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.
= 180
. ( , . .)
- ., 1- .
. (.- .)
, . 1 , .
.: 1. 1 . ,
- .
2.
| | 3-, | | .
.

   
Ѡ Ġ
Ġ Ѡ
Ð
Ð
. . (1)
Ð
Ð
. . (2)
Ð Ð . (3)
1. . 2-
. . Ð =, .
2.
2- ,ð| |.
-
() (b)
- . Ð1=Ð2
Ð1=Ð3 ()ðÐ3=Ð2.
Ð2
Ð3-.ð
1 a | | bn
3. . 2-
, . . Ð=180,
| |n

1-3 .
4. 2 . 3-
,
. Ð=, -
.Ð=, .Ð=180.
- - Ð90.
1.
. ^ , 1.
2.

) ^
1.
3.
^
3- .
4. ^ 1- | | , ^
.
(n-)
. -
. (R- ., r-
.)
R = a / 2sin(180/n); r = a / 2 tg (180)
NB! 1. 3 Ñ
. 1 ().
2.
3 . 1 ( ) - . 2:1
(. ).
3.
3 . Ñ
. 1 -

. .
4. 3 ^,
Ñ,
. 1 - . .
5.
| | = ½
H(. . a) = 2√p(p-a)(p-b)(p-c)

a
M( a) = ½
√ 2b2+2c2 -a2
B (--)= 2√ bcp(p-a) / b+c
p -
a²=b²+c²-2bx, - 1-
Ñ: 2Ñ=,
= .
1. 2
Ð
.
2. 2 Ð
.
3. 2 Ð
, . 1- Ð
4.

5. 2 Ð ,
.
Ñ C=90 a²+b²=c²
NB! TgA= a/b;
tgB =b/a;
sinA=cosB=a/c; sinB=cosA=b/c

Ñ H= √3
* a/2
S Ñ= ½
h a =½ a b sin C

d²+d`²=2a²+ 2b²
S =h a=a b sinA(
b)
= ½ d d` sinB ( d d`)

S= (a+b) h/2 =½uvsinZ= Mh
S=a h =a²sinA= ½ d d`
L= pRn
/ 180,n-Ð
..Ð= ½ L , L-, Ð
S(c)= ½ R²a= pR²n / 360
..

``b=|`a| |`b| cos (`a Ù`b),

|`a| |`b| -

|`a|{x`; y`} |`b|{x``; y``},
-, =
|`a| |`b| = x` ×
y` + x`` × y``

1. .

2.
(^)
3. .
- (^)
4.
( `` 1
. ``=k OX, k>0
- - . .
5.
( . )
6.
7.
- - . , :
-

-
Ï
ð` ,
` Î
a,
a^, Ð` = j= const, - . a .

2- = .
8. e. (x,y,z)ð(x+a,y=b,x=c)
9.
- . - k
=1 - .
-
.
1. Î(); A`B`C` Î(a`)
2. (p) ð (p`); [p)ð[p`); aða`; ÐAðÐA`
3. -

NB! S` = k² S``;
V ` = k 3 V ``
.
. , Ï .-. a , | | .-. , Î
a, | | a
. () | | (b), () (b) ,
.| | () (b)
T. ( . 2- .). 2 . 1- a | | .
b,
a
| | b.
. 2 . - . 3-,
| |.
.
- | |
1.
. . ,
2- , =.
.
^ -.
, - -, ^ 2- - , - ^.
. 2 ^ - | |.
. 1 2- . ^, ^ .
.
^ 2- -.
- ^ . -, ^ -.
[a)^ b,[a) Îa,a Èb= (p).-: a ^ b
-.
[a)^ b=.
(b) , (b)^(p). (a)Ù(b) - Ð
a b.
[a)^ bð(a)^(b)ð (a)Ù(b)=90ða ^ bn
. 2 - ^,
1- -
^
. -, ^ 2- -.
.
3- ^..
, , - -,, ^ , -
, ^ .

. V = S × a -
a - , S -
S ^-
V = S × -
V = S. - + 2S.

. = , - .
V=h S. ; V.- = abc
S=2(ab+ac+bc)

V= 1/3 * S . S=S Ñ.

V=pR²H; S= 2pR (R+H)
V= 1/3 * S = 1/3 * pR²H
S= S+ S= pR (r + L); L-
S= 4pR²
= 4/3 pR3
sin cos
sin(ab)=sin acosbsinbcosa

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

tg (ab) =
= ctg
a ctg b`+ 1 = 1 tg a tg b
ctg b ctg a tg a tg 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 cos3
x - 3 cos
:
, -
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= 2sin xy 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(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
. . . . . . . . . . . . . . .

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