Find the exact value of $\sin ^{-1}\left(\sin \left(-\frac{4 \pi}{9}\right)\right)$.
Write your answer in radians in terms of $\pi$. If necessary, click on "Undefined."

Step 1 ：The inverse sine function, or arcsine, is defined as the inverse of the sine function. It returns the angle whose sine is a given number. In this case, we are asked to find the angle whose sine is \(\sin \left(-\frac{4 \pi}{9}\right)\).

Step 2 ：The sine function is periodic with a period of \(2\pi\), so \(\sin \left(-\frac{4 \pi}{9}\right)\) is the same as \(\sin \left(\frac{5 \pi}{9}\right)\).

Step 3 ：The range of the inverse sine function is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). Since \(\frac{5 \pi}{9}\) is not in this range, we need to find an equivalent angle that is in this range.

Step 4 ：We can do this by subtracting \(\pi\) from \(\frac{5 \pi}{9}\), which gives us \(-\frac{4 \pi}{9}\).

Step 5 ：The equivalent angle in the range of the inverse sine function is \(-\frac{4 \pi}{9}\), which is the same as the original angle. This means that \(\sin ^{-1}\left(\sin \left(-\frac{4 \pi}{9}\right)\right) = -\frac{4 \pi}{9}\).

Step 6 ：Final Answer: \(\boxed{-\frac{4 \pi}{9}}\)