Inverse Trigonometric Functions Set-3

$\dfrac{3}{4}$

$\dfrac{4}{3}$

$\dfrac{4}{5}$

$\dfrac{5}{3}$

$-\dfrac{12}{5}$

$0$

$\dfrac{5}{12}$

$\dfrac{12}{5}$

$\displaystyle \frac{2}{1+x^{2}}\ \text{for}\ |x|<1\ \text{and}\ -\frac{2}{1+x^{2}}\ \text{for}\ |x|>1$

$\displaystyle \frac{2}{1+x^{2}}$

$\displaystyle -\frac{2}{1+x^{2}}$

$\displaystyle \frac{2(1-x^{2})}{(1+x^{2})^{2}}$

all real $x$

$|x|<1$

$|x|\le 1$

$|x|\ge 1$

$\dfrac{\pi}{6}$

$\dfrac{\pi}{2}$

$\dfrac{3\pi}{4}$

$\dfrac{\pi}{4}$

$\dfrac{7}{5\sqrt{13}}$

$\dfrac{1}{\sqrt{13}}$

$\dfrac{17}{5\sqrt{13}}$

$\dfrac{17}{13}$

$\dfrac{1}{\sqrt{5}}$

$-\dfrac{1}{\sqrt{5}}$

$0$

$\pm\dfrac{1}{\sqrt{5}}$

$(-1,1)$

$[-1,1]$

$\mathbb{R}$

$(-\infty,-1]\cup[1,\infty)$

$0\quad\text{for all real }x$

$\pi\ \text{for }x<0\ \text{and}\ 0\ \text{for }x\ge 0$

$2\big|\tan^{-1}x\big|\quad\text{for all real }x$

$0\ \text{if }x\ge 0,\quad -4\tan^{-1}x\ \text{if }x<0$

$\displaystyle [0,\tfrac{\pi}{4}]\cup\{\pi\}$

$\displaystyle [0,\tfrac{\pi}{2}]$

$\displaystyle \{\tfrac{\pi}{4},\pi\}$

$\displaystyle [0,\pi]$

$\dfrac{7}{25}$

$\dfrac{24}{25}$

$\dfrac{3}{5}$

$\dfrac{16}{25}$

$2\arctan x$

$\pi-2\arctan x$

$2\arctan\!\left(\dfrac{1}{x}\right)$

$x=0$

$x=-1$

$x=\pm 1$

$x=1$

$x=\dfrac{1}{2}$

$x=-\dfrac{1}{\sqrt{5}}$

$x=\dfrac{1}{\sqrt{5}}$

$x=0$

$[-1,1]$

$(-1,1)$

$\mathbb{R}$

$(-\infty,-1]\cup[1,\infty)$