Evaluate \( \sin^{-1}(\sin 3) \)
Step-by-Step Solution
We need to evaluate:
\[ \sin^{-1}(\sin 3) \]
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 \( 3 \notin \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we cannot directly write:
\[ \sin^{-1}(\sin 3) \neq 3 \]
Step 3: Use identity
\[ \sin(\pi – x) = \sin x \]
\[ \sin(3) = \sin(\pi – 3) \]
Now \( \pi – 3 \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \)
Step 4: Apply inverse sine
\[ \sin^{-1}(\sin 3) = \sin^{-1}(\sin(\pi – 3)) = \pi – 3 \]
Final Answer
\[ \boxed{\pi – 3} \]