The value of tan^-1 (tan 7(tan 7π/6) is

Question

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

A. \(\frac{7\pi}{6}\) 

B. \(\frac{5\pi}{6}\)

C. \(\frac{\pi}{6}\)

D. none of these

Answer 1

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