Finding interval for arcsecant, arcsine, and acrtangent How did my text book get these intervals? I have an idea of how to get it for arcsine -- just look at the graph of sine.

 A: As you may know already, because the trig functions are not one-to-one, before we can invert them it's necessary to restrict their domains so they become one-to-one, while also maintaining the their full range. Arctangent's domain comes from restricting the tangent function to its first period $(-\pi/2,\ \pi/2)$ which still covers the full range of tangent $(-\infty,\infty)$. This is a good choice since it also contains the origin.
Arcsine's range comes from restricting sine's domain to $[-\pi/2,\ \pi/2]$ which covers the whole range of $[-1,1]$ and also includes the origin.
Arcsecant's range comes from the arccosine's range. To make cosine invertible, the domain is restricted to $[0,\pi]$. We will keep this restriction for secant as well. Note that since the range of cosine is $[-1,1]$, secant's range is always $(-\infty, -1) \cup (1, \infty)$. Secant is the reciprocal of cosine, and reciprocals don't change the sign of their inputs. Hence secant is greater than $1$ precisely whenever cosine is positive, and secant is less than $-1$ whenever cosine is negative. On the restricted domain $[0, \pi]$, when is cosine positive? On $[0, \pi/2)$. When is it negative? On $(\pi/2, \pi]$.
