Evaluate each of the following:tan-1(tan \(\frac{7\pi}6\))

1.	Evaluate each of the following:tan-1(tan \(\frac{7\pi}6\))
Answer» 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 \(\frac{\pi}6\).

Answer»

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 \(\frac{\pi}6\).

Discussion