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MathGroup Archive 1995

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Re: Q: ArcTan[ Tan[ x ] ] = x

  • To: mathgroup at christensen.cybernetics.net
  • Subject: [mg1956] Re: Q: ArcTan[ Tan[ x ] ] = x
  • From: pehowland at taz.dra.hmg.gb ()
  • Date: Sat, 26 Aug 1995 00:10:39 -0400
  • Organization: Defence Research Agency

In article <DDuMDu.Joy at wri.com>, elsner at avalon.msfc.nasa.gov (Ron Elsner) writes:

|>Suppose I really want Mma to set ArcTan[ Tan[ x ] ] = x, or 
|>ArcSin[ Sin[ x ] ] = x.  How do I do this?

Hmm.  Mathematica appears particularly stubborn over this one.  I guess
the mathematicians amongst us would argue that, strictly speaking,
ArcSin[Sin[x]] =!= x, but rather ArcSin[Sin[x]] === x + (2 n Pi), due
to the periodicity of the trig functions.  However, it does seem that
in certain circumstances it would be nice to be able to ignore this
mathematical niceity.

Anyway, the two obvious approaches are
	Simplify[expr, Trig->True]
and
	<<Algebra`Trigonometry`
	TrigReduce[expr]

but neither of these work, so I guess the only approach is to teach
Mathematica some not-strictly-correct mathematics:

In[1]  := BadTrigRules = {ArcSin[Sin[a_]]->a, ArcTan[Tan[a_]]->a};
In[2]  := ArcSin[Sin[x]] /. BadTrigRules
Out[2] := x

which forces the answer for you.  You could write your own Simplify function
that did this sort of thing first, and then applied Mathematica's built
in Simplify fuction.

It might be nice, however, for Mathematica's trig functions to have an
option like "IgnorePeriodicity->True", or something, to force the above
behaviour.

Paul Howland
Defence Research Agency
UK


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