Question

Evaluate: ${\tan }^{-1}\left(1\right)+{\cos }^{-1}\left(\frac{1}{2}\right)+{\sin }^{-1}\left(\frac{1}{2}\right)$ which lies in the interval $[0,\pi ]$

Answer 1

${\tan }^{-1}+{\cos }^{-1}\left(1/2\right)+{\sin }^{-1}\left(1/2\right)0\le x\le \pi$

Let $x={\tan }^{-1}1\Rightarrow \tan x=1$

$\Rightarrow x=\pi /4$

Let $y={\cos }^{-1}\left(1/2\right)\Rightarrow \cos y=1/2$

$y=\pi /3$

Let $={\sin }^{-1}\left(1/2\right)\Rightarrow \sin z=1/2$

$\Rightarrow z=\pi /6$

$\therefore x+y+z=\pi /4+\pi /3+\pi /6$

$=\frac{3\pi +4\pi +2\pi }{12}$

$=\frac{9\pi }{12}=\frac{3\pi }{4}$