Trigonometric Identities

Basic Identities

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

Double angle identities (use these to reduce the power of sine or cosine)

  • $\sin ^2 x=\frac{1-\cos (2 x)}{2} $
  • $\cos ^2 x=\frac{1+\cos (2 x)}{2}$

Derivatives and Integrals of Trig Functions

Derivatives of Trig Functions

  • $ \frac{d}{d x}(\sin x)=\cos x $
  • $ \frac{d}{d x}(\cos x)=-\sin x $
  • $ \frac{d}{d x}(\tan x)=\sec ^2 x $
  • $ \frac{d}{d x}(\cot x)=-\csc^2 x $
  • $ \frac{d}{d x}(\sec x)=\sec x \tan x $
  • $ \frac{d}{d x}(\csc x)=-\csc x \cot x$

Integrals of Trig Functions

  • $\int \cos x dx = \sin x + C$
  • $\int \sin x dx = – \cos x + C$
  • $ \int \sec ^2 x d x=\tan x+C $
  • $ \int \sec x \tan x d x=\sec x+C $
  • $ \int \tan x d x=\ln |\sec x|+C $
  • $ \int \sec x d x=\ln |\sec x+\tan x|+C$

Example: $\int \cos^3 x \sin x dx$