Question

Find the values of the following questions:

$co{s}^{-1}\left(\frac{1}{2}\right)+2si{n}^{-1}\left(\frac{1}{2}\right)$

Answer 1

We can find the value of given expression by simplifying the individual terms.

Let $co{s}^{-1}\left(\frac{1}{2}\right)=x\Rightarrow cosx=\frac{1}{2}=cos\frac{\pi }{3}$
$\Rightarrow x=\frac{\pi }{3}ϵ[0,\pi ]$ [principal interval]
Again, let $si{n}^{-1}\left(\frac{1}{2}\right)=y\Rightarrow siny=\frac{1}{2}=sin\frac{\pi }{6}$
$\Rightarrow y=\frac{\pi }{6}\epsilon [-\frac{\pi }{2},\frac{\pi }{2}]$ (principal interval)
$\therefore co{s}^{-1}\left(\frac{1}{2}\right)+2si{n}^{-1}\left(\frac{1}{2}\right)=x+2y=\frac{\pi }{3}+2\times \frac{\pi }{6}=\frac{2\pi }{3}$