ArcCot--relationship with ArcTan

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

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 http://mathworld.wolfram.com/InverseCotangent.html 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 http://mathworld.wolfram.com/InverseTrigonometricFunctions.html 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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**Re: ArcCot--relationship with ArcTan**