Question

Find the principal value of the following: tan^-1(– 1)

Answer 1

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

Let tan^-1(–1) = α, where `(-pi)/(2) ≤ α ≤ pi/(2)`

∴ tan α = – 1 = `-tan  pi/(4)`

∴ tan α = `tan(- pi/4)`  ...[ ∵ tan(– θ) = – tan θ]

∴ α = `- pi/(4)       ...[ ∵ - pi/2 ≤ - pi/4 ≤ pi/2 ]`

∴ the principal value of tan^-1(–1) is `-pi/(4)`.