Question

Find the value of $co{s}^{-1}\left(\frac{1}{2}\right)+2si{n}^{-1}\left(\frac{1}{2}\right)$.

Answer 1

Let $co{s}^{-1}\left(\frac{1}{2}\right)=x$

Then,

$\cos x=\frac{1}{2}=\cos \frac{\pi }{3}$

Therefore, $co{s}^{-1}\left(\frac{1}{2}\right)=\frac{\pi }{3}$

Let $si{n}^{-1}\left(\frac{1}{2}\right)=y$

Then,

$\sin y=\frac{1}{2}=\sin \frac{\pi }{6}$

Therefore, $si{n}^{-1}\left(\frac{1}{2}\right)=\frac{\pi }{6}$

Hence, $co{s}^{-1}\left(\frac{1}{2}\right)+2si{n}^{-1}\left(\frac{1}{2}\right)=\frac{\pi }{3}+\frac{\pi }{3}=\frac{2\pi }{3}$