Trigonometric Identities

Pythagorean identities

${\sin }^2\theta +{\cos }^2\theta =1$ ${\tan }^2\theta +1={\sec }^2\theta$ $1+{\cot }^2\theta ={\csc }^2\theta$

Double-Angle Formulas

$\sin (2\theta )=2\sin \theta \cos \theta$ $\cos (2\theta )=1-2{\sin }^2\theta =2{\cos }^2\theta -1={\cos }^2\theta -{\sin }^2\theta$ $\tan (2\theta )=\frac{2\tan \theta }{1-{\tan }^2\theta }$

Sum and Difference Formulas

$\sin (x\pm y)=\sin x\cos y\pm \cos x\sin y$ $\cos (x\pm y)=\cos x\cos y\mp \sin x\sin y$ $\tan (x\pm y)=\frac{\tan x\pm \tan y}{1\mp \tan x\tan y}$

Half-Angle Formulas

$\sin \left(\frac{\theta }{2}\right)=\pm \sqrt{\frac{1-\cos \theta }{2}}$ $\cos \left(\frac{\theta }{2}\right)=\pm \sqrt{\frac{1+\cos \theta }{2}}$ $\tan \left(\frac{\theta }{2}\right)=\frac{1-\cos \theta }{\sin \theta }$

Reciprocal Identities

$\csc \theta =\frac{1}{\sin \theta }$ $\sec \theta =\frac{1}{\cos \theta }$ $\cot \theta =\frac{1}{\tan \theta }$

Even-odd Identities (just think about trig circle)

$\sin (-\theta )=-\sin (\theta )$ $\cos (-\theta )=\cos (\theta )$ $\tan (-\theta )=-\tan (\theta )$

Law of Sines

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

Law of Cosines

$c^2=a^2+b^2-2ab\cos C$

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