\(\sin^{-1}\left(\cos\displaystyle\frac{3\pi}{5}\right)\) equals
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{-\pi}{10}\)
Since \(\cos\theta = \sin\left(\frac{\pi}{2} - \theta\right)\),
\(\displaystyle \cos\frac{3\pi}{5} = \sin\left(\frac{\pi}{2} - \frac{3\pi}{5}\right) = \sin\left(-\frac{\pi}{10}\right)\)
The principal value branch of \(\sin^{-1}\) is \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), and \(-\frac{\pi}{10}\) lies in it, so
\(\displaystyle \sin^{-1}\left(\cos\frac{3\pi}{5}\right) = \sin^{-1}\left(\sin\left(-\frac{\pi}{10}\right)\right) = -\frac{\pi}{10}\)