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(π2−x)=cos(x)

cos(π2−x)=sin(x)

sin(π2+x)=cos(x)

cos(π2+x)=−sin(x)

sin(π−x)=sin(x)

cos(π−x)=−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)=sinα⋅cosβ+sinβ⋅cosα

sin(α−β)=sinα⋅cosβ−sinβ⋅cosα

cos(α+β)=cosα⋅cosβ−sinα⋅cosβ

cos(α−β)=cosα⋅cosβ+sinα⋅cosβ

tan(α+β)=tanα+tanβ1−tanα⋅tanβ

tan(α−β)=tanα−tanβ1+tanα⋅tanβ

Double Angle and Half Angle Formulas

sin(2α)=2⋅sinα⋅cosα

cos(2α)=cos2α−sin2α

tan(2α)=2tanα1−tan2α

sinα2=±1−cosα2−−−−−−−−√

cosα2=±1+cosα2−−−−−−−−√

tanα2=1−cosαsinα=sinα1−cosα

tanα2=±1+cosα1−cosα−−−−−−−−√

Other Useful Trig Formulas

Law of sines

sinαα=sinββ=sinγγ

Law of cosines

a2=b2+c2−2⋅b⋅c⋅cosαb2=a2+c2−2⋅a⋅c⋅cosβc2=a2+b2−2⋅a⋅b⋅cosγ

Area of triangle

A=12absinγ