Appendix | Precalculus

Graphs of the Parent Functions

Three graphs side-by-side. From left to right, graph of the identify function, square function, and square root function. All three graphs extend from -4 to 4 on each axis.

Three graphs side-by-side. From left to right, graph of the cubic function, cube root function, and reciprocal function. All three graphs extend from -4 to 4 on each axis.

Three graphs side-by-side. From left to right, graph of the absolute value function, exponential function, and natural logarithm function. All three graphs extend from -4 to 4 on each axis.

Graphs of the Trigonometric Functions

Three graphs of trigonometric functions side-by-side. From left to right, graph of the sine function, cosine function, and tangent function. Graphs of the sine and cosine functions extend from negative two pi to two pi on the x-axis and two to negative two on the y-axis. Graph of tangent extends from negative pi to pi on the x-axis and four to negative 4 on the y-axis.

Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse cosecant function, inverse secant function, and inverse cotangent function.

Trigonometric Identities

Pythagorean Identities	$\begin{cases}{\cos }^{2}t+{\sin }^{2}t=1\\ 1+{\tan }^{2}t={\sec }^{2}t\\ 1+{\cot }^{2}t={\csc }^{2}t\end{cases}$
Even-Odd Identities	$\begin{cases}\cos \left(-t\right)=\cos t\hfill \\ \sec \left(-t\right)=\sec t\hfill \\ \sin \left(-t\right)=-\sin t\hfill \\ \tan \left(-t\right)=-\tan t\hfill \\ \csc \left(-t\right)=-\csc t\hfill \\ \cot \left(-t\right)=-\cot t\hfill \end{cases}$
Cofunction Identities	$\begin{cases}\cos t=\sin \left(\frac{\pi }{2}-t\right)\hfill \\ \sin t=\cos \left(\frac{\pi }{2}-t\right)\hfill \\ \tan t=\cot \left(\frac{\pi }{2}-t\right)\hfill \\ \cot t=\tan \left(\frac{\pi }{2}-t\right)\hfill \\ \sec t=\csc \left(\frac{\pi }{2}-t\right)\hfill \\ \csc t=\sec \left(\frac{\pi }{2}-t\right)\hfill \end{cases}$
Fundamental Identities	$\begin{cases}\tan t=\frac{\sin t}{\cos t}\hfill \\ \sec t=\frac{1}{\cos t}\hfill \\ \csc t=\frac{1}{\sin t}\hfill \\ \text{cot}t=\frac{1}{\text{tan}t}=\frac{\text{cos}t}{\text{sin}t}\hfill \end{cases}$
Sum and Difference Identities	$\begin{cases}\cos \left(\alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta \hfill \\ \cos \left(\alpha -\beta \right)=\cos \alpha \cos \beta +\sin \alpha \sin \beta \hfill \\ \sin \left(\alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \beta \hfill \\ \sin \left(\alpha -\beta \right)=\sin \alpha \cos \beta -\cos \alpha \sin \beta \hfill \\ \tan \left(\alpha +\beta \right)=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }\hfill \\ \tan \left(\alpha -\beta \right)=\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }\hfill \end{cases}$
Double-Angle Formulas	$\begin{cases}\sin \left(2\theta \right)=2\sin \theta \cos \theta \hfill \\ \cos \left(2\theta \right)={\cos }^{2}\theta -{\sin }^{2}\theta \hfill \\ \cos \left(2\theta \right)=1 - 2{\sin }^{2}\theta \hfill \\ \cos \left(2\theta \right)=2{\cos }^{2}\theta -1\hfill \\ \tan \left(2\theta \right)=\frac{2\tan \theta }{1-{\tan }^{2}\theta }\hfill \end{cases}$
Half-Angle Formulas	$\begin{cases}\sin \frac{\alpha }{2}=\pm \sqrt{\frac{1-\cos \alpha }{2}}\hfill \\ \cos \frac{\alpha }{2}=\pm \sqrt{\frac{1+\cos \alpha }{2}}\hfill \\ \tan \frac{\alpha }{2}=\pm \sqrt{\frac{1-\cos \alpha }{1+\cos \alpha }}\hfill \\ \tan \frac{\alpha }{2}=\frac{\sin \alpha }{1+\cos \alpha }\hfill \\ \tan \frac{\alpha }{2}=\frac{1-\cos \alpha }{\sin \alpha }\hfill \end{cases}$
Reduction Formulas	$\begin{cases}{\sin }^{2}\theta =\frac{1-\cos \left(2\theta \right)}{2}\\ {\cos }^{2}\theta =\frac{1+\cos \left(2\theta \right)}{2}\\ {\tan }^{2}\theta =\frac{1-\cos \left(2\theta \right)}{1+\cos \left(2\theta \right)}\end{cases}$
Product-to-Sum Formulas	$\begin{cases}\cos \alpha \cos \beta =\frac{1}{2}\left[\cos \left(\alpha -\beta \right)+\cos \left(\alpha +\beta \right)\right]\hfill \\ \sin \alpha \cos \beta =\frac{1}{2}\left[\sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)\right]\hfill \\ \sin \alpha \sin \beta =\frac{1}{2}\left[\cos \left(\alpha -\beta \right)-\cos \left(\alpha +\beta \right)\right]\hfill \\ \cos \alpha \sin \beta =\frac{1}{2}\left[\sin \left(\alpha +\beta \right)-\sin \left(\alpha -\beta \right)\right]\hfill \end{cases}$
Sum-to-Product Formulas	$\begin{cases}\sin \alpha +\sin \beta =2\sin \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha -\beta }{2}\right)\hfill \\ \sin \alpha -\sin \beta =2\sin \left(\frac{\alpha -\beta }{2}\right)\cos \left(\frac{\alpha +\beta }{2}\right)\hfill \\ \cos \alpha -\cos \beta =-2\sin \left(\frac{\alpha +\beta }{2}\right)\sin \left(\frac{\alpha -\beta }{2}\right)\hfill \\ \cos \alpha +\cos \beta =2\cos \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha -\beta }{2}\right)\hfill \end{cases}$
Law of Sines	$\begin{cases}\frac{\sin \alpha }{a}=\frac{\sin \beta }{b}=\frac{\sin \gamma }{c}\hfill \\ \frac{a}{\sin \alpha }=\frac{b}{\sin \beta }=\frac{c}{\sin \gamma }\hfill \end{cases}$
Law of Cosines	$\begin{cases}{a}^{2}={b}^{2}+{c}^{2}-2bc\cos \alpha \hfill \\ {b}^{2}={a}^{2}+{c}^{2}-2ac\cos \beta \hfill \\ {c}^{2}={a}^{2}+{b}^{2}-2ab\text{cos}\gamma \hfill \end{cases}$