Calculus/Table of Trigonometry

Definitions

• $\tan (x)=\frac{\sin x}{\cos x}$
• $\sec (x)=\frac{1}{\cos x}$
• $\cot (x)=\frac{\cos x}{\sin x}=\frac{1}{\tan x}$
• $\csc (x)=\frac{1}{\sin x}$

Pythagorean Identities

• ${\sin }^{2}x+{\cos }^{2}x=1$
• $1+{\tan }^2(x)={\sec }^{2}x$
• $1+{\cot }^2(x)={\csc }^{2}x$

Double Angle Identities

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

Angle Sum Identities

$\sin (x+y)=\sin x\cos y+\cos x\sin y$

$\sin (x-y)=\sin x\cos y-\cos x\sin y$

$\cos (x+y)=\cos x\cos y-\sin x\sin y$

$\cos (x-y)=\cos x\cos y+\sin x\sin y$

$\sin x+\sin y=2\sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$

$\sin x-\sin y=2\cos \left(\frac{x+y}{2}\right)\sin \left(\frac{x-y}{2}\right)$

$\cos x+\cos y=2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$

$\cos x-\cos y=-2\sin \left(\frac{x+y}{2}\right)\sin \left(\frac{x-y}{2}\right)$

$\tan x+\tan y=\frac{\sin (x+y)}{\cos x\cos y}$

$\tan x-\tan y=\frac{\sin (x-y)}{\cos x\cos y}$

$\cot x+\cot y=\frac{\sin (x+y)}{\sin x\sin y}$

$\cot x-\cot y=\frac{-\sin (x-y)}{\sin x\sin y}$

Product-to-sum identities

$\cos \left(x\right)\cos \left(y\right)=\frac{\cos (x+y)+\cos (x-y)}{2}$

$\sin \left(x\right)\sin \left(y\right)=\frac{\cos (x-y)-\cos (x+y)}{2}$

$\sin \left(x\right)\cos \left(y\right)=\frac{\sin (x+y)+\sin (x-y)}{2}$

$\cos \left(x\right)\sin \left(y\right)=\frac{\sin (x+y)-\sin (x-y)}{2}$