Domain of Arccot

Date: 05/23/2002 at 21:47:58
From: Lester Penning
Subject: ArcCot Range

I have a good understanding of trig, but I just want to know why the 
range of the arccot is between 0 and Pi and not Pi/2 and -Pi/2 like 
the range of arctan is.....

Date: 05/24/2002 at 08:42:03
From: Doctor Peterson
Subject: Re: ArcCot Range

Hi, Lester.

In order to invert a trig function, we first restrict it to a domain 
on which it takes all its possible values, once each; then we invert 
the restricted function, whose range is then that restricted domain.

Look at a graph of the cotangent function, and you will see that 
although between -pi/2 and pi/2 it takes all its possible values, and 
takes each value only once, there is one problem with this choice: it 
is not continuous (or even defined) on this entire domain, but is 
undefined at 0. The domain would then have to be

    -pi/2 < x < 0 or 0 < x <= pi/2

To avoid this, we instead choose the domain

    0 < x < pi

which is cleaner to work with, making a continuous function defined 
over the entire domain. Inverting that, we get the arccot function 
you know.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/24/2002 at 16:22:26
From: Lester Penning
Subject: ArcCot Range

Question submitted via WWW:
Thank you Doctor Peterson for your quick response.  The ArcCot range 
is now clear to me.  Thanks again.