Ôîðìóëè ñêîðî÷åíîãî ìíîæåííÿ ³ ðîçêëàäàííÿ íà ìíîæíèêè:

(a±b)²=a²±2ab+b²

(a±b)³=a³±3a²b+3ab²±b³

a²-b²=(a+b)(a-b)

a³±b³=(a±b)(a²‡ab+b²),

äå çíàê ‡ îçí. ïðîòèëåæí. çíàê

xⁿ-aⁿ=(x-a)(x^n-1+ax^n-2+a²x^n-3+...+a^n-1)

ax²+bx+c=a(x-x₁)(x-x₂)

ãäå x₁ ³ x — êîðåí³ ð³âíÿííÿ

ax²+bx+c=0

Ñòåïåí³ òà êîðí³ :

a^p·a^g = a^p+g

a^p:a^g=a ^p-g

(a^p)^g=a ^pg

a^p /b^p = (a/b)^p

a^p×b^p = ab^p

a⁰=1; a¹=a

a^-p = 1/a

^pÖa =b => b^p=a

^pÖa^pÖb = ^pÖab

Öa ; a = 0

____

/ __ _

^pÖ ^gÖa = ^pgÖa

___ __

^pkÖa^gk = ^pÖa^g

_p ____

/ a ^pÖa

/ ¾¾ = ¾¾¾¾

Ö b ^pÖb

a ^1/p = ^pÖa

^pÖa^g = a^p/g

Êâàäðàòíå ð³âíÿííÿ:

ax²+bx+c=0; (a¹0)

x_1,2= (-b±ÖD)/2a; D=b² -4ac

D>0® x₁¹x₂ ;D=0® x₁=x₂

D<0, êîðåí³â íåìà.

Òåîðåìà ³ºòà:

x₁+x₂ = -b/a

x₁× x₂ = c/a

Ïðèâåäåíå êâàäðàòíå ð³âíÿííÿ:

x² + px+q =0

x₁+x₂ = -p

x₁×x₂ = q

ßêùî p=2k (p-÷åòí.)

è x²+2kx+q=0, òî x_1,2 = -k±Ö(k²-q)

Ëîãàðèôìè:

log_a x = b => a^b = x; a>0,a¹0

a ^{loga x} = x, log_aa =1; log_a 1 = 0

log_a x = b; x=a^b

log_a b = 1/(log _b a)

log_axy = log_ax + log_a y

log_a x/y = log_a x - log_a y

log_a x^k =k log_a x (x >0)

log_a^k x =1/k log_a x

log_a x = (log_c x)/( log_ca); c>0,c¹1

Ïðîãðåñ³¿

Àðèôìåòè÷íà

a_n = a_n-1 +d

2a_n= a_n-1 + a_n+1

a_n = a₁ + d(n-1)

S_n = n(a₁ + a_n )/2

^S_n ^{= (a}₁^+d(n-1))n/2

_Sn= a_{1 +} a₂ +...+a_n

Ãåîìåòðè÷íà

b_n = b_n-1 × q

b²_n = b_n-1× b_n+1

b_n = b₁×q^n-1

S_n= (b_nq- b₁)/(q-1)

S_n = b₁ (qⁿ-1)/(q-1)

S= b₁/(1-q)

Òðèãîíîìåòð³ÿ.

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

tg 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

Ôîðìóëè äîäàâàííÿ:

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

Ôîðìóëè ïîäâ³éíîãî àðãóìåíòó.

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))

Ôîðìóëè ïîëîâèííîãî àðãóìåíòó.

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

Ôîðìóëè ïðèâåäåííÿ ñóììè â äîáóòîê.

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

Ôîðìóëè ïðèâåä. äîáóòêà. â ñóìó

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))

Ñï³ââ³äíîø. ì³æ ô-ÿìè

2 tg x/2

sin x = ——————

^{1+ tg² x/2}

_{1-tg 2/x}

_{cos x = —————}

_{1+ tg² x/2}

Òðèãîíîìåòðè÷í³ ð³âíÿíèÿ

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)

Ãåîìåòð³ÿ

Òðèêóòíèêè

a + b + g =180

Òåîðåìà ñèíóñ³â

a² = b²+c² - 2bc cos a

b² = a²+c² - 2ac cos b

c² = a² + b² - 2ab cos g

Ìåä³àíà ä³ëå òðèêóã. íà äâà ð³âíîâåëèê³. Ìåä³àíà ä³ëå

ïðîòèëåã. ñòîðîíó íà äâà ð³âí³ â³äð³çêè. Á³ñåêòðèñà - êóò.

Âèñîòà ïàäຠíà ïð. ñòîðîíó

ïîä ïðÿìèì êóòîì.

Ôîðìóëà Ãåðîíà :

p=?(a+b+c)

_____________

S = Öp(p-a)(p-b)(p-c)

S = ?ab sin a

S_ðàâí.=(a²Ö3)/4

S = bh/2

S=abc/4R

S=pr

Òðàïåö³ÿ.

S = (a+b)/2× h

Êîëî

S= pR²

S_{ñåêòîðà}=(pR²a)/360

Ñòåðåîìåòð³ÿ

Ïàðàëåëèïèïåä

V=S_îñí×Ð

Ïðÿìîêóòíèé

V=abc

ϳðàì³äà

V =1/3S_îñí.×H

S_ïîëí.= S_áîê.+ S_îñí.

Çð³çàíà :

H . _____

V = 3 (S₁+S₂+ÖS₁S₂)

S₁ è S₂ — ïëîù³ îñí.

S_ïîëí.=S_áîê.+S₁+S₂

Êîíóñ

V=1/3 pR²H

S_áîê. =pRl

S_áîê.= pR(R+1)

Çð³çàíèé

S_áîê.= pl(R₁+R₂)

V=1/3pH(R₁²+R₁R₂+R₂²)

Ïðèçìà

V=S_îñí.×H

ïðÿìà: S_áîê.=P_îñí.×H

S_ïîëí.=S_áîê+2S_îñí.

ïîõèëà :

S_áîê.=P_ïñ×a

V = S_ïñ×a, à -á³÷. ðåáðî.

P_ïñ — ïåðèìåòð

S_ïñ — ïë. ïåðïåíä. ïåðåð³çó

Öèë³íäð.

V=pR²H

S_ïîëí=2pR(H+R)

S_áîê.= 2pRH

Ñôåðà è êóëÿ .

V = 4/3 pR³ - øàð

S = 4pR³ - ñôåðà

Êóëüîâèé ñåêòîð

V = 2/3 pR³H

H - âûñîòà ñåãì.

Êóëüîâèé ñåãìåíò

V=pH²(R-H/3)

S=2pRH

ãðàä

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