Question

Find the principal value of the following: cos^{- 1}`(-1/2)`

Answer 1

The principal value branch of cos^{- 1}x [0, π].

Let cos^{- 1}`(-1/2)`= α, where 0 ≤ α ≤ π

∴ cos α = `-1/2 = -cos  pi/(3)`

∴ cos α = `cos(pi - pi/3)`     ...[∵ cos(π – θ) = – cosθ]

∴ cos α = cos `(2pi)/(3)`

∴ α = `(2pi)/(3)            ...[∵ 0 ≤ (2pi)/(3) ≤ pi]`

∴ the principal value of cos^{- 1}`(-1/2) "is" (2pi)/(3)`.