Trigonometry/Trigonometric Formula Reference

The principal identities in trigonometry are:

${\sin }^2\theta +{\cos }^2\theta =1$

$\tan \theta =\frac{\sin \theta }{\cos \theta }$

Four trigonometric functions are $2\pi$ periodic:

$\sin \theta =\sin (\theta +2\pi )$

$\cos \theta =\cos (\theta +2\pi )$

$\csc \theta =\csc (\theta +2\pi )$

$\sec \theta =\sec (\theta +2\pi )$

Two trigonometric functions are $\pi$ periodic:

$\tan \theta =\tan (\theta +\pi )$

$\cot \theta =\cot (\theta +\pi )$

Formulas involving sums of angles are as follows:

$\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$

$\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$

Substituting $\beta =\alpha$ gives the double angle formulae

$\sin (2\alpha )=2\sin \alpha \cos \alpha$

$\cos (2\alpha )={\cos }^2\alpha -{\sin }^2\alpha$

Substituting ${\sin }^2\alpha +{\cos }^2\alpha =1$ gives

$\cos (2\alpha )=2{\cos }^2\alpha -1$

$\cos (2\alpha )=1-2{\sin }^2\alpha$