Question

Write the value of sin⁻¹ \[\left( \cos\frac{\pi}{9} \right)\]

Answer 1

Consider,
\[\sin^{- 1} \left( \cos\frac{\pi}{9} \right) = \sin^{- 1} \left\{ \sin\left( \frac{\pi}{2} - \frac{\pi}{9} \right) \right\} \left[ \because \cos x = \sin\left( \frac{\pi}{2} - x \right) \right]\]
\[ = \sin^{- 1} \left\{ \sin\left( \frac{7\pi}{18} \right) \right\}\]
\[ = \frac{7\pi}{18} \left[ \because \sin^{- 1} \left( \sin{x} \right) = x \right]\]
∴ \[\sin^{- 1} \left( \cos\frac{\pi}{9} \right) = \frac{7\pi}{18}\]