Question

Write the principal value of \[\sin^{- 1} \left\{ \cos\left( \sin^{- 1} \frac{1}{2} \right) \right\}\]

Answer 1

\[\sin^{- 1} \left\{ \cos\left( \sin^{- 1} \frac{1}{2} \right) \right\} = \sin^{- 1} \left\{ \cos\left[ \sin^{- 1} \left( \sin\frac{\pi}{3} \right) \right] \right\}\]
\[ = \sin^{- 1} \left[ \cos\left( \frac{\pi}{3} \right) \right]\]
\[ = \sin^{- 1} \left[ \frac{1}{2} \right]\]
\[ = \sin^{- 1} \left[ \sin\left( \frac{\pi}{3} \right) \right]\]
\[ = \frac{\pi}{3}\]