Review of Pre-Calculus

Gilbert Strang; Edwin; Scott Scruggs

Formulas from Geometry

$A=\text{area},$ [latex]V=\text{Volume},\text{ and }[/latex] $S=\text{lateral surface area}$

Formulas from Algebra

Laws of Exponents

$\begin{array}{ccccccccccccc}\hfill {x}^{m}{x}^{n}& =\hfill & {x}^{m+n}\hfill & & & \hfill \frac{{x}^{m}}{{x}^{n}}& =\hfill & {x}^{m-n}\hfill & & & \hfill {({x}^{m})}^{n}& =\hfill & {x}^{mn}\hfill \\ \hfill {x}^{-n}& =\hfill & \frac{1}{{x}^{n}}\hfill & & & \hfill {(xy)}^{n}& =\hfill & {x}^{n}{y}^{n}\hfill & & & \hfill {(\frac{x}{y})}^{n}& =\hfill & \frac{{x}^{n}}{{y}^{n}}\hfill \\ \hfill {x}^{1\text{/}n}& =\hfill & \sqrt[n]{x}\hfill & & & \hfill \sqrt[n]{xy}& =\hfill & \sqrt[n]{x}\sqrt[n]{y}\hfill & & & \hfill \sqrt[n]{\frac{x}{y}}& =\hfill & \frac{\sqrt[n]{x}}{\sqrt[n]{y}}\hfill \\ \hfill {x}^{m\text{/}n}& =\hfill & \sqrt[n]{{x}^{m}}={(\sqrt[n]{x})}^{m}\hfill & & & & & & & & & & \end{array}$

Special Factorizations

$\begin{array}{ccc}\hfill {x}^{2}-{y}^{2}& =\hfill & (x+y)(x-y)\hfill \\ \hfill {x}^{3}+{y}^{3}& =\hfill & (x+y)({x}^{2}-xy+{y}^{2})\hfill \\ \hfill {x}^{3}-{y}^{3}& =\hfill & (x-y)({x}^{2}+xy+{y}^{2})\hfill \end{array}$

Quadratic Formula

If $a{x}^{2}+bx+c=0,$ then $x=\frac{-b±\sqrt{{b}^{2}-4ca}}{2a}.$

Binomial Theorem

${(a+b)}^{n}={a}^{n}+(\begin{array}{l}n\\ 1\end{array}){a}^{n-1}b+(\begin{array}{l}n\\ 2\end{array}){a}^{n-2}{b}^{2}+\cdots +(\begin{array}{c}n\\ n-1\end{array})a{b}^{n-1}+{b}^{n},$

where $(\begin{array}{l}n\\ k\end{array})=\frac{n(n-1)(n-2)\cdots (n-k+1)}{k(k-1)(k-2)\cdots 3\cdot 2\cdot 1}=\frac{n!}{k!(n-k)!}$

Formulas from Trigonometry

Right-Angle Trigonometry

$\begin{array}{cccc} \sin \theta =\frac{\text{opp}}{\text{hyp}}\hfill & & & \csc \theta =\frac{\text{hyp}}{\text{opp}}\hfill \\ \cos \theta =\frac{\text{adj}}{\text{hyp}}\hfill & & & \sec \theta =\frac{\text{hyp}}{\text{adj}}\hfill \\ \tan \theta =\frac{\text{opp}}{\text{adj}}\hfill & & & \cot \theta =\frac{\text{adj}}{\text{opp}}\hfill \end{array}$

The figure shows a right triangle with the longest side labeled hyp, the shorter leg labeled as opp, and the longer leg labeled as adj. The angle between the hypotenuse and the adjacent side is labeled theta.

Trigonometric Functions of Important Angles

$\theta$	$\text{Radians}$	$\sin \theta$	$\cos \theta$	$\tan \theta$
0°	0	0	1	0
30°	$\text{π}\text{/}\text{6}$	$1\text{/}2$	$\sqrt{3}\text{/}2$	$\sqrt{3}\text{/}3$
45°	$\text{π}\text{/}\text{4}$	$\sqrt{2}\text{/}2$	$\sqrt{2}\text{/}2$	1
60°	$\text{π}\text{/}\text{3}$	$\sqrt{3}\text{/}2$	$1\text{/}2$	$\sqrt{3}$
90°	$\text{π}\text{/}2$	1	0	—

Fundamental Identities

$\begin{array}{cccccccc}\hfill { \sin }^{2}\theta +{ \cos }^{2}\theta & =\hfill & 1\hfill & & & \hfill \sin (-\theta )& =\hfill & - \sin \theta \hfill \\ \hfill 1+{ \tan }^{2}\theta & =\hfill & { \sec }^{2}\theta \hfill & & & \hfill \cos (-\theta )& =\hfill & \cos \theta \hfill \\ \hfill 1+{ \cot }^{2}\theta & =\hfill & { \csc }^{2}\theta \hfill & & & \hfill \tan (-\theta )& =\hfill & - \tan \theta \hfill \\ \hfill \sin (\frac{\pi }{2}-\theta )& =\hfill & \cos \theta \hfill & & & \hfill \sin (\theta +2\pi )& =\hfill & \sin \theta \hfill \\ \hfill \cos (\frac{\pi }{2}-\theta )& =\hfill & \sin \theta \hfill & & & \hfill \cos (\theta +2\pi )& =\hfill & \cos \theta \hfill \\ \hfill \tan (\frac{\pi }{2}-\theta )& =\hfill & \cot \theta \hfill & & & \hfill \tan (\theta +\pi )& =\hfill & \tan \theta \hfill \end{array}$

Law of Sines

$\frac{ \sin A}{a}=\frac{ \sin B}{b}=\frac{ \sin C}{c}$

The figure shows a nonright triangle with vertices labeled A, B, and C. The side opposite angle A is labeled a. The side opposite angle B is labeled b. The side opposite angle C is labeled c.

Law of Cosines

$\begin{array}{ccc}\hfill {a}^{2}& =\hfill & {b}^{2}+{c}^{2}-2bc \cos A\hfill \\ \hfill {b}^{2}& =\hfill & {a}^{2}+{c}^{2}-2ac \cos B\hfill \\ \hfill {c}^{2}& =\hfill & {a}^{2}+{b}^{2}-2ab \cos C\hfill \end{array}$

Addition and Subtraction Formulas

$\begin{array}{ccc}\hfill \sin (x+y)& =\hfill & \sin x \cos y+ \cos x \sin y\hfill \\ \hfill \sin (x-y)& =\hfill & \sin x \cos y- \cos x \sin y\hfill \\ \hfill \cos (x+y)& =\hfill & \cos x \cos y- \sin x \sin y\hfill \\ \hfill \cos (x-y)& =\hfill & \cos x \cos y+ \sin x \sin y\hfill \\ \hfill \tan (x+y)& =\hfill & \frac{ \tan x+ \tan y}{1- \tan x \tan y}\hfill \\ \hfill \tan (x-y)& =\hfill & \frac{ \tan x- \tan y}{1+ \tan x \tan y}\hfill \end{array}$

Double-Angle Formulas

$\begin{array}{ccc}\hfill \sin 2x& =\hfill & 2 \sin x \cos x\hfill \\ \hfill \cos 2x& =\hfill & { \cos }^{2}x-{ \sin }^{2}x=2{ \cos }^{2}x-1=1-2{ \sin }^{2}x\hfill \\ \hfill \tan 2x& =\hfill & \frac{2 \tan x}{1-{ \tan }^{2}x}\hfill \end{array}$

Half-Angle Formulas

$\begin{array}{ccc}\hfill { \sin }^{2}x& =\hfill & \frac{1- \cos 2x}{2}\hfill \\ \hfill { \cos }^{2}x& =\hfill & \frac{1+ \cos 2x}{2}\hfill \end{array}$

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