$\sinh ^{-1}(\operatorname{cosec} \theta)$

$\sinh ^{-1}(\sec \theta)$

$\cosh ^{-1}(\operatorname{cosec} \theta)$

$\cosh ^{-1}(\sec \theta)$

$\left(0, \frac{\pi}{2}\right)$

$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

$\left(-\frac{3 \pi}{4}, \frac{\pi}{4}\right)$

$\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$

$\left(0, \frac{\pi}{4}\right]$

$\left[\frac{\pi}{4}, \frac{\pi}{2}\right)$

$[0, \pi]$

$\left[\frac{\pi}{6}, \frac{\pi}{2}\right)$

$(-1,2)$

$(1, \infty)$

$(-\infty, \infty)$

$(-\infty, 1)$