The value of \(sec^{-1}\left(sec\frac{8\pi}{5}\right)\)isA. \(\frac{2

1.	The value of \(sec^{-1}\left(sec\frac{8\pi}{5}\right)\)isA. \(\frac{2\pi}{5}\)B. \(\frac{3\pi}{5}\)C. \(\frac{8\pi}{5}\)D. none of these
Answer» Correct Answer is \(\frac{2\pi}{5}\) Now, let x = \(sec^{-1}\left(sec\frac{8\pi}{5}\right)\) ⇒ sec x =sec ( \(\frac{8\pi}{5}\)) Here range of principle value of sec is [0, π] ⇒ x = \(\frac{8\pi}{5}\)∉ [0, π] Hence for all values of x in range [0, π] ,the value of \(sec^{-1}\left(sec\frac{8\pi}{5}\right)\) is ⇒ sec x =sec (2π - \(\frac{8\pi}{5}\) ) ( \(\because\)sec ( \(\frac{8\pi}{5}\))= sec (2π - \(\frac{2\pi}{5}\)) ) ⇒ sec x =sec ( \(\frac{2\pi}{5}\)) ( \(\because\)sec (2π - θ)= sec θ) ⇒ x= \(\frac{2\pi}{5}\)

The value of \(sec^{-1}\left(sec\frac{8\pi}{5}\right)\)isA. \(\frac{2\pi}{5}\)B. \(\frac{3\pi}{5}\)C. \(\frac{8\pi}{5}\)D. none of these

Answer»

Correct Answer is \(\frac{2\pi}{5}\)

Now, let x = \(sec^{-1}\left(sec\frac{8\pi}{5}\right)\)

⇒ sec x =sec ( \(\frac{8\pi}{5}\))

Here range of principle value of sec is [0, π]

⇒ x = \(\frac{8\pi}{5}\)∉ [0, π]

Hence for all values of x in range [0, π] ,the value of

\(sec^{-1}\left(sec\frac{8\pi}{5}\right)\) is

⇒ sec x =sec (2π - \(\frac{8\pi}{5}\) ) ( \(\because\)sec ( \(\frac{8\pi}{5}\))= sec (2π - \(\frac{2\pi}{5}\)) )

⇒ sec x =sec ( \(\frac{2\pi}{5}\)) ( \(\because\)sec (2π - θ)= sec θ)

⇒ x= \(\frac{2\pi}{5}\)