Appendix C: Review of Pre-Calculus

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

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

Quadratic Formula

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

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

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

Trigonometric Functions of Important Angles

This is a table with five columns and six rows. The first row is a header row, and has the entries theta, Raidians, sin theta, cos theta, tan theta. The second row has the entries 0 degrees, 0, 0, 1, and 0. The third row has the entries 30 degrees, pi/6, ½, (the square root of 3)/2, and (the square root of 3)/3. The fourth row reads 45 degrees, pi/4, (the square root of 2)/2, (the square root of 2)/2, and 1. The fifth row reads 60 degrees, pi/3, (the square root of 3)/2, 1/2, and the square root of 3. The sixth row reads 90 degrees, pi/2, 1, 0, and blank.

Fundamental Identities

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

Law of Sines

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

Law of Cosines

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

Addition and Subtraction Formulas

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

Double-Angle Formulas

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

Half-Angle Formulas

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