Mark the tick against the correct answer in the following: The value of $\sec ^{-1}\left(\sec \frac{8 \pi}{5}\right)$ is A. $\frac{2 \pi}{5}$ B. $\frac{3 \pi}{5}$ C. $\frac{8 \pi}{5}$ D. none of these

Home » Mark the tick against the correct answer in the following: The value of is A. B. C. D. none of these

Mark the tick against the correct answer in the following: The value of $\sec ^{-1}\left(\sec \frac{8 \pi}{5}\right)$ is
A. $\frac{2 \pi}{5}$
B. $\frac{3 \pi}{5}$
C. $\frac{8 \pi}{5}$
D. none of these

CBSE | Class 12 | Inverse Trigonometric Functions | Maths | RS Aggarwal

Mark the tick against the correct answer in the following: The value of $\sec ^{-1}\left(\sec \frac{8 \pi}{5}\right)$ is
A. $\frac{2 \pi}{5}$
B. $\frac{3 \pi}{5}$
C. $\frac{8 \pi}{5}$
D. none of these

Solution:

Option(A) is correct.
To Find: The value of $\sec ^{-1}\left(\sec \left(\frac{8 \pi}{5}\right)\right)$
Now, let $x=\sec ^{-1}\left(\sec \left(\frac{8 \pi}{5}\right)\right)$
$\Rightarrow \sec x=\sec \left(\frac{8 \pi}{5}\right)$
Here range of principle value of sec is $[0, \pi]$
$\Rightarrow \mathrm{x}=\frac{\ln }{5} \notin[0, \pi]$
Hence for all values of $x$ in range $[0, \pi]$ , the value of $\sec ^{-1}\left(\sec \left(\frac{8 \pi}{5}\right)\right)$ is
$\begin{array}{l} \Rightarrow \sec x=\sec \left(2 \pi-\frac{2 \pi}{5}\right)\left(\because \sec \left(\frac{8 \pi}{5}\right)=\sec \left(2 \pi-\frac{2 \pi}{5}\right)\right) \\ \Rightarrow \sec x=\sec \left(\frac{2 \pi}{5}\right)(\because \sec (2 \pi-\theta)=\sec \theta) \\ \Rightarrow x=\frac{2 \pi}{5} \end{array}$