. . . ( ), , , ', , , , . : . : , , , . . , , . . . sin x = a, cosx = a, tgx = a, ctgx = a. , , , . , . 1

2

2x 2sin

.. 3

. , , . 4

. 1. , 10 , , 5 ( ).

cos x = a ?

cos x = a ? cos x = 0 ? cos x = 1 ? cos x = -1 ? arccos ? arccos (-) ?

6

sin x = a ?

sin x = a ? sin x = 0 ? sin x = 1 ? sin x = -1 ?

os

arcsin ? arcsin (-) ?

C 7

tg x = a ?

x= cos sin os x

tg x = 0 ? arctg ? arctg (-) ?

8

tg x = a ?

ctg x = 0 ? arctg ? arctg (-) ?

9

x=

arcsin x, arccos x, arcctg x, arctg x. 10

2.

11

0 12

arcsin xarcsin 450 = 2 2

1 2 arccos = 3 3 2

0

arctg 3 = arctg1 3 =

arctg 1 = arctg 4 4

arccos x)

3 3 = 4 4

5 arcctg 3 = 66

(

3.

13

) sin x =

1 ; 2 x = (1) n + n, n Z . 6

) cos x =

x=

+ 2n, n Z . 6

3 ; 2

) 2osx 1 = 0;x=

)tg 2 x = 1;x=

+ 2n, n Z . 4

n + , n Z. 8 2

. , , 14

3 sin 2 x 5 sin x 2 = 03 cos 2 x 5 cos x = 1

.

1. 2 sin2x + sinx 1 = 0 , , . 15

1+ 3 1 = , 4 2 1 3 t2 = = 1 . 4 t1 =

16

= (1) n arcsin x = (1) n

+ n, n Z . 6

1 + n, n Z , 2

=

+ 2k , k Z . 2

: (1) n

+ n, n Z , 6

2. 6sin2x + 5cosx 2 = 0

, , .

1) sin x =

+ 2k , k Z . 2

17

2 18= 2 + 2n, n Z . 3

:

2 + 2n, n Z . 3

3. tg x + 2 tg x = 3.

? . ? .

, ,

, os x s 2 6 ( 1 cos

2

19

1 , tgx

tgx +

2 = 3. tgxt+ 2 = 3, t(t 0)

20

1)tgx = 2, x = arctg 2 + n, n Z .2)tgx = 1, x = arctg1 + k , k Z , x=

+ k , k Z . 4

: arctg 2 + n, n Z ,

+ k , k Z . 4

21

2 3t + t t1 = 2,

IV. . 22

cos 2 x = 7 8 sin x 2

2 cos 3 x + sin( 3 x) 1 = 0 2 2 (tgx + ctgx) + 3(tgx + ctgx) = 42 cos 2 x 5 cos( x) + 2 = 0

1.

cos2x = 7 8sinx

. os2x = 1 2sin2 x, 1 2sin2 x = 7 8sinx, 1 2sin2 x 7 + 8sinx = 0, 2sin2 x + 8sinx 6 = 0, sin2 x 4sinx + 3 = 0, sin x = t, t2 4t + 3 = 0, t1 = 1, t2 = 3. sin x = 1, x = /2 + 2k, k Z,

1. 2.sin x = 3, .

. /2 + 2k, k Z.2.

2os23x + sin( 2 3x) 1 = 0

. sin( 2 3x) = os3x, 2os23x + os3x 1 = 0, cos3x = t, 2t2 + t 1 = 0, D = 1 + 8 = 9, t1 =1 3 4

= 1,

t2 =

1 + 3 4

=

1 2

,1 2

cos3x = 1, 3x = + 2k, k Z, x=3

cos3x =

,1

3x = arccos 2 + 2n, n Z, 3x = 3 + 2n, n Z, x=9 +2 n 3

+

2 k 3

, k Z,

, n Z.

.3.

3

+

2 k 3

, k Z; 9 +

2 n 3

, n Z.

(tgx + ctgx)2 + 3(tgx + ctgx) = 4

. tgx + ctgx = t, t2 + 3t 4 = 0, t1 = 4, t2 = 1, tgx + ctgx = 4, tgx + ctgx = 1,

tgx +

1 tg x

+ 4 = 0,

tgx + tgx = z

1 tg x

1 = 0,

tgx = y +1

+4=0 0,

z+

1 z

1=0 = 0,

2 + 4 +1 =

z 2 z +1 z

0

z0

2 + 4 +1 = 0, D = 16 4 = 12, y1 = 4 2 y2 = 4 + 212

z2 z + 1 = 0, D= 1 4 = 3 < 0, = 2 = 2 +3

= =3

4 2 3 2 4 +2 3 2

12

3

tgx = 2

tgx = 2 +3

3 3

x = arctg(2 x = arctg(2 +

) + n, n Z, ) + n, n Z.3

x = arctg(2 +

) + k, k Z,

3

. arctg(2 +

) + k, k Z, arctg(2 +

3

) + n, n Z.

4. 2 cos2 x 5cos( x) + 2 = 0 . 2 cos2 x 5cos( x) + 2 = 0 cos( x) = osx, 2 cos2 x + 5cosx + 2 = 0, cos x = t,

2t2 + 5t + 2 = 0, D = 25 16 = 9, t1 =5 3 4

= 2,

t2 =

5 +3 4

=1 2

1 2

,

cos x = 2

cos x =

x = arccos( 2 ) + 2n, n Z, x = ( x=2 3

1

3

) + 2n, n Z,

+ 2n, n Z.

.

2 3

+ 2n, n Z.

5. cos 2 +sinx +sin = 0,25 . cos sinx +sinx +sin 0,25 = 0, 1 sinx +sin 0,25 = 0, 4sinx 4sin 3 = 0, sin x = t, 4t 4t 3=0, D = 16 + 48 = 64, t1 = 1/2, sin = 1/21 x = (1) n arcsin( ) + n, n Z 2

t2=3/2 sin x = 3/2

x = (1) ( arcsin

n

1 ) + n, n Z 2

x = ( 1) n +1

6

+ n, n Zx = ( 1) n +1

.

6

+ n, n Z

.

V. 23

3-4 , . 3sinx + 2cos x 2 = 0 .2k , k Z ;( arccos 1 ) + 2n, n Z 3

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

6

+n, n Z ;

2

+ 2k , k Z

2sinx cos x 1= 0 + 2k , k Z ; 3 + 2n, n Z

() ()

. V tg x 2 ctg x + 1 = 0 .4 +k , k Z ;arctg 2 +n, n Z

()

()

V cos 2x sin x = 0 .

2

+ 2k , k Z ; ( 1) n

6

+ n, n Z

()

V tg x + 5 ctg x = 6 .4 + n, n Z ; arctg 5 + k , k Z

()

24

+ n, n Z ; arctg 5 + k , k Z 4 + 2k , k Z ; (1) n + n, n Z 2 6 1 2k , k Z ;( arccos ) + 2n, n Z 3

+ k , k Z ;arctg 2 + n, n Z 4

( 1) n +1

+ n, n Z ; + 2k , k Z 6 2 + 2k , k Z ; + 2n, n Z2k , k Z ; (1) n

3 + n, n Z 6

+ 2k , k Z ;

2 + 2n, n Z 3

25

-

V. 26

. : , , - . , , . 27

V. 28

29