List of all Mathematics formulas intermediate to advanced level.

Comprehensive Mathematics Formula Guide
Class 6th to class 12th and college level.

Calculus:

Rational Numbers Property
Commutative Property	Addition	`a + b = b + a`	For any rational numbers a and b.
	Subtraction	`a – b != b – a`	For any rational numbers a and b.
	Multiplication	`(a xx b) = (b xx a)`	For any rational numbers a and b.
	Division	`(a-:b) != (b-:a)`	For any rational numbers a and b.
Associative Property	Addition	`(a + b) + c = a + (b + c)`	For any rational numbers a, b, and c.
	Subtraction	`(a – b) – c != a – (b – c)`	For any rational numbers a, b, and c.
	Multiplication	`(a xx b) xx c = a xx (b xx c)`	For any rational numbers a, b, and c.
	Division	`(a -: b) -: c != a -: (b -: c)`	For any rational numbers a, b, and c.
Distributive Property		`a xx ( b + c ) = (a xx b) +( a xx c)`	For any three rational numbers a, b and c.

Number Formation

Number Formation
Number	Mathematical Representation	Example
One Digit Number	`a=a`	`a=5` `5=5`
Two Digit Number	`ab = 10a + b`	`ab=15` `a=1` `b=5` `15=10xx1 + 5`
Three Digit Number	`abc=100a+10b+c`	`abc=251` `a=2` `b=5` `c=1` `251=100xx2 + 10xx5+1`
Four Digit Number	`abcd=1000a+100b+10c+d`	`abcd=3121` `a=3` `b=1` `c=2` `d=1` `3121=1000xx3 + 100xx1+10xx2+1`
`...oo`	`...oo`	So on...

Laws of Exponents Property

Laws of Exponents Property
Number	Mathematical Representation
`x^0`	`x^0=1`
`(1)^n`	`(1)^n=1`
`x^-n`	`x^-n= 1/x^n`
`(x^n)^m`	`(x^n)^m= x^(nm)`
`x^n / x^m`	`x^n / x^m= x^(n-m)`
`x^n / y^n`	`x^n / y^n= (x/y)^n`
`x^n xx y^n`	`x^n xx y^n= (xy)^n`
`(x / y)^(-n)`	`(x/ y)^(-n)= (y/x)^n`

Algebraic Identity

Algebraic Identity Property
Expression	Identity
`(a + b)^2`	`(a + b)^2 = a^2 + b^2 + 2ab`
`(a – b)^2`	`(a – b)^2 = a^2 + b^2 – 2ab`
`(a + b) (a – b)`	`(a + b) (a – b) = a^2 – b^2`
`(x + a) (x + b)`	`(x + a) (x + b) = x^2 + (a + b)x + ab`
`(x + a) (x – b)`	`(x + a) (x – b) = x^2 + (a – b)x – ab`
`(x – a) (x + b)`	`(x – a) (x + b) = x^2 + (b – a)x – ab`
`(x – a) (x – b)`	`(x – a) (x – b) = x^2 – (a + b)x + ab`
`(a + b)^3`	`(a + b)^3 = a^3 + b^3 + 3ab(a + b)`
`(a – b)^3`	`(a – b)^3 = a^3 – b^3 – 3ab(a – b)`

Square Roots Property:

A square number always ends with 0, 1, 4, 5, 6, and 9 at its units place. And it is the inverse operation of the square number.

Geometry Formula

Geometry Formula
Cuboid	H=Height, L=Length, B=Breadth, S=Side
	Total Surface area of Cuboid	`A=2((LxxB) + (BxxH) + (LxxH))`
	Lateral Surface area of Cuboid	`A=2xx H(L+B)`
	Volume of Cuboid	`V=L xx B xx H`
	Perimeter of Cuboid	`P=4 xx (L+B+H)`
	Space Diagonals of Cuboid	`D=sqrt (L^2+B^2+H^2)`
	Face Diagonals of Cuboid	`D=sqrt (L^2+B^2)`
Cube	S= Side
	Surface area of Cube	`A=6(S)^2`
	Lateral Surface area of Cube	`A=4(S)^2`
	Volume of Cube	`V=(S)^3`
	Perimeter of Cube	`P=12 xx (S)`
	Space Diagonals of Cube	`D=sqrt 3 (S)`
Sphere	r= Radius
	Surface area of Sphere	`S=4pir^2`
	Diameter of a Sphere	`D=2r`
	Volume of Sphere	`V=(4/3)pir^3`
	Circumference of Sphere	`2pir`
Hemisphere	r= Radius
	Total Surface area of Hemisphere	`C=3pir^2`
	Curved Surface area of Hemisphere	`C=2pir^2`
	Area of Hollow Hemisphere	`A=2pi(r_2^2 + r_1^2)+pi(r_2^2 - r_1^2)`
	Volume of Hemisphere	`V=(2/3)pir^3`
Hexagon	a= Length of One Side
	Area of Hexagon	`A=(3sqrt3)/2xx a^2`
	Permiter of a Hexagon	`P=6xxa`
	Short Diagonal of Hexagon	`D=sqrt3xxa`
	Long Diagonal of Hexagon	`D=2xxa`
Prism	L= Base Length, W= Base Width, H= Height, A= Apothem Length
	The surface area of a prism = (2×BaseArea) +Lateral Surface Area. The volume of a prism = Base Area× Height
	Rectangular Prism (A Rectangular Prism has 2 parallel rectangular bases and 4 rectangular faces.)	Surface area of Prism	`A=2xx(LWxxWHxxHL)`
		Volume of Prism	`V=WxxLxxH`
		Base Area of Prism	`A=LxxW`
	Triangular Prism A triangular prism has 3 rectangular faces and 2 parallel triangular bases.	Surface area of Prism	`A=AL+3LH`
		Volume of Prism	`V=(1/2)ALH`
		Base Area of Prism	`A=(1/2)AL`
	Pentagonal Prism A pentagonal prism has 5 rectangular faces and 2 parallel pentagonal bases.	Surface area of Prism	`A=5AL+5LH`
		Volume of Prism	`V=(5/2)ALH`
		Base Area of Prism	`A=(5/2)AL`
	Hexagonal Prism A hexagonal prism has six rectangular faces and two parallel hexagonal bases.	Surface area of Prism	`A=6AL+6LH`
		Volume of Prism	`V=3ALH`
		Base Area of Prism	`A=3AL`

Algebra:

Euler's Formula:

`cos(theta) + isin(theta) = e^(itheta)`

Polar Form to Quadratic Form:

`"Convert " 5e^(ipi/3) " to the imaginary from " fr"a+bi."`
`"Sol: "`
`e^(itheta) = cos(theta) + isin(theta)`
`theta = pi/3`
`rArr5(cos(pi/3) + isin(pi/3))`
`rArr5(1/2 + isqrt3/2)`
`rArr5/2 + 5sqrt3/2i`

Quadratic Form to Polar Form:

`"Convert " sqrt2 + sqrt2i " to the Ploar from " re^(itheta).`
`Solution:`
`a=sqrt2, b=sqrt2`
`r = sqrt(a^2 + b^2)` `" "rArrsqrt((sqrt(2))^2+(sqrt(2))^2) = 2`
`tantheta = sintheta/costheta `
`tantheta = sqrt2/sqrt2 rArr 1 rArr theta = pi/4`
`rArr2e^ipi/4`

Trigonometry Formula

All Trigonometry Formulas
Number	Mathematical Representation
`cos^2theta = 1- sin^2 theta`	`tantheta = sintheta / costheta`
`sec theta = 1/cos theta`	`tan^2theta = sec^2theta – 1`
`"Ploar Form" = re^(itheta)` ` r= sqrt(a^2 + b^2)`	``

Trigonometric table(sin-cos-tan table) for 0 to 360.

Trigonometric table 0 to 360 Degree with (sin-cos-tan table)
`Degrees`	`0^@`	`30^@`	`45^@`	`60^@`	`90^@`	`120^@`	`135^@`	`150^@`	`180^@`	`210^@`	`225^@`	`240^@`	`270^@`	`300^@`	`315^@`	`330^@`	`360^@`
`Radians`	`0`	`frac{pi}{6}`	`frac{pi}{4}`	`frac{pi}{3}`	`frac{pi}{2}`	`frac{2pi}{3}`	`frac{3pi}{4}`	`frac{5pi}{6}`	`pi`	`frac{7pi}{6}`	`frac{5pi}{4}`	`frac{4pi}{3}`	`frac{3pi}{2}`	`frac{5pi}{3}`	`frac{7pi}{4}`	`frac{11pi}{6}`	`2pi`
`sin`	`0`	`1/2`	`1/sqrt2`	`sqrt3/2`	`1`	`sqrt3/2`	`1/sqrt2`	`1/2`	`0`	`-1/2`	`-1/sqrt2`	`-sqrt3/2`	`-1`	`-sqrt3/2`	`-1/sqrt2`	`1/2`	`0`
`cos`	`1`	`sqrt3/2`	`1/sqrt2`	`1/2`	`0`	`-1/2`	`-1/sqrt2`	`-sqrt3/2`	`-1`	`-sqrt3/2`	`-1/sqrt2`	`-1/2`	`0`	`1/2`	`1/sqrt2`	`sqrt3/2`	`1`
`tan`	`0`	`1/sqrt3`	`1`	`sqrt3`	`UD`	`-sqrt3`	`-1`	`-1/sqrt3`	`0`	`1/sqrt3`	`1`	`sqrt3`	`UD`	`-sqrt3`	`-1`	`-1/sqrt3`	`0`
`cosec`	`UD`	`2`	`sqrt2`	`2/sqrt3`	`1`	`2/sqrt3`	`sqrt2`	`2`	`UD`	`-2`	`-sqrt2`	`2/sqrt3`	`-1`	`-2/sqrt3`	`-sqrt2`	`-2`	`UD`
`sec`	`1`	`2/sqrt3`	`sqrt2`	`2`	`UD`	`-2`	`-sqrt2`	`2/sqrt3`	`-1`	`-2/sqrt3`	`-sqrt2`	`2`	`UD`	`2`	`sqrt2`	`2/sqrt3`	`1`
`cot`	`UD`	`sqrt3`	`1`	`frac{1}{sqrt3}`	`0`	`-frac{1}{sqrt3}`	`-1`	`-sqrt3`	`UD`	`sqrt3`	`1`	`frac{1}{sqrt3}`	`0`	`-frac{1}{sqrt3}`	`-1`	`-sqrt3`	`UD`

Rotations in 2D

Matrix for rotation of a point about the origin by an angle 'Theta'.
`[[x_2], [y_2]] = [[cos(theta), -sin(theta)], [+sin(theta), cos(theta)]] [[x_1], [y_1]]`