Question

Find the principal value of the following: tan^{- 1}( - √3)

Answer 1

The principal value branch of tan^-1x  is `(- π/2, π/2)`.

Let tan^{- 1}(- √3) = α, where `(-π)/(2) ≤ α ≤ π/(2)`

∴ tan α = - √3 = - tan  `pi/(3)`

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

∴ α = `- pi/(3)                 ...[∵ - pi/2 < (-pi)/3 < pi/2 ]`

∴ the principal value of tan^{- 1}( - √3) is `-pi/(3)`.