Evaluate \( \sin^{-1}(\sin 12) \)
Step-by-Step Solution
We need to evaluate:
\[ \sin^{-1}(\sin 12) \]
Step 1: Principal value range
The principal value range of \( \sin^{-1}x \) is:
\[ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
Step 2: Reduce the angle
Since 12 is not in the principal range, we find an equivalent angle:
\[ 12 – 2\pi \approx 12 – 6.283 = 5.717 \]
Still not in range, so subtract again:
\[ 12 – 4\pi \approx 12 – 12.566 = -0.566 \]
This lies in \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \)
Step 3: Apply inverse sine
\[ \sin^{-1}(\sin 12) = -0.566 \text{ (approx)} \]
Final Answer
\[ \boxed{12 – 4\pi} \]