Iifomula zezibalo zamabanga aphakamileyo. Iifomula zeMathematika zimele ukuhlanganiswa kophuhliso lokuqiqa kwaye zenziwe ngamanani kunye neeleta.

Ukubazi kubalulekile ukusombulula iingxaki ezininzi ezihlawuliswe kukhuphiswano nakwi-Enem, ikakhulu kuba ihlala inciphisa ixesha lokusombulula ingxaki.

Nangona kunjalo, ukuhombisa iifomula akwanele ukuba uphumelele kwisicelo sabo. Ukwazi intsingiselo yobungakanani nganye kunye nokuqonda imeko ekufuneka kusetyenziswe ifomula nganye kubalulekile.

Kule tekisi siqokelela iifomyula eziphambili ezisetyenziswa kwizikolo eziziisekondari, zihlelwe ngumxholo.

## Imisebenzi

Imisebenzi ibonisa ubudlelwane phakathi kwezinto ezimbini eziguqukayo, ke ixabiso elinikezwe enye yazo liya kudibana nexabiso elikhethekileyo lenye.

Izinto ezimbini ezinokuthi zinxulunyaniswe ngeendlela ezahlukeneyo kwaye ngokomgaqo wolawulo lwazo, bafumana ukwahlulwa okwahlukileyo.

### Cokisa umsebenzi

f (x) = izembe + b

a: ithambeka
b: ulungelelwaniso lomgama

f (x) = izembe2+ b x + c , apho ≠ 0

a, b no c: isidanga sokusebenza semlingani

#### Ingqungquthela yomzekeliso.

Disc: ucalucalulo lwe-quadratic equation ( B = b2 - 4.ac)

a, b kunye c: coefficients ze-quadratic equation

### Umsebenzi obonakalayo

f(x) = ax, Nge> 0 kunye no ≠ 0

### Umsebenzi weLogarithmic

f (x) = ilogun x , ngokuqinisekileyo kunye no-1

f (x) = isono x

f(x) = cos x

### Umsebenzi wePolynomial

f(x) = an . xn + an-1. xn-1+… + A2 . x2 + a1 . x1 + a0 0

unneln-1,…, Yena2el1el0 0 amanani amanani
n: inani elipheleleyo
x: umahluko ontsonkothileyo

## Inkqubela phambili

Ukuqhubela phambili kukulandelelana kwamanani apho, ukuqala kwikota yokuqala, zonke ezinye zifunyanwa ngokudibanisa okanye ukuphindaphinda ngexabiso elifanayo.

Kwiinkqubela phambili ezibizwa ngokuba yi-arithmetic, amagama alandelayo afunyanwa ngokudibanisa igama langaphambili kunye nenani elifanayo (ratio)

Kwinkqubela phambili yejiyometri, ulandelelwano lwenziwa ngokuphindaphinda ixesha elidlulileyo ngomlinganiso.

### Ukuqhubela phambili kweArithmetic

#### Ixesha ngokubanzi

unn = a1 + (n - 1) r

unn: Ixesha eliqhelekileyo
un1: Kwikota yokuqala
n: inani lamagama
r: Umyinge weBP

#### Isishwankathelo se-PA esiphelileyo

Sn: isishwankathelo semigaqo
un1: Kwikota yokuqala
unn: nth ixesha
n: inani lamagama

### Ukuqhubela phambili kweJometri

#### Ixesha ngokubanzi

unn = a1 . Intonin-1

unn: nth ixesha
un1: Kwikota yokuqala
q: Umlinganiso we-PG
n: inani lamagama

#### Ingqokelela ye PG

Sn: isishwankathelo semigaqo
un1: Kwikota yokuqala
q: Umlinganiso we-PG
n: inani lamagama

#### Umda wesixa seGP engapheliyo

Umda: xa umda wenani lisiya infinito
un1: Kwikota yokuqala
q: Umlinganiso we-PG
n: inani lamagama

Bona kwakhona:

## Iplani yejometri

Iplane geometry yinxalenye yemathematics efunda iipropathi zamanani ejiyometri kwindiza. Ukufundwa kwejiyometri kuthetha ukusetyenziswa kwee-postulates, ii-axioms kunye neethiyori.

### Ubungakanani bee-angles zangaphakathi zepoligoni.

Syo = (n - 2). 180º

SyoIsixa seengalo zangaphakathi
n: inani lamacala e-polygon

### Ithiyori yebali

I-AB kunye neCD: amacandelo omgca ogqitywe ngokusikwa ngenqwaba yemigca efanayo
I-A´B´ kunye ne-C´D´: amacandelo omnye umgca othe tye, ojikelezayo ukuya kwelokuqala, ogqitywe ngokusika ngengqimba efanayo yemigca efanayo

### Ubudlelwane beetriki kunxantathu ofanelekileyo

b2 = a. n

a: hypotenuse
b: icala
n: ingqikelelo ye-catheter b ngaphezulu kwe-hypotenuse

c2 = a. m

a: hypotenuse
c: icala
m: ingqikelelo yecala c kwi-hypotenuse

ah = b. c

a: hypotenuse
b kunye c: abaqokeleli
h: ukuphakama okuhambelana ne-hypotenuse

h2 = m. n

h: ukuphakama okuhambelana ne-hypotenuse
m: ingqikelelo yecala c kwi-hypotenuse
n: ingqikelelo ye-catheter b ngaphezulu kwe-hypotenuse

un2 =b2 + c2 (Ithiyori kaPythagoras)

a: hypotenuse
b kunye c: abaqokeleli

### I-Polygon ibhalwe kwisangqa.

#### Unxantathu obhalwe ngokulinganayo

: ilinganiswe kwicala lonxantathu obhaliweyo
r: irediyasi yomjikelezo

r: irediyasi yomjikelezo
un3: apothem kanxantathu obhalwe ngokulinganayo

#### Isikwere esibhalisiweyo

: ilinganiswe kwicala lesikwere esibhaliweyo
r: irediyasi yomjikelezo

un4 4: apothem yesikwere esibhaliweyo
r: irediyasi yomjikelezo

#### I-hexagon ebhaliweyo rhoqo

linganisa kwicala leheksagoni ebhaliweyo
r: irediyasi yomjikelezo

un6 6ukufakwa kweheksagoni ebhaliweyo
r: irediyasi yomjikelezo

### Ubude bokujikeleza

C = 2.r

C: ubude bomjikelezo
r: irediyasi yomjikelezo

### Indawo yemilo yesicwangciso

#### Indawo yoonxantathu

A: indawo kanxantathu
b: umlinganiso wesiseko
h: imilinganiselo yokuphakama ehambelana nesiseko

#### Ifomula yeHeron yommandla kanxantathu

p: isemiperimeter
a, b no c: macala onxantathu

#### Indawo engunxantathu elinganayo

A: indawo kanxantathu onamacala alinganayo
imilinganiselo kwicala lonxantathu onamacala alinganayo

#### Indawo yoxande

A = bh

A: indawo yoxande
b: umlinganiso wesiseko
h: umlinganiso wokuphakama

#### Indawo yesikwere

A = L2

A: indawo yesikwere
L: imilinganiselo esecaleni

#### Indawo yeParallelogram

A = bh

A: indawo yeparallelogram
b: isiseko
h: ukuphakama

#### Indawo yeTrapezoidal

A: Indawo yetrapezoidal
B: umlinganiso wesiseko esiphambili
b: umlinganiso wesiseko esincinci
h: umlinganiso wokuphakama

#### Indawo yaseRhombus

Indawo yeRhombus
D: umlinganiso wediagonal omkhulu
d: Umlinganiso omncinci wediagonal

#### Ummandla oqhelekileyo weheksagoni

A: Indawo yeheksagoni rhoqo
Imilinganiselo yehexagon esecaleni

#### Indawo ejikelezayo

A = p. r2

A: indawo yesangqa

#### Icandelo leSetyhula

A: Ummandla wecandelo lesetyhula
R: unomathotholo
αizidanga: i-engile ngeedigri

Bona ngakumbi:

## I-Trigonometry

I-Trigonometry yinxalenye yemathematics efunda ubudlelwane phakathi kwamacala kunye nee-engile zoonxantathu.

Ikwasetyenziswa nakwezinye iindawo zokufunda, ezinje ngefiziksi, iJografi, inzululwazi ngeenkwenkwezi, ubunjineli, phakathi kwabanye.

### Ubudlelwane beTrigonometric

isono: i-sine ye-angle B
b: kwicala elingqonge icala B
a: hypotenuse

cos: cosine ye-engile B
c: icala elikufutshane nekona B
a: hypotenuse

I-tg: ubungakanani be-engile B
b: kwicala elingqonge icala B
c: icala elikufutshane nekona B

sen sen2 α + cos2 α = 1

isono α: sine ye-engile α
cos α: cosine ye-engile α

I-tg α: ukujija kweekona α
isono α: sine ye-engile α
cos α: cosine ye-engile α

cotg α: cotangent ye-engile α
I-tg α: ukujija kweekona α
isono α: sine ye-engile α
cos α: cosine ye-engile α

sec α: secant yeengile α
cos α: cosine ye-engile α

α cossec: i-angular cosecant α
isono α: sine ye-engile α

tg2 α + 1 = umzuzwana2 α

I-tg α: ukujija kweekona α
sec α: secant yeengile α

ikota2 + 1 = icosec2 α

cotg α: cotangent ye-engile α
α cossec: i-angular cosecant α

### Umthetho weSine

umlinganiso osecaleni
isono: i-sine yekona ejongene necala a
b: imilinganiselo esecaleni
isono: i-sine yekona ejongene necala b
c: imilinganiselo esecaleni
isono: isine yekona ejongene necala c

### Umthetho wecosine

un2 =b2 + c2 -2bccos

a, b no c: macala onxantathu
cos: cosine yecala elijonge kwicala a

### Utshintsho kwiTrigonometric

#### Isine sesixa see-arcs ezimbini

isono (a + b) = isono a. cos b + isono b.cos a

isono (a + b): sine yokongezwa kwe-arc kunye ne-arc b
ngaphandle kwe-sine ye-arc a
cos b: cosine yearc b
isono b: sine arc b
cos a: cosine yearc a

#### Sine umahluko kweengqameko ezimbini

isono (a-b) = isono a. cos b - isono b.cos a

isono (a-b): isihlunu sokukhupha i-arc kunye ne-arc b
ngaphandle kwe-sine ye-arc a
cos b: cosine yearc b
isono b: sine arc b
cos a: cosine yearc a

#### I-cosine yesixa see-arcs ezimbini.

cos (a + b) = cos a. cos b - isono a. isono b

cos (a + b): cosine yesixa se-arc a ukuya kwi-arc b
cos a: cosine yearc a
cos b: cosine yearc b
ngaphandle kwe-sine ye-arc a
isono b: sine arc b

#### Cosine umahluko yee-arcs ezimbini.

cos (a - b) = cos a. cos b + isono a. isono b

cos (a-b): cosine yokuthabatha i-arc a kunye ne-arc b
cos a: cosine yearc a
cos b: cosine yearc b
ngaphandle kwe-sine ye-arc a
isono b: sine arc b

#### Ukudibanisa isixa see-arcs ezimbini.

I-tg (a + b): isixa se-arc ukuya kwi-arc b (ii-arcs apho kuchazwe khona itangent)
I-tg a: ubume be-arc a
I-tg b: ubushushu be-arc b

#### Umahluko wee-arcs ezimbini.

I-tg (a-b): ukubambeka kokukhupha i-arc a kunye ne-arc b (ii-arcs apho kuchazwe khona itangent)
I-tg a: ubume be-arc a
I-tg b: ubushushu be-arc b

## Uhlalutyo oludibeneyo

Kuhlalutyo oludibeneyo sifunda iindlela kunye neendlela ezisivumela ukuba sisombulule iingxaki ezinxulumene nokubala.

Iifomula ezisetyenziswe kulo mxholo zihlala zisetyenziselwa ukusombulula iingxaki ezinokwenzeka.

### Imvume elula

P = n!

n!: n. (n - 1) (n - 2)… 3) 2) 1

### Ubuncinane bukaNewton

Tk + 1: Ixesha eliqhelekileyo

Jonga kwakhona uhlalutyo lomdibaniso wokuzilolonga.

## Amathuba

Isifundo esinokwenzeka sivumela ukufumana ixabiso leziganeko ezinokubakho kuvavanyo olungahleliwe (into engahleliwe). Ngamanye amagama, ukubonwa kunokwenzeka "amathuba" okufumana iziphumo ezithile.

p (A): ukubakho komsitho A
n (A): inani leziphumo ezilungileyo
n (Ω): inani leziphumo ezinokubakho

### Amathuba okujoyina imisitho emibini.

p (AUB) = p (A) + p (B) - p (A ∩ B)

p (AUB): amathuba omsitho A okanye umcimbi B owenzekayo
p (A): amathuba omsitho A
p (B): amathuba omsitho B owenzekayo
p (A ∩ B): amathuba omsitho A kunye nomsitho B owenzekayo

### Ukubakho kweziganeko ezizodwa.

p (I-AUB) = p (A) + p (B)

p (AUB): amathuba omsitho A okanye umcimbi B owenzekayo
p (A): amathuba omsitho A
p (B): amathuba omsitho B owenzekayo

### Amathuba athile

p (A / B): kunokwenzeka ukuba isiganeko A senzekile, isiganeko B
p (A ∩ B): amathuba omsitho A kunye nomsitho B owenzekayo
p (B): amathuba omsitho B owenzekayo

### Ukubakho kweziganeko ezizimeleyo.

p (A ∩ B) = p (A) p (B)

p (A ∩ B): amathuba omsitho A kunye nomsitho B owenzekayo
p (A): amathuba omsitho A
p (B): amathuba omsitho B owenzekayo

## Izibalo

Kwizibalo, sifunda ngokuqokelela, ukurekhoda, ukulungiselela kunye nohlalutyo lwedatha yophando.

Sebenzisa iifomula zemathematics, kunokwenzeka ukuba wazi ulwazi olunxulumene noluntu olunikiweyo ukusuka kwidatha yesampulu yabemi.

### Umndilili weArithmetic

MUnUmndilili wezibalo
: Isambuku sazo zonke iisampulu zamaxabiso
n: isixa sedatha yesampulu

### obanzi

V: umahluko
(xyo -MUnUkuphambuka kwamaxabiso x kwi-arithmetic kuthetha
n: isixa sedatha yesampulu

### Ukusuka kumngangatho

I-SD: ukuphambuka okuqhelekileyo
V: umahluko

Jonga kwakhona Ubalo kunye nezibalo- Ukuzivocavoca

## Imathematika yezemali

Ukufunda ukulingana kwemali ngokuhamba kwexesha kugxilwe kwimathematics, kusetyenziswa iifomula esivumela ukuba sazi ukuba ixabiso lemali lihluka njani ngokuhamba kwexesha.

### Inzala elula

J = C. i. t

J: umdla
C: inkunzi
i: inzala
t: ixesha lesicelo

M = C + J

M: ubungakanani
C: inkunzi
J: umdla

### Isifungo esimbaxa

M = C(1 + i)t

M. ubungakanani
C: inkunzi
i: inzala
t: ixesha lesicelo

J = M - C

J: umdla
M: ubungakanani
C: inkunzi

## Ijometri yendawo

Ijometri yendawo ihambelana nendawo yemathematics enoxanduva lokufunda amanani esithubeni, oko kukuthi, lawo anemilinganiselo engaphezulu kwesibini.

### Ulwalamano lwe-Euler

V - A + F = 2

V: inani lezinto
A: inani lemiphetho
F: inani lobuso

### Prism

d: idiagonal yepavile
a, b no c: imilinganiselo yemilinganiselo yepavita

V = B.h

V: umthamo weprism
B: indawo yesiseko
h: ukuphakama kweprism

### Pirámide

V: ivolumu yepiramidi
B: indawo yesiseko
h: ukuphakama kwepiramidi

#### Umboko wePyramidal

V: umthamo womboko wepiramidi
h: ukuphakama komboko wepiramidi
B: indawo yesiseko esikhulu
b: indawo yesiseko esincinci

### Isilinda

UnL= 2.πR

UnL: Indawo esecaleni
R: unomathotholo
h: ukuphakama kwesilinda

UnB = 2.πR2

UnB: Isiseko sendawo
R: unomathotholo

UnT = 2.R (h + R) =

UnT: indawo iyonke
R: unomathotholo
h: ukuphakama

V = R2.h

V: ivolumu
R: unomathotholo

### Ikhona

UnL = RR g

UnL: Indawo esecaleni
R: unomathotholo
g: imveliso

UnB = π R2

UnB: Isiseko sendawo
R: unomathotholo

UnT = RR (g + R)

UnT : indawo iyonke
R: unomathotholo
g: imveliso

V: ivolumu
UnB: Isiseko sendawo
h: ukuphakama

### Umboko wendlela

UnL = π. (R + r)

UnL: Indawo esecaleni
g: imveliso
R: irediyasi enkulu
r: irediyasi encinci

V: ivolumu
h: ukuphakama
R: irediyasi enkulu
r: irediyasi encinci

### Icandelo

I-A = 4. π R2

A: indawo yesigaba
R: unomathotholo

V: umthamo wenqanaba
R: unomathotholo

Bona ngakumbi:

## Uhlalutyo lwejiyometri

Kwi-geometry yohlalutyo simele imigca, isangqa, i-ellipses, phakathi kwezinye kwi-Cartesian plane. Ke ngoko, kunokwenzeka ukuba uchaze ezi milo zejiyometri usebenzisa ii-equations.

d (A, B): umgama phakathi kwamanqaku A no-B
x1: I-abscissa yenqaku A
x2: i-abscissa yenqaku B
y1: I-abscissa yenqaku A
y2: i-abscissa yenqaku B

m: ithambeka lomgca
x1: I-abscissa yenqaku A
x2: i-abscissa yenqaku B
y1: I-abscissa yenqaku A
y2: i-abscissa yenqaku B

### Ukulingana ngokubanzi kumgca ochanekileyo.

izembe + ngo + c = 0

a, b kunye c: engxamisekileyo

### Ukuncitshiswa kokulingana okulinganayo

y = mx + b

m: ithambeka
b: ulungelelwaniso lomgama

### Ukwahlulahlula umgca kumgca

a: Ixabiso apho umgca unqamleza khona i-x-axis
b: ixabiso apho umgca unqumla khona i-y-axis

### Umgama phakathi kwenqaku kunye nomgca

d: umgama phakathi kwenqaku kunye nomgca
a, b kunye c: coefficients yomgca
x: inqaku le-abscissa
y: ukulungiswa kwenqaku

### I-Angle phakathi kwemigca emibini

m1: ithambeka lomgca 1
m2: ithambeka lomgca 2

### Ukujikeleza

#### Umlinganiso wokujikeleza

(x - xc)2 + (Kwaye-kwayec)2 =R2

x no-y: uququzelelo lwalo naliphi na inqaku elisesangqa
xc yyc: uququzelelo lweziko lesangqa
R: unomathotholo

#### Umlinganiso oqhelekileyo wokujikeleza

x2 + kwaye2 - 2.xc.x - 2.yc.y + (xc2 + kwayec2 -R2= 0

x no-y: uququzelelo lwalo naliphi na inqaku elisesangqa
xc yyc: uququzelelo lweziko lesangqa
R: unomathotholo

### Ellipse

(i-axis enkulu eye-x axis)

x kunye no-y: ulungelelwaniso lwalo naliphi na inqaku elllipse
umlinganiso we-semi-axis ephambili
b: umlinganiso we-axis yesiqingatha esincinci

(i-axis enkulu eyeye-axis y)

x kunye no-y: ulungelelwaniso lwalo naliphi na inqaku elllipse
umlinganiso we-semi-axis ephambili
b: umlinganiso we-axis yesiqingatha esincinci

### Isibaxo

(i-axis yokwenyani yeka-axis x)

x kunye no-y: ulungelelwaniso lwalo naliphi na inqaku elikwi-hyperbola
a: umlinganiso we-semi-axis yokwenyani
b: umlinganiso we-semi-axis yentelekelelo

(eyona axis yokwenyani yey ye axis y)

x kunye no-y: ulungelelwaniso lwalo naliphi na inqaku elikwi-hyperbola
a: umlinganiso we-semi-axis yokwenyani
b: umlinganiso we-semi-axis yentelekelelo

### Umzekeliso

y2 = 2.px (i-vertex kwimvelaphi kwaye kugxilwe kwi-axcissa axis)

x kunye no-y: uququzelelo lwalo naliphi na inqaku elilelparabola
p: ipharamitha

x2 = 2.py (i-vertex kwimvelaphi kwaye kugxilwe kwi-axis ebekiweyo)

x kunye no-y: uququzelelo lwalo naliphi na inqaku elilelparabola
p: ipharamitha

## Amanani Complex

Iinombolo ezintsonkothileyo ngamanani ayilwe yinxalenye yokwenyani kunye nentelekelelo. Icandelo lokucinga elicingelwayo limelwe ngoonobumba okt libonisa isiphumo sokulingana i2 = -1.

### Ifom yeAlgebra

z = a + bi

z: inombolo entsonkothileyo
a: Inxalenye yokwenyani
bi: icandelo lokucinga (apho i = √ - 1)

### Ifom yeTrigonometric

z: inombolo entsonkothileyo
Imodyuli yenani elintsonkothileyo ()
Argument: impikiswano z

(Ifomula yeMoivre)

z: inombolo entsonkothileyo
module: Imodyuli yenani entsonkothileyo
n: isicatshulwa
Argument: impikiswano z