Question

Write the value of

\[\cos^{- 1} \left( \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)\].

Answer 1

We have

\[\cos^{- 1} \frac{1}{2} + 2 \sin^{- 1} \frac{1}{2}\]
\[ = \cos^{- 1} \left( \cos\frac{\pi}{3} \right) + 2 \sin^{- 1} \left( \sin\frac{\pi}{6} \right)\]

`[because"The range of sine is" [-pi/2,pi/2];  pi/6 in[-pi/2,pi/2] "and the range of cosine is"  [0,pi] ;  pi/3 in  [0,pi]]`
\[ = \frac{\pi}{3} + 2\left( \frac{\pi}{6} \right)\]
\[ = \frac{\pi}{3} + \frac{\pi}{3}\]
\[ = \frac{2\pi}{3}\]

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