:

:

. :

(ab)²=a²2ab+b²

(ab)³=a³3a²b+3ab²b³

a²-b²=(a+b)(a-b)

a³b³=(ab)(a²∓ab+b²),

(a+b)³=a³+b³+3ab(a+b)

(a-b)³=a³-b³-3ab(a-b)

xn-an=(x-a)(xn-1+axn-2+a²xn-3+...+an-1)

ax²+bx+c=a(x-x1)(x-x2)

x1 x2

ax²+bx+c=0

:

apag = ap+g

ap:ag=a p-g

(ap)g=a pg

ap /bp = (a/b)p

ap×bp = abp

a0=1; a1=a

a-p = 1/a

pÖa =b => bp=a

pÖapÖb = pÖab

Öa ; a = 0

____

/ __ _

pÖ gÖa = pgÖa

___ __

pkÖagk = pÖag

p ____

/ a pÖa

/ ¾¾ = ¾¾¾¾

Ö b pÖb

a 1/p = pÖa

pÖag = ag/p

ax²+bx+c=0; (a¹0)

x1,2= (-bÖD)/2a; D=b² -4ac

D>0 x1¹x2 ;D=0 x1=x2

D<0, .

:

x1+x2 = -b/a

x1× x2 = c/a

. :

x² + px+q =0

x1+x2 = -p

x1×x2 = q

p=2k (p-.)

x²+2kx+q=0, x1,2 = -kÖ(k²-q)

-

Ö((x2-x1)²-(y2-y1)²)

:

loga x = b => ab = x; a>0,a¹0

a loga x = x, logaa =1; loga 1 = 0

loga x = b; x = ab

loga b = 1/(log b a)

logaxy = logax + loga y

loga x/y = loga x - loga y

loga xk =k loga x (x >0)

logak x =1/k loga x

loga x = (logc x)/( logca); c>0,c¹1

logbx = (logax)/(logab)

an = a1 +d(n-1)

Sn = ((2a1+d(n-1))/2)n

bn = bn-1 × q

b2n = bn-1× bn+1

bn = b1×qn-1

Sn = b1 (1- qn)/(1-q)

S= b1/(1-q)

.

sin x = a/c

cos x = b/c

tg x = a/b=sinx/cos x

ctg x = b/a = cos x/sin x

sin (p-a) = sin a

sin (p/2 -a) = cos a

cos (p/2 -a) = sin a

cos (a + 2pk) = cos a

sin (a + 2pk) = sin a

tg (a + pk) = tg a

ctg (a + pk) = ctg a

sin² a + cos² a =1

ctg a = cosa / sina , a ¹ pn, nÎZ

tga × ctga = 1, a ¹ (pn)/2, nÎZ

1+tg²a = 1/cos²a , a¹p(2n+1)/2

1+ ctg²a =1/sin²a , a¹ pn

:

sin(x+y) = sin x cos y + cos x sin y

sin (x-y) = sin x cos y - cos x sin y

cos (x+y) = cos x cos y - sin x sin y

cos (x-y) = cos x cos y + sin x sin y

tg(x+y) = (tg x + tg y)/ (1-tg x tg y )

x, y, x + y ¹ p/2 + pn

tg(x-y) = (tg x - tg y)/ (1+tg x tg y)

x, y, x - y ¹ p/2 + pn

.

sin 2a = 2sin a cos a

cos 2a = cos² a - sin² a = 2 cos² a - 1 =

= 1-2 sin²a

tg 2a = (2 tga)/ (1-tg²a)

1+ cos a = 2 cos² a/2

1-cosa = 2 sin² a/2

tga = (2 tg (a/2))/(1-tg²(a/2))

- .

sin² a/2 = (1 - cos a)/2

cos²a/2 = (1 + cosa)/2

tg a/2 = sina/(1 + cosa ) = (1-cos a)/sin a

a¹ p + 2pn, n ÎZ

- .

sin x + sin y = 2 sin ((x+y)/2) cos ((x-y)/2)

sin x - sin y = 2 cos ((x+y)/2) sin ((x-y)/2)

cos x + cos y = 2cos (x+y)/2 cos (x-y)/2

cos x - cos y = -2sin (x+y)/2 sin (x-y)/2

sin (x+y)

tg x + tg y =

cos x cos y

sin (x - y)

tg x - tgy =

cos x cos y

. .

sin x sin y = (cos (x-y) - cos (x+y))

cos x cos y = (cos (x-y)+ cos (x+y))

sin x cos y = (sin (x-y)+ sin (x+y))

. -

sin x = (2 tg x/2)/(1+tg2x/2)

cos x = (1-tg2 2/x)/ (1+ tg² x/2)

sin2x = (2tgx)/(1+tg2x)

sin²a = 1/(1+ctg²a) = tg²a/(1+tg²a)

cos²a = 1/(1+tg²a) = ctg²a / (1+ctg²a)

ctg2a = (ctg²a-1)/ 2ctga

sin3a = 3sina -4sin³a = 3cos²asina-sin³a

cos3a = 4cos³a-3 cosa=

= cos³a-3cosasin²a

tg3a = (3tga-tg³a)/(1-3tg²a)

ctg3a = (ctg³a-3ctga)/(3ctg²a-1)

sin a/2 = Ö((1-cosa)/2)

cos a/2 = Ö((1+cosa)/2)

tga/2 = Ö((1-cosa)/(1+cosa))=

sina/(1+cosa)=(1-cosa)/sina

ctga/2 = Ö((1+cosa)/(1-cosa))=

sina/(1-cosa)= (1+cosa)/sina

sin(arcsin a) = a

cos( arccos a) = a

tg ( arctg a) = a

ctg ( arcctg a) = a

arcsin (sina) = a ; aÎ [-p/2 ; p/2]

arccos(cos a) = a ; a Î [0 ; p]

arctg (tg a) = a ; a Î[-p/2 ; p/2]

arcctg (ctg a) = a ; a Î [ 0 ; p]

arcsin(sina)=

1)a - 2pk; aÎ[-p/2 +2pk;p/2+2pk]

2) (2k+1)p - a; aÎ[p/2+2pk;3p/2+2pk]

arccos (cosa) =

1) a-2pk ; aÎ[2pk;(2k+1)p]

2) 2pk-a ; aÎ[(2k-1)p; 2pk]

arctg(tga)= a-pk

aÎ(-p/2 +pk;p/2+pk)

arcctg(ctga) = a -pk

aÎ(pk; (k+1)p)

arcsina = -arcsin (-a)= p/2-arccosa =

= arctg a/Ö(1-a²)

arccosa = p-arccos(-a)=p/2-arcsin a=

= arc ctga/Ö(1-a²)

arctga =-arctg(-a) = p/2 -arcctga =

= arcsin a/Ö(1+a²)

arc ctg a = p-arc cctg(-a) =

= arc cos a/Ö(1-a²)

arctg a = arc ctg1/a =

= arcsin a/Ö(1+a²)= arccos1/Ö(1+a²)

arcsin a + arccos = p/2

arcctg a + arctga = p/2

sin x = m ; |m| = 1

x = (-1)n arcsin m + pk, kÎ Z

sin x =1 sin x = 0

x = p/2 + 2pk x = pk

sin x = -1

x = -p/2 + 2 pk

cos x = m; |m| = 1

x = arccos m + 2pk

cos x = 1 cos x = 0

x = 2pk x = p/2+pk

cos x = -1

x = p+ 2pk

tg x = m

x = arctg m + pk

ctg x = m

x = arcctg m +pk

sin x/2 = 2t/(1+t2); t - tg

cos x/2 = (1-t²)/(1+t²)

.

: af(x)>(<) a()

1) a>1, .

2) a<1, .

: :

logaf(x) >(<) log a j(x)

1. a>1, : f(x) >0

j(x)>0

f(x)>j(x)

2. 0<a<1, : f(x) >0

j(x)>0

f(x)<j(x)

3. log f(x) j(x) = a

: j(x) > 0

f(x) >0

f(x ) ¹ 1

:

1. :

sin 2x - Ö3 cos x = 0

2sin x cos x -Ö3 cos x = 0

cos x(2 sin x - Ö3) = 0

....

2. ....

3.

sin² x - sin 2x + 3 cos² x =2

sin² x - 2 sin x cos x + 3 cos ² x = 2 sin² x + cos² x

sin x = 0, cos x = 0,

, => cos x

- :

sin a ³ m

2pk+a1 = a = a2+ 2pk

2pk+a2 = a= (a1+2p)+ 2pk

:

I cos (p/8+x) < Ö3/2

pk+ 5p/6< p/8 +x< 7p/6 + 2pk

2pk+ 17p/24 < x< p/24+2pk;;;;

II sin a = 1/2

2pk +5p/6 =a= 13p/6 + 2pk

cos a ³(=) m

2pk + a1 < a< a2+2 pk

2pk+a2< a< (a1+2p) + 2pk

cos a ³ - Ö2/2

2pk+5p/4 =a= 11p/4 +2pk

tg a³(=) m

pk+ arctg m =a= arctg m + pk

ctg ³(=) m

pk+arcctg m < a< p+pk

:

(xn) = n× xn-1

(ax) = ax× ln a

(lg ax )= 1/(x×ln a)

(sin x) = cos x

(cos x) = -sin x

(tg x) = 1/cos² x

(ctg x) = - 1/sin²x

(arcsin x) = 1/ Ö(1-x²)

(arccos x) = - 1/ Ö(1-x²)

(arctg x) = 1/ Ö(1+x²)

(arcctg x) = - 1/ Ö(1+x²)

-:

(u × v) = u×v + u×v

(u/v) = (uv - uv)/ v²

.

y = f(x0)+ f (x0)(x-x0)

x

1.

2. k =

= x

3. X0, f(x0), f (x0),

:

ò xn dx = xn+1/(n+1) + c

ò ax dx = ax/ln a + c

ò ex dx = ex + c

ò cos x dx = sin x + cos

ò sin x dx = - cos x + c

ò 1/x dx = ln|x| + c

ò 1/cos² x = tg x + c

ò 1/sin² x = - ctg x + c

ò 1/Ö(1-x²) dx = arcsin x +c

ò 1/Ö(1-x²) dx = - arccos x +c

ò 1/1+ x² dx = arctg x + c

ò 1/1+ x² dx = - arcctg x + c

.

a + b + g =180

a² = b²+c² - 2bc cos a

b² = a²+c² - 2ac cos b

c² = a² + b² - 2ab cos g

. .

. .

- .

.

.

:

p=(a+b+c)

_____________

S = Öp(p-a)(p-b)(p-c)

S = ab sin a

S.=(a²Ö3)/4

S = bh/2

S=abc/4R

S=pr

.

S = (a+b)/2× h

S= pR²

S=(pR²a)/360

V=S×

V=abc

V =1/3S.×H

S.= S.+ S.

:

H . _____

V = 3 (S1+S2+ÖS1S2)

S1 S2 .

S.=S.+S1+S2

V=1/3 pR²H

S. =pRl

S.= pR(R+1)

S.= pl(R1+R2)

V=1/3pH(R12+R1R2+R22)

V=S.×H

: S.=P.×H

S.=S+2S.

:

S.=P×a

V = S×a, -. .

P

S . .

.

V=pR²H ; S.= 2pRH

S.=2pR(H+R)

S.= 2pRH

.

V = 4/3 pR³ -

S = 4pR³ -

V = 2/3 pR³H

H - .

V=pH²(R-H/3)

S=2pRH

0 30 45 60 90 120 135 180
a -p/2 -p/3 -p/4 -p/6 0 p/6 p/4 p/3 p/2 2p/3 3p/4 3p/6 p
sina -1 -Ö3/2 -Ö2/2 - 0 Ö2/2 Ö3/2 1 - 0
cosa 1 Ö3/2 Ö2/2 0 - -Ö2/2 - Ö3/2 -1
tga Ï -Ö3 -1 -1/Ö3 0 1/Ö3 1 Ö3 Î -Ö3 -1 0
ctga --- Ö3 1 1/Ö3 0 -1/Ö3 -1 --
n 2 3 4 5 6 7 8 9
2 4 9 16 25 36 49 64 81
3 8 27 64 125 216 343 512 729
4 16 81 256 625 1296 2401 4096 6561
5 32 243 1024 3125 7776 16807 32768 59049
6 64 729 4096 15625 46656

 

7 128 2181

 

8 256 6561

 

-a p-a p+a p/2-a p/2+a 3p/2 - a 3p/2+a
sin -sina sina -sina cosa cosa -cosa -cosa
cos cosa -cosa -cosa sina -sina -sina sina
tg -tga -tga tga ctga -ctga ctga -ctga
ctg -ctga -ctga ctga tga -tga tga -tga




2010