# Trigonometric Function

$sin(θ)=cos(π2−θ)=1csc(θ).$$cos(θ)=sin(π2−θ)=1sec(θ).$$tan(θ)=sin(θ)cos(θ)=cot(π2−θ)=1cot(θ).$$sin2(θ)+cos2(θ)=1.$$sin(θ)=sin(θ+2 k π),$$cos(θ)=cos(θ+2 k π),$$tan(θ)=tan(θ+k π).$

## Values

$θsin(θ)cos(θ)tan(θ)0010π6123213π412121π332123π210-$

## Differential Equations

$ddx sin(x)=cos(x),$$ddx cos(x)=−sin(x),$

## Power Series Expansion

$sin(x)=x−x33!+x55!−x77!+⋯=Σn=0∞(−1)n(2 n+1)!x2 n+1.$$cos(x)=1−x22!+x44!−x66!+⋯=Σn=0∞ (−1)n(2 n)! x2 n.$

## Infinite Product Expansion

$sin(z)=z Πn=1∞ (1−z2n2 π2), z:ℂ,$$cos(z)=Πn=1∞ (1−z2(n−1 / 2)2 π2), z:ℂ,$

## Euler's Formula

$ei x=cos(x)+i sin(x).$

## Functional Equations

$cos(x−y)=cos(x) cos(y)+sin(x) sin(y).$

## Identities

### Parity

$sin(−x)=−sin(x),cos(−x)=cos(x),tan(−x)=−tan(x).$

### Sum

$sin(x+y)=sin(x) cos(y)+cos(x) sin(y),cos(x+y)=cos(x) cos(y)−sin(x) sin(y),tan(x+y)=tan(x)+tan(y)1−tan(x) tan(y).$

### Difference

$sin(x−y)=sin(x) cos(y)−cos(x) sin(y),cos(x−y)=cos(x) cos(y)+sin(x) sin(y),tan(x−y)=tan(x)−tan(y)1+tan(x) tan(y).$

### Double-Angle Formulae

$sin(2 x)=sin(x) cos(x)+cos(x) sin(x)=2 sin(x) cos(x)=,cos(2 x)=cos(x) cos(x)−sin(x) sin(x)=cos2(x)−sin2(x)=,tan(2 x)=tan(x)+tan(x)1−tan(x) tan(x)=2 tan(x)1−tan2(x).$$sin(t)=2 t1+t2,cos(t)=1−t21+t2,tan(t)=2 t1−t2.$