Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>$\csc^{-1}(\csc \theta) = \theta$, for all $\theta$ belonging to</p>
<p>(a) $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}$</p>
<p>(b) $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) - \{0\}$</p>
<p>(c) (c) option not fully visible</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: The inverse cosecant function excludes 0 from its principal branch since $\csc(0)$ is undefined.
<p>The principal value branch of $\csc^{-1}$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}$. Therefore, $\csc^{-1}(\csc \theta) = \theta$ holds for all $\theta$ in this interval, excluding 0 where $\csc$ is undefined.</p>
Correct Answer: A