Evaluate \( \sin^{-1}(\sin 4) \)
Step-by-Step Solution
We need to evaluate:
\[ \sin^{-1}(\sin 4) \]
Step 1: Principal value range
The principal value range of \( \sin^{-1}x \) is:
\[ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
Step 2: Identify interval of 4
Since \( 4 \in (\pi, \frac{3\pi}{2}) \), we write:
\[ 4 = \pi + \theta \]
Step 3: Use identity
\[ \sin(\pi + \theta) = -\sin \theta \]
\[ \sin(4) = -\sin(4 – \pi) \]
Step 4: Apply inverse sine
\[ \sin^{-1}(\sin 4) = \sin^{-1}[-\sin(4 – \pi)] \]
Since \( (4 – \pi) \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we get:
\[ \sin^{-1}(\sin 4) = \pi – 4 \]
Final Answer
\[ \boxed{\pi – 4} \]