Question

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

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{6} \)

Hint:

Always ensure your final answer for an inverse trig function lies within its principal range. For \( \sin^{-1} \), if the argument is positive, the answer will be in \( [0, \pi/2] \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The principal value is the value of the inverse function that falls within its designated principal value branch.
For \( \sin^{-1} x \), the branch is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).
Step 2: Detailed Explanation:
Let \( \theta = \sin^{-1}(1) \).
This implies \( \sin \theta = 1 \).
We know from the trigonometric table that \( \sin(\frac{\pi}{2}) = 1 \).
Since \( \frac{\pi}{2} \) lies within the interval \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), it is the principal value.
Step 3: Final Answer:
The principal value of \( \sin^{-1}(1) \) is \( \frac{\pi}{2} \).