Evaluate \( \sin^{-1}(\sin 2) \)
Step-by-Step Solution
We need to evaluate:
\[ \sin^{-1}(\sin 2) \]
Step 1: Principal value range
The principal value range of \( \sin^{-1}x \) is:
\[ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
Step 2: Check the angle
Since \( 2 \notin \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we convert it using identity:
\[ \sin(\pi – x) = \sin x \]
\[ \sin(2) = \sin(\pi – 2) \]
Step 3: Apply inverse sine
\[ \sin^{-1}(\sin 2) = \sin^{-1}(\sin(\pi – 2)) = \pi – 2 \]
Now \( \pi – 2 \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), so it is valid.
Final Answer
\[ \boxed{\pi – 2} \]