Evaluate each of the following: tan^-1(tan 7π/6)

Question

Evaluate each of the following:

tan^-1(tan \(\frac{7\pi}6\))

Answer 1

The value of tan\(\frac{7\pi}6\) = \(\frac{1}{\sqrt3}\)

∴ The question becomes tan^–1\((\frac{1}{\sqrt3})\)

Let,

tan^–1\((\frac{1}{\sqrt3})\) = y

⇒ tan y =\((\frac{1}{\sqrt3})\)

⇒ tan\((\frac \pi{6})=(\frac{1}{\sqrt3})\)

The range of the principal value of tan^–1 is \((\frac{-\pi}2,\frac{\pi}2) \) and tan\((\frac \pi{6})=(\frac{1}{\sqrt3})\).

∴ The value of tan^–1(tan\(\frac{7\pi}6\)) is \(\frac{\pi}6\).