Question

Find the principal value of \( \sin^{-1} \left( \sin \frac{7\pi}{4} \right) \).

Hint:

The principal value of \( \sin^{-1} x \) always lies in the range \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \).

Answer 1

Step 1: Identify the principal value range for \( \sin^{-1} \). \[ -\frac{\pi}{2} \leq \sin^{-1} x \leq \frac{\pi}{2} \] Step 2: Convert \( \sin \frac{7\pi}{4} \) to its equivalent angle. \[ \sin \frac{7\pi}{4} = \sin \left( -\frac{\pi}{4} \right) \] Step 3: Apply inverse sine. \[ \sin^{-1} \left( \sin \left( -\frac{\pi}{4} \right) \right) = -\frac{\pi}{4} \]