For the principal value branch, the value of \(\sin\left(\displaystyle\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{1}{2}\)
The principal value branch of \(\sin^{-1}\) is \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), and \(\sin^{-1}(-x) = -\sin^{-1}x\):
\( \displaystyle \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \)
So the expression becomes:
\( \displaystyle \sin\left(\frac{\pi}{2} + \frac{\pi}{3}\right) = \sin\frac{5\pi}{6} = \frac{1}{2} \)
Equivalently, \(\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta\), and \(\cos\left(-\frac{\pi}{3}\right) = \frac{1}{2}\).