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Question
Find the principal value of the following: tan- 1( - √3)
Solution
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)`.
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