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The principal value of $\sin^{-1}(x)$ lies in the interval:
A$[0, 2\pi]$
B$\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$
C$[-\pi, \pi]$
D$[0, \pi]$
Answer & Solution
Correct answer: B. $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$
The principal-value range of $\sin^{-1}$ is $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$. This is the unique interval on which $\sin$ is monotonic and covers its full range $[-1, 1]$.
Quick reference:
- $\sin^{-1}, \tan^{-1}, \csc^{-1}$: range $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ (with closed/open variations).
- $\cos^{-1}, \sec^{-1}, \cot^{-1}$: range $[0, \pi]$ (option A is for these).
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