Pre-calculus Math References – Perfectly Logical

$a(b+c)=ab+ac$	$\frac{a}{b} + \frac{c}{d}$
$\frac{a+c}{b} = \frac{a}{b} + \frac{c}{b}$	$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \centerdot \frac{d}{c} =\frac{ad}{bc}$

Arithmetic Operations

$x^m +x^n = x^{m+n}$	$\frac{x^m}{x^n} = x^{m-n}$
$( x^m ) ^n = x^{mn}$	$x^{-n} = \frac{1}{x^n}$
$x^{\frac{1}{n}} = ^n \sqrt{x}$	$x^{\frac{m}{n}} = ^n \sqrt{x^m} = ( ^n \sqrt{x} ) ^m$
$^n \sqrt{xy} = ^n \sqrt{x} ^n \sqrt{y}$	$^n \sqrt{\frac{x}{y}} = \frac{ ^n \sqrt{x}}{ ^n \sqrt{y}}$

Exponents and Radicals

$x^2 - y^2 = ( x + y ) ( x - y )$	$x^3 + y^3 = ( x + y ) ( x^2 - xy + y^2 )$
$x^3 - y^3 = ( x - y ) ( x^2 + xy + y^2 )$

Factoring Special Polynomials

$( x + y ) ^2 = x^2 + 2xy + y^2$	$( x - y ) ^2 = x^2 - 2xy + y^2$
$( x +y ) ^3 = x^3 + 3x^2 y + 3xy^2 + y^3$	$( x - y ) ^3 = x^3 - 3x^2 y + 3xy^2 - y^3$
$( x +y ) ^n = x^n + nx^{n-1} y + \frac{n(n-1)}{2} x^{n-2} y^2 +$ $\cdots + \binom{n}{k} x^{n-k} y^k + \cdots + nxy^{n-1} + y^n$ where $\binom{n}{k} = \frac{n(n-1) \cdots (n-k-1)}{k!}$

Inequalities and Absolute Values

Triangle	$A = \frac{1}{2} b h = \frac{1}{2} a b sin \theta$ where $\theta$ is the angle made by a and b. Also, b is the base of the triangle and h is the height of the triangle.
Circle	$A = \pi r^2$ $C = 2 \pi r$
Sector of Circle	$A = \frac{1}{2} r^2 \theta$ $s = r \theta$ ( $\theta$ in radians)
Sphere	$V = \frac{4}{3} \pi r^3$ $A = 4 \pi r^2$
Cylinder	$V = \pi r^2 h$
Cone	$V = \frac{1}{3} \pi r^2 h$ $A = \pi r \sqrt{r^2 + h^2}$

Formulas for area A, circumference C, and volume V

Distance between points (x₁,y₁) and (x₂,y₂):	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Midpoint between points (x₁,y₁) and (x₂,y₂):	$(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2} )$

Distance and Midpoint Formulas

Slope of a line through points (x₁,y₁) and (x₂,y₂):	$m = \frac{y_2 - y_1}{x_2 - x_1}$
Point-slope equation of a line through point (x₁,y₁) with slope m:	$y - y_1 = m(x - x_1)$
Slope-intercept equation of a line with slope m and y-intercept b:	$y = mx + b$

Properties of Lines

The equation of a circle with center at (h,k) and radius r:

$(x-h)^2 + (y - k)^2 = r^2$

Circle Equation

Right Angle Trigonometry

$csc(\theta) = \frac{1}{sin(\theta)}$	$sec(\theta) = \frac{1}{cos(\theta)}$
$tan(\theta) = \frac{sin(\theta)}{cos(\theta)}$	$cot(\theta) = \frac{cos(\theta)}{sin(\theta)}$
$cot(\theta) = \frac{1}{tan(\theta)}$	$sin^2 (\theta) + cos^2 (\theta) = 1$
$1 + tan^2 (\theta) = sec^2 (\theta)$	$1 + cot^2 (\theta) = csc^2 (\theta)$
$sin(- \theta) = - sin(\theta)$	$cos(- \theta) = cos(\theta)$
$tan(- \theta) = - tan(\theta)$	$sin(\frac{\pi}{2} - \theta) = cos(\theta)$
$cos(\frac{\pi}{2} - \theta) = sin(\theta)$	$tan(\frac{\pi}{2} - \theta) = cot(\theta)$