Question

Find the principal value of the following:

${\displaystyle {\sec }^{-1}\left(2\right)}$

Answer 1

Let ${\displaystyle {\sec }^{-1}\left(2\right)=y}$
Then,
sec y = 2
We know that the range of the principal value branch is ${\displaystyle [0,\pi ]-\left\{\frac{\pi }{2}\right\}.}$
Thus,
${\displaystyle \sec y=2=\sec \left(\frac{\pi }{3}\right)}$

${\displaystyle \Rightarrow y=\frac{\pi }{3}\in [0,\pi ],y\ne \frac{\pi }{2}}$

Hence, the principal value of ${\displaystyle {\sec }^{-1}\left(2\right) is \frac{\pi }{3}.}$