Question

Find the principal value of \( \sin^{-1} \left( \frac{1}{\sqrt{2}} \right)\)

Hint:

For values of \( x \) in the domain of the arcsine function, the principal value is the angle within the interval \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \).

Answer 1

Step 1: Understanding the principal value of \( \sin^{-1} \).
The principal value of \( \sin^{-1} x \) is the value of the angle \( \theta \) such that \( \sin \theta = x \) and \( \theta \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \).

Step 2: Applying the value.
We are given \( \sin^{-1} \left( \frac{1}{\sqrt{2}} \right) \), which corresponds to the angle \( \theta = \frac{\pi}{4} \) because \( \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \).

Step 3: Conclusion.
Thus, the principal value of \( \sin^{-1} \left( \frac{1}{\sqrt{2}} \right) \) is \( \frac{\pi}{4} \).