Inverse Trigonometric Functions — Chapter Summary

Trigonometric functions are periodic

Trigonometric functions are periodic, so they are neither one-one nor onto on their full natural domains. By restricting their domains and codomains,

The principal value branches are sin^

The principal value branches are sin^-1:[-1,1]→[-π/2,π/2], cos^-1:[-1,1]→[0,π], tan^-1:R→(-π/2,π/2), cot^-1:R→(0,π), sec^-1:R-(-1,1)→[0,π]-{π/2}, and

sin(sin^

sin(sin^-1 x)=x for x in [-1,1], cos(cos^-1 x)=x for x in [-1,1], and tan(tan^-1 x)=x for all real x. The reverse compositions are true only when the

sin^

sin^-1(sin θ)=θ only for θ in [-π/2,π/2], cos^-1(cos θ)=θ only for θ in [0,π], tan^-1(tan θ)=θ only for θ in (-π/2,π/2), cot^-1(cot θ)=θ only for θ in

Odd properties

Odd properties: sin^-1(-x)=-sin^-1 x, tan^-1(-x)=-tan^-1 x, and cosec^-1(-x)=-cosec^-1 x. Supplementary properties: cos^-1(-x)=π-cos^-1 x, cot^-1(-x)=