.

sin e cos noiiu e ?aciinoe aeaoo a?aoiaioia

sin(((()=sin ((cos((sin((cos(

cos(((()=cos((cos((+sin ( (sin(

tg ( ( tg (

tg (((() = 1 ( tg ( ( tg (

tg (((() = ctg ( ( ctg ((+ 1 =

ctg ( ( ctg (

= 1 ( tg ( ( tg (

tg ( ( tg (

O?eaiiiiao?e/aneea ooieoeee aeaieiiai a?aoiaioa

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 cos3 x – 3 cos

AAAEII: ciae ia?aae ei?iai caaeneo io oiai, aaea iao-ny oaie 1/2 x:

sin 1/2 x= ( 1-cosx

2

cos 1/2 x= ( 1+cosx

2

NB! Neaaeothuea oi?ioeu ni?aaaaeeeau i?e ciaiaiaoaea ( 0 e nouanoaiaaiey
ooieoeee, aoiaeyueo a yoe oi?ioeu (tg, ctg)

tg 1/2 x=sinx =1-cosx =( 1-cosx

1+cosx sinx 1+cosx

ntg1/2 x=sinx =1+cosx =(1+cosx

1-cosx sinx 1-cosx

Oi?ioeu iiieaeaiey noaiaie:

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

I?aia?aciaaiea i?iecaaaeaiey aeaoo ooieoeee a noiio:

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! Auoaia?a/eneaiiua oi?ioeu ni?aaaaeeeau i?e ciaiaiaoaea ( 0 e
nouanoaiaaiey ooieoeee, aoiaeyueo a yoe oi?ioeu (tg, ctg)

sinx ( siny= 2sin x(y cos x(+ y

2 2

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

2 2

cosx – cosy = – 2sin x+y sin x-y

2 2

tgx ( tgy= sin(x(y)

cosx cosy

tgx + tgy= cos(x-y)

cosx siny

ctgx – ctgy= cos(x+y)

sinx cosy

ctgx(ctgy= sin(y(x)

sinx siny

sin x = 1 x= 1/2 ( +2(n, n( Z

sin x = 0 x= (n, n( Z

sin x = -1 x= – 1/2 ( +2(n, n( Z

sin x = a , (a(( 1

x = (-1)karcsin a + (k, k( Z

tg x= a x= arctg a +(n, n( Z

cos=1 x=2(n, n( Z

cosx=0 x= 1/2 ( +(n, n( Z

cosx=-1 x=( +2(n, n( Z

cosx = a , (a(( 1

x=arccos a + 2(n, n( Z

ctg x = a x=arcctg a + (n, n( Z

Ciaee o?eaiiiiao?e/aneeo ooieoeee a /aoaa?oyo:

?\f(() sin cos tg ctg

I + + + +

II + ( ( (

III ( ( + +

IY ( + ( +

Oi?ioeu i?eaaaeaiey

– ( (/2 ( ( ( ( ( 3/2 ( ( ( 2( – (

sin ( sin ( cos ( (+sin ( – cos ( – sin (

cos cos ( (+sin ( – cos ( ( sin ( cos (

tg – tg ( (+ ctg ( ( tg ( (+ ctg ( – tg (

ctg – ctg ( (+ tg ( ( ctg ( (+ tg ( – ctg (

Cia/aiey o?eaiiiiao?e/aneeo

ooieoeee iniiaiuo oaeia:

\oaie

f\ 0 30(=(

6 45(=(

4 60(=(

3 90(=(

2

180(=( 270(=3(

2

sin 0 1/2 (2 / 2 (3 / 2 1 0 – 1

cos 1 (3 / 2 (2 / 2 1/2 0 (1 0

tg 0 (3 / 3 1 (3 ( 0 (

ctg – (3 1 (3 / 3 0 ( 0

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09 Ноя 2009
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