Re: ArcCot--relationship with ArcTan
- To: mathgroup at smc.vnet.net
- Subject: [mg38071] Re: ArcCot--relationship with ArcTan
- From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
- Date: Thu, 28 Nov 2002 14:09:54 -0500 (EST)
- Organization: NewsReader.Com Subscriber
- References: <arv3l0$av9$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
"Chen Yaohan" <chen1han at prodigy.net> wrote: > I argued over the more appropriate definition of (or algorithm for) the > inverse cotangent function with someone, based on what math we have > learned. Such arguments could probably be continued _ad infinitum_! > (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 Right: Closer to, but not the same as, B. It is not the same as B because, in Mathematica, ArcCot[0] yields Pi/2, while ArcTan[1/0] yields Indeterminate. FWIW, however, note that FullSimplify[ArcTan[1/x]] does give ArcCot[x]. > 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? Since I favor definition A (at least for the purposes of real analysis), I'll leave this for proponents of Mathematica's definition. Just realize that either definition is "correct". [BTW, although it's probably a bit above your level at present, you might be interested to look for information about "branch cuts".] > 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! Right you are! Almost undoubtedly, the two MathWorld entries which you cited were written by Eric _long ago_. Many of those early entries had errors. And, as you see, some of those errors have yet to be corrected. [I'll be sending an expanded version of this message to Eric, and so these errors will soon be eliminated.] Anyway, you're absolutely correct that the table should say Range, instead of Domain (because the first column is talking about the inverse functions, rather than the original trig functions). Furthermore, if you wish to take the reals as the domain of the inverse cotangent, then the range which is appropriate for Mathematica's definition is -Pi/2 < y < 0 or 0 < y <= Pi/2 and the range which is appropriate using definition A would be 0 < y < Pi. > The range for ArcCsc listed there is strange too. Correct again. It's the same sort of error. If the domain of the inverse cosecant is taken to be a subset of the reals, then the range should, to correspond with Mathematica's definition, be -Pi/2 <= y < 0 or 0 < y <= Pi/2. > 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. It is not a good idea to expect _any_ computer algebra system to say such things! The following might (or might not) amuse you: In[12]:= D[Floor[x],x] Out[12]= Floor'[x] In[13]:= %/.x->1 Out[13]= Floor'[1] In[14]:= N[%] Out[14]= 12.5946 Regards, David -- -------------------- http://NewsReader.Com/ -------------------- Usenet Newsgroup Service New Rate! $9.95/Month 50GB