Question

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

${\sin }^{-1}\left(\sin \frac{11\mathrm{\pi }}{5}\right)$

Short answer

The exact value of the given expression is $\frac{\mathrm{\pi }}{5}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

${\sin }^{-1}\left(\sin \frac{11\mathrm{\pi }}{5}\right)$

The composite function follows the form $f^{-1}\left(f\left(x\right)\right)$, where $-\frac{\mathrm{\pi }}{2}\le x\le \frac{\mathrm{\pi }}{2}$.

The value $\frac{11\mathrm{\pi }}{5}$does not lie in the interval $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$.

Step 2. Determine where the quadrant lies.

The angle $\frac{11\mathrm{\pi }}{5}$lies in the fourth quadrant. Therefore, the angle of $\frac{11\mathrm{\pi }}{5}$ is given by,

role="math" localid="1646502703272" $$\begin{gathered} =\frac{11\mathrm{\pi }}{5}-2\mathrm{\pi } \\ =\frac{\mathrm{\pi }}{5} \end{gathered}$$

The angle $\frac{\mathrm{\pi }}{5}$is in the interval $\left[-\frac{\mathrm{\pi }}{2},-\frac{\mathrm{\pi }}{2}\right]$and $\sin \frac{11\mathrm{\pi }}{5}=\sin \left(\frac{\mathrm{\pi }}{5}\right)$.

Hence,${\sin }^{-1}\left(\sin \frac{11\mathrm{\pi }}{5}\right)=\frac{\mathrm{\pi }}{5}$