Inverse Trigonometric Functions — Formula Sheet

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}

sin^-1(-x) = -sin^-1 x

cos^-1(-x) = π - cos^-1 x

tan^-1(-x) = -tan^-1 x

cot^-1(-x) = π - cot^-1 x

sec^-1(-x) = π - sec^-1 x

sin^-1(P/H) = cos^-1(B/H) = tan^-1(P/B) = cot^-1(B/P) = sec^-1(H/B) = cosec^-1(H/P)

sin^-1 x = cos^-1√(1-x²) = tan^-1[x/√(1-x²)]

cos^-1 x = sin^-1√(1-x²) = tan^-1[√(1-x²)/x]

tan^-1 x = sin^-1[x/√(1+x²)] = cos^-1[1/√(1+x²)]

cosec^-1(1/x) = sin^-1 x

Domain of sin^-1 x: [-1,1]

Domain of cos^-1 x: [-1,1]

Domain of tan^-1 x: R

Domain of cot^-1 x: R

Domain of sec^-1 x: R-(-1,1)

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.

Odd properties

sin^-1(-x)=-sin^-1 x, tan^-1(-x)=-tan^-1 x, and cosec^-1(-x)=-cosec^-1 x.