
3. tg x = a
4. tg x = a
1. sinx = 0
2. sinx = 1
x = ar
x = ar
3. sinx = 1
4.
sx = 0
5.
sx = 1
6.
sx = 1
x=
2
x=
x=
2
2
53
V
sin x = 0
sx=1
tg x = 1
s x = 1/2
sin x = -1/2
sin x = - /2
sin x = /2
tg x = 0
sin x = 1
s x = /2
sin x = 1/2
sin x = -1
s x = -1/2
sin x = /2
tg x = -
sx
1 =0
tg x = /3
2 sin x 1 =0
VI
2
s x = /2
tg x = - /3
tg x = 2 sin x + 1 =0
.
.
cos 2 x
1.
1.
sin 2 x
3
2
1.
tg2 x
3
3
3.
.
sin x
4
1.
1.
2
tg
2.
x
0.
2cos3 x sin 3 x
3.
3.
2
.
2sin 2 x
1
3
sin 2 x
2
4 sin x 5 tg x
3
cos 2 x
4
3 tg x sin 2 x
2
2
cos x 1 0 .
2 sin 2 x 3cos x .
2
2 cos x 8sin x 5 .
54
.
1
3.
sin 2 x
2sin 2 x
2sin 2 2 x 3sin 2 x 1 0 ,
1)
sin 2 x 0.
sin 2 x 1
2x
sin 2 x 1
sin 2 x
x
k, k
2
k
1
12
4
k;
1
2
sin 2 x
2 k, k
1
2
1
2
Z
2x
x
1
k
6
4
k, k Z .
k, k Z
Z.
k
12
k, k
2
Z.
2
4 sin x 5 tg x
arcsin 1
4
k, k
2
cos x 1 0 .
Z.
.
2
cos x
1) 4 sin 2 x 5 sin x
cos x
2)
cos x ,
sin x
,
cos x
tg x
1 0, cos x
0.
0
sin x
1
sin x
1
4
arcsin 1
4
sin x
k, k
1
4
x
arcsin 1
4
k, k
Z.
Z.
55
2
3.
2
cos x 2 sin x 3cos x .
2
cos x
2
cos x
2cos x 3cos x 1 0
cos x
1
cos x
1
2
2 k;
3
2
3
1
n
arcsin 3
5
cos x
3cos x ,
1
2
2 k, k Z .
Z.
3 tg x sin 2 x
2
:
1
2
1 cos x
2 k, k Z .
x
2 k, k
4.
5sin x
x
2
2 cos 2 x 8sin x 5 .
n, n Z .
3 sin x 2sin x cos x
2 cos x
2 cos x 8sin x 5 ,
3sin 2 x
2 cos 2 x 8sin x 5 ,
2
cos x 0 .
8sin x 3 0
2)
:
sin x 1 ,
sin x
cos x 0
3,
5
1
x
n
arcsin 3
5
1
cos x 0 .
n
arcsin 3
5
n, n Z .
n, n Z .
VII
.
56
.
,
,
,
.
.
.
.
1.
4
sinx
5
57
3.
6
.
7
.
.
58
7
-1
0
1
.
1)
2)
59
3)
,
8
sin2x =
.
sin
= cosx
sin2x = cosx.
60
sin2x=2sinxcosx
2sinxcosx = cosx,
2sinxcosx
cosx = 0,
cosx (2sinx
1) = 0,
cosx = 0
x=
2sinx = 1,
n
.
n,
.
1)
2)
1.
.
.
,
,
,
.
61
,
.
,
:
.
n.
n=0
x=0
n=1
x=
n=2
x=2
n.
n=0:
n = 1:
n=2:
,
.
2)
.
.
3)
.
62
IV
.
.
,
,
,
.
1)
,
2)
, [
3)
;
;
; [-3; 3].
63
1)
:
.
,
,
.
.
, D=144.
.
.
.
0
:
1
:
2
:
.
64
2)
.
,
.
.
2
3
4
5
.
3)
.
.
65
.
1
0
1
3;
.
4.
66
.
2 sin 2 x sin x 1 0 ;
6 sin 2 x 7 sin x 2
0;
2tg 2 x 3tgx 1 0 ;
4
s2x
3 s2 x
4sinx = 0;
10 s x + 7 = 0;
6 s2 x + 7sin x
1.
1 = 0.
sin 2 x
3
1 0;
2.
;
;
3
;
.
67
x
2
2 sin
cos
2
tg (6 x
1;
6
4
3
)
1 0;
0;
;
*
*
3sin22x + 7 s2x
s2x + 3sinx
3 = 0;
sx 4sin2x = 0.
-
.
68
-
;
.
,
3
.
69
1.
2.
,
sin 0,5x
1
1)
2)
4 k, k
2
1
k
Z
4)
Z
4 k, k
Z
cos 1,5 x 1 .
1)
4 k, k
3
2)
1
Z
k
3
3)
4 k, k
3
Z
cos x
3
3.
2)
4 k, k Z
3)
4 k, k
2
1)
1.
5 6k , k
1
k
1
2
Z
6k , k Z
4)
4 k, k
3
3
2 k, k
3
Z
Z
3.
2
3)
5
2
4)
1
6k , k
k
Z
6k , k
Z
70
tg3 x
4
1.
3.
1)
4
9
8 k, k
3
Z
3)
4
9
8 k, k
3
2)
4
9
4 k, k
3
Z
4)
4
9
4 k, k Z
3
tg
2.
1)
2)
4
x
1.
2 k, k Z
2
k, k
2
Z
2cos x sin x
3.
1)
1
k
2)
1
k
8
8
2
3)
2 k, k Z
4)
k, k Z
2.
2
2
Z
3)
1
k
k, k Z
4)
1
k
2 sin 1 x
2
1
k, k
Z
2
12
2 k, k
Z
k, k
Z
2
.
1.
sin 1 x
2
2
4 cos x
5 ctg x
3.
2
sin x
1 0.
71
72
-
,
.
1.
2.
.
-11
-11
73
11.
4.
1.
2.
3.
.
4.
.
5.
6.
.
a cos x b sin x
c( a b c
0) .
5.
74
6.
10
.
.
75
1.
2001.
-22.
2.
8-32.
3.
-42.
4.
. 2005.
. 23-27.
5.
10
.
6.
7.
//
.10-15.
8.
.
-
9.
.:
10.
.
B.
.12 -18.
11.
-77.
12.
13.
76
14.
-
.
15.
1947.
.18-25.
16.
. //
. 1955
1
. 26-
30.
17.
9
18.
.
. . 26-28.
19.
.
. 68-70.
20.
.
. ...
.
.
-
, 1996.
21.
-
-
22.
.
:
23.
24.
:
25.
(
) //
.
-42.
77
26.
27.
-
-
28.
29.
17-
-39.
30.
17-
-39
31.
. //
2009.
ati n and S ien
. 2009.
32.
-
ati n and S ien
-20
33.
-
78
-244.
34.
//
.-
92
35.
//
2839.
36.
-
.
37.
-33.
38.
.
-27.
39.
asinx+b sx= //
40.
41.
2
2009.
42.
.,
.
43.
, 1987.
44.
-18.
79
45.
46.
-
.
47.
1981.
48.
49.
//
2006.
50.
51.
.
52.
.
53.
-
, 1995.
54.
55.
56.
4-
.
57.
.,
58.
80
59.
60.
//
. 1999.
61.
1972.
62.
:
63.
:
.
:
1.
64.
.
65.
-
66.
-
67.
68.
-26.
69.
-
-13.
70.
, 2002.
.
71.
72.
-59.
81
73.
-24.
74.
. ...
.
1967.
75.
76.
-
77.
VIII
.
.
78.
//
-10
, 1976
79.
.
.57.
.,
-
,
1975.
80.
./
82
1.
du ti in analysis infinit
83
sin 2 x
cos 2 x 1
84
sin
sin x
sin x
x
cos x 1
x3
3!
x2
2!
x5
5!
x4
4!
x7
7!
x6
6!
1
2
...
n
, tgn
cos x
...
...
85
-
86
I
sinx
sx
x
/2 +
sx +
2 sinx
1
2
3
4
5
1
2
3
4
5
sx
1=0
sx = 1
s3x = 0
s5x
s5x = 0
2sinx sx
sinx = 0
3
s2x = 1
sx = 1
sin3x = sin17x
87
1
1
2
2
3
5
+
1
2
1
+
+
4
+
4
5
4
5
+
3
+
3
+
2
3
5
4
+
+
+
88
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