       Re: Domain of Sin[ArcSin[x]] ?

• To: mathgroup at smc.vnet.net
• Subject: [mg39642] Re: Domain of Sin[ArcSin[x]] ?
• From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
• Date: Thu, 27 Feb 2003 00:28:07 -0500 (EST)
• Organization: NewsReader.Com Subscriber
• References: <b3hs25\$ik5\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

mbuescher at hb.edu (Michael Buescher) wrote:
> I want to demonstrate to my students that in the real number system,
> Sin[ArcSin[x]] is only defined on [-1,1] because that is the domain of
> ArcSin[x].

OK. But realize that if you view the reals as a subsystem of the complexes,
then it certainly can be argued that, even specifying that we wish to
consider Sin[ArcSin[x]] to be a _real-valued_ function of a _real_
variable, its domain is _all_ of R. Why? Well, the only requirement is
that, given a real input, x, we also get a real output, Sin[ArcSin[x]]. And
that's essentially why Mathematica happily indicates the domain to be R
when you plot the function.

But you don't want to deal with the reals as a subsystem of the complexes.
In particular, you want to disallow any intermediate calculations which
would temporarily put us outside of R. I don't know of an elegant, general
way to do this in Mathematica. However, for your specific problem, you can
use

Plot[Sin[If[Element[ArcSin[x], Reals], ArcSin[x]]], {x, -5, 5}]

for example. After a few complaints about encountering things that aren't
machine-sized reals, Plot will indicate that the domain is [-1, 1].

David

> When I Plot the composition, however, I get Sin[ArcSin[x]] =
> x for all real numbers, not just on [-1,1].  I tried this both with and
> without the RealOnly package.
>
> Is there any way to ensure that Mathematica uses only real numbers in
> its calculations, so that Sin[ArcSin[x]] is undefined when ArcSin[x] is
> not a real number?

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