Question

Find the exact value of the expression. Express angles in radians.

${\sin }^{-1}\left(-\frac{\sqrt{2}}{2}\right)$

Short answer

The exact value of the given expression is $-\frac{\mathrm{\pi }}{4}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

${\sin }^{-1}\left(-\frac{\sqrt{2}}{2}\right)$

Assume $\theta ={\sin }^{-1}\left(-\frac{\sqrt{2}}{2}\right)$

As $y={\sin }^{-1}x$means $x=\sin y$, where,

$-\frac{\mathrm{\pi }}{2}\le \theta \le \frac{\mathrm{\pi }}{2}$

Rewrite the function,

$\sin \theta =-\frac{\sqrt{2}}{2}$

Step 2. Determine the interval.

The only angle $\theta$within the interval $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$whose sine is $-\frac{\sqrt{2}}{2}$is given below,

$-\frac{\mathrm{\pi }}{4}$

Then, we can say,

localid="1646501876925" $$\begin{gathered} \sin \theta =-\frac{\sqrt{2}}{2} \\ \sin \theta =\sin \left(-\frac{\mathrm{\pi }}{4}\right) \\ \theta =-\frac{\mathrm{\pi }}{4} \end{gathered}$$