Classwork: Trigonometric Integrals

Trigonometric Identities

Basic Identities

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

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

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

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

Derivatives and Integrals of Trig Functions

Derivatives of Trig Functions

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

Integrals of Trig Functions

• $\int \cos xdx=\sin x+C$
• $\int \sin xdx=–\cos x+C$
• $\int {\sec }^{2}xdx=\tan x+C$
• $\int \sec x\tan xdx=\sec x+C$
• $\int \tan xdx=\ln |\sec x|+C$
• $\int \sec xdx=\ln |\sec x+\tan x|+C$

Example: $\int {\cos }^{3}x\sin xdx$