Laws of Exponents

[latex]\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}^{\text{−}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}[/latex]

Special Factorizations

[latex]\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}[/latex]

Quadratic Formula

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

Binomial Theorem

[latex]{(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},[/latex]

where [latex](\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)!}[/latex]

Right-Angle Trigonometry

[latex]\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}[/latex]

Trigonometric Functions of Important Angles

Fundamental Identities

[latex]\begin{array}{cccccccc}\hfill { \sin }^{2}\theta +{ \cos }^{2}\theta & =\hfill & 1\hfill & & & \hfill \sin (\text{−}\theta )& =\hfill & \text{−} \sin \theta \hfill \\ \hfill 1+{ \tan }^{2}\theta & =\hfill & { \sec }^{2}\theta \hfill & & & \hfill \cos (\text{−}\theta )& =\hfill & \cos \theta \hfill \\ \hfill 1+{ \cot }^{2}\theta & =\hfill & { \csc }^{2}\theta \hfill & & & \hfill \tan (\text{−}\theta )& =\hfill & \text{−} \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}[/latex]

Law of Sines

[latex]\frac{ \sin A}{a}=\frac{ \sin B}{b}=\frac{ \sin C}{c}[/latex]

Law of Cosines

[latex]\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}[/latex]

Addition and Subtraction Formulas

[latex]\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}[/latex]

Double-Angle Formulas

[latex]\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}[/latex]

Half-Angle Formulas

[latex]\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}[/latex]