Question

Find the principal values of the following questions:

$si{n}^{-1}(-\frac{1}{2})$

Answer 1

Let $si{n}^{-1}(-\frac{1}{2})=\theta \Rightarrow sin\theta =-\frac{1}{2}$
We know that the range of principal value of $si{n}^{-1}\theta is[-\frac{\pi }{2},\frac{\pi }{2}]\therefore sin\theta =-\frac{1}{2}=-sin\frac{\pi }{6}=sin(-\frac{\pi }{6})(\because sin(-\theta )=-sin\theta )$
$\Rightarrow \theta =-\frac{\pi }{6}$, where $\theta ϵ[-\frac{\pi }{2},\frac{\pi }{2}]\Rightarrow si{n}^{-1}(-\frac{1}{2})=-\frac{\pi }{6}$
Hence, the principal value of $si{n}^{-1}(-\frac{1}{2})$ is $-\frac{\pi }{6}$
Note : Principal value of any inverse function is unique