:

:

= 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. (Ï ) ^ 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) = 2p(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

ARCSIN a

-p/2£arcsin a £p/2 sin(arcsin a)=a

arcsin (-a)= -arcsin a

a 0 1/2 Ö2/2 Ö3/2 1
arcsin a 0 p/6 p/4 p/3 p/2

SIN X= A

x=(-1)n arcsin a +pk

sin x=0 x=pk
sin x=1 x=p/2+2pk
sin x=-1 x=-p/2+2pk

ARCCOS a

0 £arccos a £p cos(arccos a)=a

arccos (-a)=p -arccos a

a 0 1/2 Ö2/2 Ö3/2 1
arccos a p/2 p/3 p/4 p/6 0

COS X= A

x= arccos a +2pk

cos x=0 x=p/2+pk
cos x=1 x=2pk
cos x=-1 x=p+2pk

ARCTG a

-p/2£arctg a £p/2 tg(arctg a)=a

arctg (-a)= -arctg a

a 0 Ö3/3 1 Ö3
tg a 0 p/6 p/4 p/3

TG X= A

x= arctg a +pk

sina*cosb=1/2[sin(a-b)+sin(a+b)]

sina*sinb=1/2[cos(a-b)-cos(a+b)]

cosa*cosb=1/2[cos(a-b)+cos(a+b)]

sina*cosb=1/2[sin(a-b)+sin(a+b)]

sina*sinb=1/2[cos(a-b)-cos(a+b)]

cosa*cosb=1/2[cos(a-b)+cos(a+b)]

sina+sinb=2sin(a+b)/2 * cos(a-b)/2

sina-sinb=2sin(a-b)/2 * cos(a+b)/2

cosa+cosb=2cos(a+b)/2 * cos(a-b)/2

cosa-cosb=-2sin(a+b)/2 * sin(a-b)/2

(a+b)2=a2+2ab+b2

(a-b)2=a2+2ab+b2

(a+b+c)2=a2+b2+c2+2ab+2ac+2bc

a2-b2=(a-b)(a+b)

(a+b)3=a3+3a2b+3ab2+b3

(a-b)3=a3-3a2b+3ab2-b3

a3+b3=(a+b)(a2-ab+b2)

a3-b3=(a-b)(a2+ab+ b2)

0 p/6 p/4 p/3 p/2 p 2/3p 3/4p 5/6p 3/2p

0 30 45 60 90 180 120 135 150 270
sin 0 1/2 Ö2/2 Ö3/2 1 0 Ö3/2 Ö2/2 1/2 -1
cos 1 Ö3/2 Ö2/2 1/2 0 -1 -1/2 -Ö2/2 -Ö3/2 0
tg 0 1/Ö3 1 Ö3 - 0 -Ö3 -1 -1/Ö3 -
ctg - Ö3 1 1/Ö3 0 - -1/Ö3 -1 -Ö3 0

sin2+cos2=1 sin=Ö1-cos2 sin(-a)=-sina tg(-a)=-tga

tgctg=1 cos=Ö1-sin2 cos(-a)=cosa ctg(-g)=-ctga

tg=1/ctg ctg=1/tg 1+tg2=1/cos2=sec2

sin2=(1-cos)(1+cos) 1+ctg2=1/sin2=cosec2 sin2a=2sinacosa

cos2=(1-sin)(1+sin) 1-tg2/(1+tg2)=cos4-sin4 cos2a=cos2 a-sin2 a

cos/(1-sin)=1+sin/cos 1/(tg+ctg)=sincos tg2a=2tga/1-tga

cos(a+b)=cosacosb-sinasinb sin3a=3sina-4sin3a

cos(a-b)=cosacosb+sinasinb cos3a=4cos3a-3cosa

sin(a+b)=sinacosb+cosasinb tg(a+b)=tga+tgb

sin(a-b)=sinacosb-cosasinb 1-tgatgb

2cos2a/2=1+cosa 2sin2a/2=1-cosa

0 p/6 p/4 p/3 p/2 p 2/3p 3/4p 5/6p 3/2p
0 30 45 60 90 180 120 135 150 270
sin 0 1/2 Ö2/2 Ö3/2 1 0 Ö3/2 Ö2/2 1/2 -1

2cos2a/2=1+cosa

2sin2a/2=1-cosa

 
cos

1 Ö3/2 Ö2/2 1/2 0 -1 -1/2 -Ö2/2 -Ö3/2 0
tg 0 1/Ö3 1 Ö3 - 0 -Ö3 -1 -1/Ö3 -
ctg - Ö3 1 1/Ö3 0 - -1/Ö3 -1 -Ö3 0

sin2+cos2=1 sin=Ö1-cos2 sin(-a)=-sina tg(-a)=-tga

tgctg=1 cos=Ö1-sin2 cos(-a)=cosa ctg(-g)=-ctga

tg=1/ctg ctg=1/tg 1+tg2=1/cos2=sec2

sin2=(1-cos)(1+cos) 1+ctg2=1/sin2=cosec2 sin2a=2sinacosa

cos2=(1-sin)(1+sin) 1-tg2/(1+tg2)=cos4-sin4 cos2a=cos2 a-sin2 a

cos/(1-sin)=1+sin/cos 1/(tg+ctg)=sincos tg2a=2tga/1-tga

cos(a+b)=cosacosb-sinasinb sin3a=3sina-4sin3a

cos(a-b)=cosacosb+sinasinb cos3a=4cos3a-3cosa

sin(a+b)=sinacosb+cosasinb tg(a+b)=tga+tgb

sin(a-b)=sinacosb-cosasinb 1-tgatgb

sin(2p-a)=-sina sin(3p/2-a)=-cosa

cos(2p-a)=cosa cos(3p/2-a)=-sina

tg(2p-a)=-tga tg(3p/2-a)=ctga

sin(p-a)=sina ctg(3p/2-a)=tga

cos(p-a)=-cosa sin(3p/2+a)=-cosa

sin(p+a)=-sina cos(3p/2+a)=sina

cos(p+a)=-cosa tg(p/2+a)=-ctga

sin(p/2-a)=cosa ctg(p/2+a)=-tga

cos(p/2-a)=sina sina+sinb=2sin(a+b)/2cos(a-b)[Ñ.Ê.Â.1] /2

tg(p/2-a)=ctga sina-sinb=2sin(a-b)/2*cos(a+b)[Ñ.Ê.Â.2] /2

ctg(p/2-a)=tga cosa+cosb=2cos(a+b)/2cos(a-b)/2

sin(p/2+a)=cosa cosa-cosb=-2sin(a+b)/2sin(a-b)/2

cos(p/2+a)=-sina

Y = S I N x

1).Ԡ D(y)=R 2).Ǡ E(y)=[-1;1]

3). 2p

4).; sin (-x)=-sin x

5). [-p/2+2pk;p/2+2pk], kÎZ

[p/2+2pk;3p/2+2pk], kÎZ

6). =1 =p/2+2pk, kÎZ

=-1 =-p/2+2pk, kÎZ

7). =pk, kÎZ

8).MAX =1 =p/2+2pk, kÎZ

MIN =-1 =-p/2+p+2pk, kÎZ

9).x>0 [2pk;p+2pk], kÎZ

x<0 [p+2pk;2p+2pk], kÎZ

Y = C O S x

1).Ԡ D(y)=R 2).Ǡ E(y)=[-1;1]

3). 2p

4).׸; cos (-x)=cos x

5). [-p+2pk;2pk], kÎZ

[2pk;p+2pk], kÎZ

6). =1 =2pk, kÎZ

=-1 =p=2pk, kÎZ

7). =p/2+pk, kÎZ

8).MAX =1 =2pk, kÎZ

MIN =-1 =p+2pk, kÎZ

9).x>0 [-p/2+2pk;p/2+2pk], kÎZ

x<0 [-p/2+2pk;p/2+2pk], kÎZ

Y = T G x

1).Ԡ D(y)-, =p/2+pk kÎZ

2).Ǡ E(y)=R

3). p

4).; tg (-x)=-tg x

5). (-p/2+pk;p/2+pk), kÎZ

6). =pk, kÎZ

7). x>0 (pk;p/2+pk), kÎZ

x<0 (-p/2+pk;pk), kÎZ

 [Ñ.Ê.Â.1]

 [Ñ.Ê.Â.2]


 
2012 , , .