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ArcCot--relationship with ArcTan

  • To: mathgroup at
  • Subject: [mg38040] ArcCot--relationship with ArcTan
  • From: "Chen Yaohan" <chen1han at>
  • Date: Tue, 26 Nov 2002 00:50:39 -0500 (EST)
  • Sender: owner-wri-mathgroup at

I argued over the more appropriate definition of (or algorithm for) the
inverse cotangent function with someone, based on what math we have learned.

(A) arccot x = Pi/2 - arctan x
(B) arccot x = arctan 1/x

My calculus book used the definition A when it talked about the derivatives
of inverse trig functions. I think it said that definition B is also used in
some situations, but I don't have the book now.

It's obvious why my calculus book and most calculus study guides found on
the Internet use the definition A--arccot would be continuous on its domain,
and the complementary angle relationship is preserved. But Mathematica uses
definition B, or a definition closer to B. See

So, is there any merit of definition B, besides "convenience"? Should we
sacrifice the continuity of inverse cotangent just to satisfy the simple
"x-reciprocal" relationship?

And this page
says the "domain" (my math teacher would say range) of ArcCot is (0, Pi/2)
or (-Pi, -Pi/2). Where does that come from? It's not even consistent with
the graph on Wolfram's own inverse cotangent page! The range for ArcCsc
listed there is strange too. Also, Wolfram doesn't say that arccot is not
differentiable because of discontinuity at x=0 when it lists the first
derivative of ArcCot[z], while definition B would require that. Is it
something in more advanced mathematics that I don't know yet?

When I argued over the two definitions few weeks ago, my opinion was that
definition B was acceptable, or even should be authoritative, because
Wolfram and possibly other systems use it. But after my opponent pointed out that
inconsistency, I didn't know what to say. Still, I believe there is a reason
why definition B not only survived but also is used in these authoritative
software. Could anybody explain to me? Thanks!

Yaohan Chen

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