Solution

Let \(sin^{-1}(\frac{1}{\sqrt{2}}) = y\). Then, \(sin y = \frac{1}{\sqrt{2}}\). We know that the range of the principal value branch of \(sin^{-1}\) is \((\frac{-\pi}{2}, \frac{\pi}{2})\) and \(sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\). Therefore, principal value of \(sin^{-1}\) is \(\frac{\pi}{4}\).