Trigonometric Identities

$$\newcommand{\cosec}{\mathrm{cosec}}$$

The Basics

Euler Equation: $$e^{i \theta}=\cos \theta + i \sin \theta$$

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

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

$$\sec \theta = \frac{1}{\cos \theta}$$

$$\cosec \theta = \frac{1}{\sin \theta}$$

Pythagorean formula

for sines and cosines: $$\sin^{2} \theta + \cos^{2} \theta =1$$

for tangents and secants: $$\sec^{2} \theta = 1+\tan^{2} \theta$$

for cotangents and cosectants: $$\cosec^{2} \theta=1+\cot^{2} \theta$$

Angle Addition

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

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

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

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

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

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

Double Angles

$$\sin 2 \theta = 2 \sin \theta \cos \theta$$

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

$$\tan 2 \theta = \frac{2 \tan \theta}{1-\tan^{2} \theta}$$

Squared

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

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

$$\tan^{2} \theta = \frac{1-\cos 2 \theta}{1+\cos 2 \theta}$$

Half Angles

$$\sin (\theta/2)= \pm \sqrt{0.5 (1-\cos \theta)}$$

$$\cos (\theta/2)= \pm \sqrt{0.5 (1+\cos \theta)}$$ <

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

Triple Angles

$$\sin 3 \theta = 3 \sin \theta – 4 \sin^{3} \theta$$

$$\cos 3 \theta = 4 \cos^{3} \theta – 3 \cos \theta$$

$$\tan 3 \theta = \frac{3 \tan \theta – \tan^{3} \theta}{1-3 \tan^{2} \theta}$$

Addition

$$\sin \theta + \sin \phi = 2 \sin \frac{\theta+\phi}{2} \cos\frac{\theta-\phi}{2}$$

$$\sin \theta – \sin \phi = 2 \cos \frac{\theta+\phi}{2} \sin\frac{\theta-\phi}{2}$$

$$\cos \theta + \cos \phi = 2 \cos \frac{\theta+\phi}{2} \cos\frac{\theta-\phi}{2}$$

$$\cos \theta – \cos \phi = -2 \sin \frac{\theta+\phi}{2} \sin\frac{\theta-\phi}{2}$$

Products

$$\sin \theta \cos \phi=\frac{\sin(\theta+\phi)+\sin(\theta-\phi)}{2}$$

$$\cos \theta \cos \phi=\frac{\cos(\theta+\phi)+\cos(\theta-\phi)}{2}$$

$$\sin \theta \sin \phi=\frac{\cos(\theta-\phi)-\cos(\theta+\phi)}{2}$$

Hyperbolic

$$\sinh \theta = \frac{e^\theta-e^{-\theta}}{2}=i \sin i \theta$$

$$\cosh \theta = \frac{e^\theta+e^{-\theta}}{2}= \cos i \theta$$

$$\tanh \theta = \frac{\sinh \theta}{\cosh \theta} = \frac{e^\theta-e^{-\theta}}{e^\theta+e^{-\theta}}= -i \tan i \theta$$

$$e^\theta=\cosh \theta + \sinh \theta$$

$$e^{-\theta}=\cosh \theta – \sinh \theta$$