Difference Between Maximum and Minimum Values of sin⁻¹x
Given:
\[
x \in [-1, 1]
\]
To Find:
Difference between maximum and minimum values of
\[
\sin^{-1}x
\]
Concept Used:
The principal value range of inverse sine function is:
\[
\sin^{-1}x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]
\]
Step 1: Maximum Value
Maximum value occurs at \( x = 1 \)
\[
\sin^{-1}(1) = \frac{\pi}{2}
\]
Step 2: Minimum Value
Minimum value occurs at \( x = -1 \)
\[
\sin^{-1}(-1) = -\frac{\pi}{2}
\]
Step 3: Required Difference
\[
\text{Difference} = \frac{\pi}{2} – \left(-\frac{\pi}{2}\right)
= \pi
\]
Final Answer:
\[
\pi
\]