A | Basic Functions and Identities

Graphs of the Parent Functions

Graphs of the Trigonometric Functions

Trigonometric Identities

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

Table A1