Question

Find the principal values of the following questions:

$cose{c}^{-1}$(2)

Answer 1

Let $cose{c}^{-1}2=\theta \Rightarrow cosec\theta =2$. We know that the range of principal value of $cose{c}^{-1}\theta$ is $[-\frac{\pi }{2},\frac{\pi }{2}]-\left\{0\right\}.cosec\theta =2=cosec\frac{\pi }{6}$
$\Rightarrow \theta =\frac{\pi }{2}$, where $\theta ϵ[-\frac{\pi }{2},\frac{\pi }{2}]-\left\{0\right\}\Rightarrow cose{c}^{-1}\left(2\right)=\frac{\pi }{6}$
Hence, the principal value of $cose{c}^{-1}\left(2\right)is\frac{\pi }{6}$