Question

Find the value of $4{\cos }^{-1}\frac{1}{2}+2{\sin }^{-1}\frac{1}{2}+3{\tan }^{-1}(-1)+2{\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Answer 1

Given,

$4{\cos }^{-1}\frac{1}{2}+2{\sin }^{-1}\frac{1}{2}+3{\tan }^{-1}1+2{\cos }^{-1}\frac{\sqrt{3}}{2}$

$=4{\cos }^{-1}\cos \frac{\pi }{3}+2{\sin }^{-1}\sin \frac{\pi }{6}+3{\tan }^{-1}\tan \frac{\pi }{4}+2{\cos }^{-1}\cos \frac{\pi }{6}$

$=4\times \frac{\pi }{3}+2\times \frac{\pi }{6}+3\times \frac{\pi }{4}+2\times \frac{\pi }{6}$

$=\frac{16\pi +4\pi +9\pi +4\pi }{12}$

$=\frac{33\pi }{12}$