When evaluating inverse trigonometric functions, make sure the result lies within the principal range of the function. If necessary, use trigonometric identities to adjust the angle to the correct range.
We are asked to evaluate \( \sin^{-1} \left( \sin \frac{3\pi}{5} \right) \).
The inverse sine function \( \sin^{-1}(x) \) gives the angle \( \theta \) such that \( \sin(\theta) = x \) and \( \theta \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \).
Since \( \frac{3\pi}{5} \) is greater than \( \frac{\pi}{2} \), we need to adjust the angle so that it lies within the principal range \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \).
We can use the identity:
\[
\sin \left( \pi - x \right) = \sin x.
\]
Thus, we have:
\[
\sin \left( \frac{3\pi}{5} \right) = \sin \left( \pi - \frac{3\pi}{5} \right) = \sin \left( \frac{2\pi}{5} \right).
\]
Therefore:
\[
\sin^{-1} \left( \sin \frac{3\pi}{5} \right) = \frac{2\pi}{5}.
\]
Thus, the value of \( \sin^{-1} \left( \sin \frac{3\pi}{5} \right) \) is \( \boxed{\frac{2\pi}{5}} \).
