# 0.3 Review of pre-calculus

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## Formulas from geometry

$A=\text{area},$ $V=\text{Volume},\phantom{\rule{0.2em}{0ex}}\text{and}$ $S=\text{lateral surface area}$

## Laws of exponents

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

## Special factorizations

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

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

## Binomial theorem

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

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

## Right-angle trigonometry

$\begin{array}{cccc}\text{sin}\phantom{\rule{0.1em}{0ex}}\theta =\frac{\text{opp}}{\text{hyp}}\hfill & & & \text{csc}\phantom{\rule{0.1em}{0ex}}\theta =\frac{\text{hyp}}{\text{opp}}\hfill \\ \text{cos}\phantom{\rule{0.1em}{0ex}}\theta =\frac{\text{adj}}{\text{hyp}}\hfill & & & \text{sec}\phantom{\rule{0.1em}{0ex}}\theta =\frac{\text{hyp}}{\text{adj}}\hfill \\ \text{tan}\phantom{\rule{0.1em}{0ex}}\theta =\frac{\text{opp}}{\text{adj}}\hfill & & & \text{cot}\phantom{\rule{0.1em}{0ex}}\theta =\frac{\text{adj}}{\text{opp}}\hfill \end{array}$

## Trigonometric functions of important angles

 $\theta$ $\text{Radians}$ $\text{sin}\phantom{\rule{0.1em}{0ex}}\theta$ $\text{cos}\phantom{\rule{0.1em}{0ex}}\theta$ $\text{tan}\phantom{\rule{0.1em}{0ex}}\theta$ $0\text{°}$ $0$ $0$ $1$ $0$ $30\text{°}$ $\text{π}\text{/}\text{6}$ $1\text{/}2$ $\sqrt{3}\text{/}2$ $\sqrt{3}\text{/}3$ $45\text{°}$ $\text{π}\text{/}\text{4}$ $\sqrt{2}\text{/}2$ $\sqrt{2}\text{/}2$ $1$ $60\text{°}$ $\text{π}\text{/}\text{3}$ $\sqrt{3}\text{/}2$ $1\text{/}2$ $\sqrt{3}$ $90\text{°}$ $\text{π}\text{/}2$ $1$ $0$ —

## Fundamental identities

$\begin{array}{cccccccc}\hfill {\text{sin}}^{2}\theta +{\text{cos}}^{2}\theta & =\hfill & 1\hfill & & & \hfill \text{sin}\left(\text{−}\phantom{\rule{0.1em}{0ex}}\theta \right)& =\hfill & \text{−}\text{sin}\phantom{\rule{0.1em}{0ex}}\theta \hfill \\ \hfill 1+{\text{tan}}^{2}\theta & =\hfill & {\text{sec}}^{2}\theta \hfill & & & \hfill \text{cos}\left(\text{−}\phantom{\rule{0.1em}{0ex}}\theta \right)& =\hfill & \text{cos}\phantom{\rule{0.1em}{0ex}}\theta \hfill \\ \hfill 1+{\text{cot}}^{2}\theta & =\hfill & {\text{csc}}^{2}\theta \hfill & & & \hfill \text{tan}\left(\text{−}\phantom{\rule{0.1em}{0ex}}\theta \right)& =\hfill & \text{−}\text{tan}\phantom{\rule{0.1em}{0ex}}\theta \hfill \\ \hfill \text{sin}\left(\frac{\pi }{2}-\theta \right)& =\hfill & \text{cos}\phantom{\rule{0.1em}{0ex}}\theta \hfill & & & \hfill \text{sin}\left(\theta +2\pi \right)& =\hfill & \text{sin}\phantom{\rule{0.1em}{0ex}}\theta \hfill \\ \hfill \text{cos}\left(\frac{\pi }{2}-\theta \right)& =\hfill & \text{sin}\phantom{\rule{0.1em}{0ex}}\theta \hfill & & & \hfill \text{cos}\left(\theta +2\pi \right)& =\hfill & \text{cos}\phantom{\rule{0.1em}{0ex}}\theta \hfill \\ \hfill \text{tan}\left(\frac{\pi }{2}-\theta \right)& =\hfill & \text{cot}\phantom{\rule{0.1em}{0ex}}\theta \hfill & & & \hfill \text{tan}\left(\theta +\pi \right)& =\hfill & \text{tan}\phantom{\rule{0.1em}{0ex}}\theta \hfill \end{array}$

## Law of sines

$\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}A}{a}=\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}B}{b}=\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}C}{c}$

## Law of cosines

$\begin{array}{ccc}\hfill {a}^{2}& =\hfill & {b}^{2}+{c}^{2}-2bc\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}A\hfill \\ \hfill {b}^{2}& =\hfill & {a}^{2}+{c}^{2}-2ac\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}B\hfill \\ \hfill {c}^{2}& =\hfill & {a}^{2}+{b}^{2}-2ab\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}C\hfill \end{array}$

$\begin{array}{ccc}\hfill \text{sin}\phantom{\rule{0.2em}{0ex}}\left(x+y\right)& =\hfill & \text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}y+\text{cos}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}y\hfill \\ \hfill \text{sin}\phantom{\rule{0.1em}{0ex}}\left(x-y\right)& =\hfill & \text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}y-\text{cos}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}y\hfill \\ \hfill \text{cos}\phantom{\rule{0.1em}{0ex}}\left(x+y\right)& =\hfill & \text{cos}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}y-\text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}y\hfill \\ \hfill \text{cos}\phantom{\rule{0.1em}{0ex}}\left(x-y\right)& =\hfill & \text{cos}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}y+\text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}y\hfill \\ \hfill \text{tan}\phantom{\rule{0.1em}{0ex}}\left(x+y\right)& =\hfill & \frac{\text{tan}\phantom{\rule{0.2em}{0ex}}x+\text{tan}y}{1-\text{tan}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{tan}y}\hfill \\ \hfill \text{tan}\left(x-y\right)& =\hfill & \frac{\text{tan}\phantom{\rule{0.2em}{0ex}}x-\text{tan}y}{1+\text{tan}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{tan}y}\hfill \end{array}$

## Double-angle formulas

$\begin{array}{ccc}\hfill \text{sin}\phantom{\rule{0.2em}{0ex}}2x& =\hfill & 2\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}x\hfill \\ \hfill \text{cos}\phantom{\rule{0.2em}{0ex}}2x& =\hfill & {\text{cos}}^{2}x-{\text{sin}}^{2}x=2\phantom{\rule{0.1em}{0ex}}{\text{cos}}^{2}x-1=1-2\phantom{\rule{0.1em}{0ex}}{\text{sin}}^{2}x\hfill \\ \hfill \text{tan}\phantom{\rule{0.2em}{0ex}}2x& =\hfill & \frac{2\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}x}{1-{\text{tan}}^{2}x}\hfill \end{array}$

## Half-angle formulas

$\begin{array}{ccc}\hfill {\text{sin}}^{2}x& =\hfill & \frac{1-\text{cos}\phantom{\rule{0.2em}{0ex}}2x}{2}\hfill \\ \hfill {\text{cos}}^{2}x& =\hfill & \frac{1+\text{cos}\phantom{\rule{0.2em}{0ex}}2x}{2}\hfill \end{array}$

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