Question

find the exact value of \\( \sin ^ { - 1 } \left( \sin \frac { 8 \pi } { 9 } \
ight) \\).
write your answer in radians in terms of \\( \pi \\).
if necessary, click on \undefined.\

Explanation:

Step1: Recall the property of inverse sine function

The property of \(y = \sin^{-1}(x)\) is that \(\sin^{-1}(\sin\theta)=\theta\) when \(\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]\).
We know that \(\sin\alpha=\sin(\pi - \alpha)\). So, \(\sin\frac{8\pi}{9}=\sin(\pi-\frac{\pi}{9})=\sin\frac{\pi}{9}\).

Step2: Check the range of the angle

Since \(\frac{\pi}{9}\in[-\frac{\pi}{2},\frac{\pi}{2}]\)

Answer:

\(\frac{\pi}{9}\)