Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>$\sin^{-1}(\sin \theta) = \theta$, for all $\theta$ belonging to</p>
<p>(a) $[0, \pi]$</p>
<p>(b) $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$</p>
<p>(c) $\left[-\frac{\pi}{2}, 0\right]$</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: The inverse sine function returns values only within its principal value branch, which is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.
<p>The principal value branch of $\sin^{-1}$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. Therefore, $\sin^{-1}(\sin \theta) = \theta$ holds for all $\theta$ in this interval.</p>
Correct Answer: B