# Trig identity reference

Look up AND understand all your favorite trig identities.

## Reciprocal and quotient identities

$\sec(\theta)= \dfrac{1}{\cos(\theta)}$

$\csc(\theta)= \dfrac{1}{\sin(\theta)}$

$\cot(\theta)= \dfrac{1}{\tan(\theta)}$

$\tan(\theta)= \dfrac{\sin(\theta)}{\cos(\theta)}$

$\cot(\theta)= \dfrac{\cos(\theta)}{\sin(\theta)}$

$\csc(\theta)= \dfrac{1}{\sin(\theta)}$

$\cot(\theta)= \dfrac{1}{\tan(\theta)}$

$\tan(\theta)= \dfrac{\sin(\theta)}{\cos(\theta)}$

$\cot(\theta)= \dfrac{\cos(\theta)}{\sin(\theta)}$

## Pythagorean identities

$\sin^2(\theta) + \cos^2(\theta)=1^2$

$\tan^2(\theta) + 1^2=\sec^2(\theta)$

$\cot^2(\theta) + 1^2=\csc^2(\theta)$

## Identities that come from sums, differences, multiples, and fractions of angles

These are all closely related, but let's go over each kind.

**Angle sum and difference identities**

$\sin(\theta+\phi)=\sin\theta\cos\phi+\cos\theta\sin\phi$

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

$\cos(\theta+\phi)=\cos\theta\cos\phi-\sin\theta\sin\phi$

$\cos(\theta-\phi)=\cos\theta\cos\phi+\sin\theta\sin\phi$

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

$\cos(\theta+\phi)=\cos\theta\cos\phi-\sin\theta\sin\phi$

$\cos(\theta-\phi)=\cos\theta\cos\phi+\sin\theta\sin\phi$

$\tan(\theta-\phi)=\dfrac{\tan\theta-\tan\phi}{1+\tan\theta\tan\phi}$

**Double angle identities**

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

$\cos(2\theta)=2\cos^2\theta-1$

$\tan(2\theta)=\dfrac{2\tan\theta}{1-\tan^2\theta}$

$\cos(2\theta)=2\cos^2\theta-1$

$\tan(2\theta)=\dfrac{2\tan\theta}{1-\tan^2\theta}$

**Half angle identities**

$\sin\dfrac\theta2=\pm\sqrt{\dfrac{1-\cos\theta}{2}}$

$\cos\dfrac\theta2=\pm\sqrt{\dfrac{1+\cos\theta}{2}}$

## Symmetry and periodicity identities

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

$\cos(-\theta)=+\cos(\theta)$

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

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

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

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

$\cos(-\theta)=+\cos(\theta)$

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

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

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

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

## Cofunction identities

$\csc\theta= \sec(\dfrac{\pi}{2}-\theta)$

## Appendix: All trig ratios in the unit circle

Use the movable point to see how the lengths of the ratios change according to the angle.