A | Basic Functions and Identities

Graphs of the Parent Functions

Figure A1

Figure A2

Figure A3

Graphs of the Trigonometric Functions

Figure A4

Figure A5

Figure A6

Figure A7

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