Write the principal value of cos^-1(cos 2π/3) + sin^-1 (sin 2π/3)

Question

Write the principal value of

\(cos^{-1}(cos\frac{2\pi}{3})+sin^{-1}(sin\frac{2\pi}{3}).\)

Answer 1

We know that sin^-1 (sin θ) = π – θ, 

if θ ∈ [π/2, 3π/2] and cos^-1 (cos θ) = θ, if θ ∈ [0, π]

Given \(cos^{-1}(cos\frac{2\pi}{3})+sin^{-1}(sin\frac{2\pi}{3}).\) 

\(=\frac{2\pi}{3}+(\pi-\frac{2\pi}{3})\)

= π

\(\therefore \)\(cos^{-1}(cos\frac{2\pi}{3})+sin^{-1}(sin\frac{2\pi}{3})\)\(=\pi\)