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Re: Simplifying ArcTan

• To: mathgroup at smc.vnet.net
• Subject: [mg55544] Re: Simplifying ArcTan
• From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
• Date: Tue, 29 Mar 2005 03:42:28 -0500 (EST)
• References: <d28ftk$n0u$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

"Florian Jaccard" <florian.jaccard at he-arc.ch> wrote:
> It does !
>
> But you have to avoid 2 mistakes :
>
> 1) The brackets on wrong place
> 2) x may not be Pi/2
>
> In[4]:=
> FullSimplify[ArcTan[Cos[x], Sin[x]],
>   x >= 0 && x < Pi/2]
>
> Out[4]=
> x

You say that "x may not be Pi/2". Apparently that is true if Mathematica is
to do the simplification now. But there's no mathematical reason we should
have to exclude x = Pi/2. Even Mathematica knows that, at x = Pi/2,
ArcTan[Cos[x], Sin[x]] is the same as x:

In[5]:= ArcTan[Cos[x], Sin[x]] /. x -> Pi/2

Out[5]= Pi/2

Indeed, it would surely be desirable if Mathematica simplified
ArcTan[Cos[x], Sin[x]] to x whenever -Pi < x <= Pi.

More generally, we may simplify ArcTan[Cos[x], Sin[x]] to an expression
involving no trig or inverse trig functions

  x + 2 Pi Floor[1/2 - x/(2 Pi)]

for all real x.

David Cantrell


> -----Message d'origine-----
> De : fizzy [mailto:fizzycist at knology.net]
> Envoyé : lun., 28. mars 2005 09:42
> À : mathgroup at smc.vnet.net
> Objet :  Simplifying ArcTan
>
> Why does   FullSimplify[ ArcTan[ Cos[x], Sin[x] , x>=0 && x <= Pi/2 ]
> not output  x ?
>
> Thanks...jerry blimbaum