Question

Principal value of \( \sin^{-1}(1) \) is:

Hint:

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

Answer 1

Step 1: Understanding the inverse sine function.
The principal value of \( \sin^{-1}(x) \) is the angle \( \theta \) in the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) such that \( \sin(\theta) = x \). Step 2: Evaluating the inverse sine.
For \( \sin^{-1}(1) \), we need to find the angle \( \theta \) such that \( \sin(\theta) = 1 \) in the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). The angle \( \theta = \frac{\pi}{2} \) satisfies this condition. Step 3: Conclusion.
Thus, the principal value of \( \sin^{-1}(1) \) is \( \frac{\pi}{2} \).