# How do you evaluate csc^-1(sqrt2)?

Jun 24, 2016

$\frac{\pi}{4}$

#### Explanation:

Let $\theta = {\csc}^{-} 1 \left(\sqrt{2}\right)$.

Since $\csc \left(x\right)$ and ${\csc}^{-} 1 \left(x\right)$ are inverse functions, this means that $\csc \left(\theta\right) = \sqrt{2}$.

Another way of reaching that fact is to take the cosecant of both sides: $\csc \left(\theta\right) = \csc \left({\csc}^{-} 1 \left(\sqrt{2}\right)\right)$, and since $\csc \left({\csc}^{-} 1 \left(x\right)\right) = x$, this becomes $\csc \left(\theta\right) = \sqrt{2}$.

Taking the reciprocal of both sides, the equation becomes $\sin \left(\theta\right) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.

So, we want to find $\theta$, or the angle where the value of sine is $\frac{\sqrt{2}}{2}$.

This is a well known value of sine. It occurs when $\theta = \frac{\pi}{4}$, which means that ${\csc}^{-} 1 \left(\sqrt{2}\right) = \frac{\pi}{4}$.