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Topic: arc cotangent function--negative values
Replies: 0

 Colin Campbell Posts: 3 From: Virginia Registered: 12/26/10
arc cotangent function--negative values
Posted: Feb 21, 2011 12:26 AM

There seems to be two competing arccot functions "out there".

The more "serious" function has a discontinuity at x=0, where y jumps between pi/2 and -pi/2. Graphs can be seen at:
http://mathworld.wolfram.com/InverseCotangent.html
http://www.intmath.com/analytic-trigonometry/7-inverse-trigo-functions.php
http://www.calculatorsoup.com/calculators/trigonometry/graphs-inversefunctions.php
Perhaps the most authoritative source is the graph at the National Institute of Standards:
http://dlmf.nist.gov/4.15.F4.mag
where it states: "arccot x is discontinuous at x=0"
The three best trig calculators on the web return values on this curve for negative values of x:
http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp
http://www.1728.com/trigcalc.htm
http://www.calculatorsoup.com/calculators/trigonometry/inversetrigonometricfunctions.php

The more "popular" function is a continuous ski-slope through y=pi/2 at x=0. It can be found in books, all the tutoring web sites, and other locations. The most prominent graph can be found at:
http://en.wikipedia.org/wiki/Inverse_trigonometric_functions
It is clear from the graph that this arccot is complementary (sums to a constant value) to the arctan function.

I understand the inverse trig functions are multi-valued and their principal values depend on somewhat arbitrary branch cuts. I suspect the differences come from the way the functions are derived--directly from the cotangent function or indirectly from another inverse function. (I have read one article that states arccot is defined as the complement of arctan.)

Any light anyone can shed on this topic will be appreciated. Thanks.