. , , ,

,,,

() —

sin cos

sin(ab)=sin acosbsinbcosa

cos(ab)=cosacosb`+sin a sinb

tg atg b

tg (ab) = 1 tg a tg b

tg (ab) =

=ctg a ctg b`+ 1 =1 tg a tg b

ctg bctg a tg atg b

sin2x=2sinx cosx

cos 2x = cos2x - sin2x=

= 2cos2x-1=1-2sin2x

tg2x= 2 tgx

1 - tg2x

sin 3x =3sin x - 4 sin3x

cos 3x= 4 cos3x - 3 cos

: , - ½ x:

sin ½ x= 1-cosx

2

cos ½ x= 1+cosx

2

NB! ¹ 0 , (tg, ctg)

tg ½ x=sinx =1-cosx = 1-cosx

1+cosx sinx 1+cosx

tg½ x=sinx =1+cosx = 1+cosx

1-cosx sinx 1-cosx

:

sin2 x = 1 cos 2x

2

cos2 x = 1+ cos 2x

2

sin3 x = 3 sin x sin 3x

4

cos3 x = 3 cos x + cos 3x

4

:

2 sinx siny = cos(x-y) cos(x+y)

2 cosx cosy = cos(x-y)+cos(x+y)

2 sinx cosy = sin(x-y) + sin (x+y)

tgx tgy = tgx + tgy

ctgx + ctgy

ctgx ctgy = ctgx + ctgy

tgx + tgy

tgx ctgy = tgx + ctgy

ctgx + tgy

NB! ¹ 0 , (tg, ctg)

sinx siny= 2sinxy cosx`+ y

2 2

cosx + cosy =2cos x+y cosx-y

2 2

cosx - cosy = - 2sin x+y sinx-y

2 2

tgx tgy= sin(xy)

cosx cosy

tgx + tgy= cos(x-y)

cosx siny

ctgx- tgy= cos(x+y)

sinx cosy

ctgxctgy= sin(yx)

sinx siny

sin x = 1 x= ½ p +2pn, nÎ Z

sin x = 0 x= pn, nÎ Z

sin x = -1 x= - ½ p +2pn, nÎ Z

sin x = a , [a]≤ 1

x = (-1)karcsin a + pk, kÎ Z

cosx=1 x=2pn, nÎ Z

cosx=0 x= ½ p +pn, nÎ Z

cosx= -1 x=p +2pn, nÎ Z

cosx= -½ x=2/3 p +2pn, nÎ Z

cosx = a , [a]≤ 1

x=arccos a + 2pn, nÎ Z

arccos(-x)= p- arccos x

arcctg(-x)= p - ctg x

tg x= 0 x= n, nÎ Z

ctg x= 0 x=½ p+ p n, nÎ Z

tg x= a x= arctg a +pn, nÎ Z

ctg x = a x=arcctg a + pn, nÎ Z

:

f(a)

sin

cos

tg

ctg

I

+

+

+

+

II

+

-

-

-

III

-

-

+

+

IY

-

+

-

+

a =p × a/180; a=a× 180/p

ïðèâåäåíèÿ

a

p/2 a

p a

3/2 p a

2p a

sin

-sin a

cos a

`+sin a

- cos a

- sin a

cos

cos a

`+sin a

- cos a

sin a

cos a

tg

- tg a

`+ctg a

tg a

`+ctg a

- tg a

ctg

- ctg a

`+tg a

ctg a

`+ tg a

-ctg a

:

0

30

45

60

90

180

270

p / 6

p /4

p /3

p /2

p

3p/2

sin

0

½

Ö2 / 2

Ö3 / 2

1

0

1

cos

1

Ö3 / 2

Ö2 / 2

½

0

-1

0

tg

0

Ö3 / 3

1

Ö3

-

0

-

ctg

Ö3

1

Ö3 / 3

0

-

0


sin cos sin(ab)=sin acosbsinbcosa cos(ab)=cosacosb`+sin a sinb tg atg b tg (ab) = 1 tg a tg b tg (ab) = =ct

 

 

 

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