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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