The value of \(tan^{-1}\left(tan\frac{7\pi}{6}\right)\)isA. \(\frac{7

1.	The value of \(tan^{-1}\left(tan\frac{7\pi}{6}\right)\)isA. \(\frac{7\pi}{6}\) B. \(\frac{5\pi}{6}\)C. \(\frac{\pi}{6}\)D. none of these
Answer» Correct Answer is \(\frac{\pi}{6}\) Now, let x = \(tan^{-1}\left(tan\frac{7\pi}{6}\right)\) ⇒ tan x =tan ( \(\frac{7\pi}{6}\)) Here range of principle value of tan is [ \(-\frac{\pi}{2},\frac{\pi}{2}\)] ⇒ x = \(\frac{7\pi}{6}\)∉ [ \(-\frac{\pi}{2},\frac{\pi}{2}\)] Hence for all values of x in range [\(-\frac{\pi}{2},\frac{\pi}{2}\) ] ,the value of \(tan^{-1}\left(tan\frac{13\pi}{6}\right)\) is ⇒ tan x =tan (π + \(\frac{\pi}{6}\) ) (\(\because\) tan ( \(\frac{7\pi}{6}\))= tan (π +\(\frac{\pi}{6}\)) ) ⇒ tan x =tan ( \(\frac{\pi}{6}\)) (\(\because\) tan (π+ θ)= tan θ) ⇒ x =\(\frac{\pi}{6}\)

The value of \(tan^{-1}\left(tan\frac{7\pi}{6}\right)\)isA. \(\frac{7\pi}{6}\) B. \(\frac{5\pi}{6}\)C. \(\frac{\pi}{6}\)D. none of these

Answer»

Correct Answer is \(\frac{\pi}{6}\)

Now, let x = \(tan^{-1}\left(tan\frac{7\pi}{6}\right)\)

⇒ tan x =tan ( \(\frac{7\pi}{6}\))

Here range of principle value of tan is [ \(-\frac{\pi}{2},\frac{\pi}{2}\)]

⇒ x = \(\frac{7\pi}{6}\)∉ [ \(-\frac{\pi}{2},\frac{\pi}{2}\)]

Hence for all values of x in range [\(-\frac{\pi}{2},\frac{\pi}{2}\) ] ,the value of

\(tan^{-1}\left(tan\frac{13\pi}{6}\right)\) is

⇒ tan x =tan (π + \(\frac{\pi}{6}\) ) (\(\because\) tan ( \(\frac{7\pi}{6}\))= tan (π +\(\frac{\pi}{6}\)) )

⇒ tan x =tan ( \(\frac{\pi}{6}\)) (\(\because\) tan (π+ θ)= tan θ)

⇒ x =\(\frac{\pi}{6}\)