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

  • To: mathgroup at
  • Subject: [mg39657] Re: [mg39620] Domain of Sin[ArcSin[x]] ?
  • From: Dr Bob <drbob at>
  • Date: Thu, 27 Feb 2003 00:33:59 -0500 (EST)
  • References: <>
  • Reply-to: drbob at
  • Sender: owner-wri-mathgroup at

If you don't want to choose the proper limits for the plot, YOU have to 
decide what you want plotted when ArcSin@x isn't real.  For instance,

Plot[Sin@Re@ArcSin@x, {x, -2, 2}]


Plot[Switch[Im@ArcSin@x, 0, x, _, 0], {x, -2, 2}]


Plot[Sin@Re@ArcSin@x, {x, -1, 1}]

Your choice.

Alternatively, just Plot the Sin function and say, "See, the values are all 
between -1 and 1, so that's the domain of ArcSin.  The range of the 
function is the (largest possible) domain of the inverse."


On Wed, 26 Feb 2003 02:42:54 -0500 (EST), Michael Buescher 
<mbuescher at> 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].  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?
> Michael Buescher
> Hathaway Brown School

majort at
Bobby R. Treat

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