Question

The principal value of \(\sin^{-1}\left\{\sin\left(\frac{5\pi}{6}\right)\right\}\) is

• A. \(\frac{\pi}{6}\)
• B. \(\frac{5\pi}{6}\)
• C. \(\frac{7\pi}{6}\)
• D. None of these

Hint:

Even if \(\sin\theta\) is computed from an angle outside \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\), \(\sin^{-1}\) always returns the principal value within this range.

Answer 1

The Correct Option is A

Step 1: Compute \(\sin\left(\frac{5\pi}{6}\right)\).
\[ \sin\left(\frac{5\pi}{6}\right)=\sin\left(\pi-\frac{\pi}{6}\right)=\sin\left(\frac{\pi}{6}\right)=\frac{1}{2} \] Step 2: Apply principal value range of \(\sin^{-1}\).
Principal value of \(\sin^{-1}(x)\) lies in:
\[ \left[-\frac{\pi}{2},\frac{\pi}{2}\right] \] Step 3: Find \(\sin^{-1}\left(\frac{1}{2}\right)\).
\[ \sin^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{6} \] Final Answer: \[ \boxed{\frac{\pi}{6}} \]