Math formulas and cheat sheets for Trigonometry Identities

Trigonometry Identities

Right-Triangle Definitions

sinα=OppositeHypotenuse
cosα=AdjacentHypotenuse
tanα=OppositeAdjacent
cscα=1sinα=HypotenuseOpposite
secα=1cosα=HypotenuseAdjacent
cotα=1tanα=AdjacentOpposite

Reduction Formulas

sin(x)=sin(x)
cos(x)=cos(x)
sin(π2x)=cos(x)
cos(π2x)=sin(x)
sin(π2+x)=cos(x)
cos(π2+x)=sin(x)
sin(πx)=sin(x)
cos(πx)=cos(x)
sin(π+x)=sin(x)
cos(π+x)=cos(x)

Basic Identities

sin2x+cos2x=1
tan2x+1=1cos2x
cot2x+1=1sin2x

Sum and Difference Formulas

sin(α+β)=sinαcosβ+sinβcosα
sin(αβ)=sinαcosβsinβcosα
cos(α+β)=cosαcosβsinαcosβ
cos(αβ)=cosαcosβ+sinαcosβ
tan(α+β)=tanα+tanβ1tanαtanβ
tan(αβ)=tanαtanβ1+tanαtanβ

Double Angle and Half Angle Formulas

sin(2α)=2sinαcosα
cos(2α)=cos2αsin2α
tan(2α)=2tanα1tan2α
sinα2=±1cosα2
cosα2=±1+cosα2
tanα2=1cosαsinα=sinα1cosα
tanα2=±1+cosα1cosα

Other Useful Trig Formulas

Law of sines

sinαα=sinββ=sinγγ

Law of cosines

a2=b2+c22bccosαb2=a2+c22accosβc2=a2+b22abcosγ

Area of triangle

A=12absinγ