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, , , , . -, .

1. : arcsin(x2 4x+ 4) = pi2.

. arcsin a = pi/2, a = 1. ,

x2 4x+ 4 = 1 x2 4x+ 3 = 0 x1 = 1, x2 = 3.

: 1, 3.

2. : 12 arctg2 x pi arctg x pi2 = 0.. t = arctg x, t:

12t2 pit pi2 = 0 t1 = pi3, t2 = pi

4.

: arctg x = pi3 ,arctg x = pi

4

[x =3 ,

x = 1.

:3, 1.

3. : arcsin2 x 2 arcsinx 3 = 0.. t = arcsinx:

t2 2t 3 = 0 t1 = 3, t2 = 1.

arcsinx = 3. : x = sin 3 ! ,

[pi2; pi2

], 3 ( 3 > pi

2).

arcsinx = 1. 1 -: 1 [pi

2; pi2

], x = sin(1) = sin 1 .

: sin 1. 4. : arccos2 x arccosx 2 = 0.. t = arccosx:

t2 t 2 = 0 t1 = 2 , t2 = 1.

arccosx = 2 . 2 -: 2 [0; pi], x = cos 2.

: arccosx = 1. , 1 / [0; pi].: cos 2.

1

5. : arccosx = arctg x.

. [0; pi]. -

(pi2; pi2

). ,

[0; pi

2

). x [1; 1].

[0; pi

2

) , .

:

cos(arccosx) = cos(arctg x).

: cos(arccosx) = x. (, [0; pi

2

) ):

cos(arctg x) =

1

1 + tg2(arctg x)=

1

1 + x2.

:

x =

1

1 + x2,

( -, , x > 0).

:

512

.

6. : arcsinx = 2arctg x.

. x [1; 1], arcsinx [pi2; pi2

], arctg x

[pi4; pi4

].

[pi

2; pi2

].

[pi

2; pi2

] :

sin(arcsinx) = sin(2 arctg x).

: sin(arcsinx) = x. -:

sin(2 arctg x) =2 tg(arctg x)

1 + tg2(arctg x)=

2x

1 + x2.

:

x =2x

1 + x2,

.

: 0, 1.

7. : sin(3 arccos x) =1

2.

. : 3 arccosx =pi

6+ 2pin,

3 arccosx =5pi

6+ 2pin

arccosx =pi

18+

2pin

3=pi(1 + 12n)

18,

arccosx =5pi

18+

2pin

3=pi(5 + 12n)

18(n Z).

2

0 6 arccosx 6 pi pi18

13pi18

, 5pi

18 17pi

18:

arccosx =pi

18,

arccosx =13pi

18,

arccosx =5pi

18,

arccosx =17pi

18.

:

x1 = cospi

18, x2 = cos

13pi

18, x3 = cos

5pi

18, x4 = cos

17pi

18.

, x4 = x1 x2 = x3.: cos pi

18, cos 5pi

18.

8. : arcsin2 x+ arccos2 x =5pi2

36.

. t = arcsinx, arccosx = pi2 t. :

t2 +(pi2 t)2

=5pi2

36 2t2 pit+ pi

2

9= 0.

: t1 = pi3 , t2 =pi6, x1 =

32, x2 = 12 .

:3

2,1

2.

3