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MathGroup Archive 2002

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Re: Wrestling with Mathematica on Trig Simplification

  • To: mathgroup at smc.vnet.net
  • Subject: [mg32317] Re: Wrestling with Mathematica on Trig Simplification
  • From: "Dana" <ng_only at hotmail.com>
  • Date: Sat, 12 Jan 2002 21:32:36 -0500 (EST)
  • References: <a1bllp$bcb$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

I am really new at this, so I am probably way off.  Taking your excellent
idea, perhaps you could rewrite it as the following rule??

myrule = a_ Sin[t_] + b_ Cos[t_] ->  Abs[a + b I]Cos[t - Arg[b + I a]]

As an example:

In[5]:=
3Sin[1/2]+4Cos[1/2]/.myrule

Out[5]=
5*Cos(1/2 - ArcTan(3/4))

On could include this following in Mathematica 4 to set a & b to Reals.
Element[{a, b}, Reals]

- - - - - - - - - - - - - - - - - - - - - -
HTH.
Dana DeLouis

--

"David Park" <djmp at earthlink.net> wrote in message
news:a1bllp$bcb$1 at smc.vnet.net...
> Steven,
>
> I think this may be a good example of what often happens with Mathematica.
> Users, especially new users, expect and hope that there will be off the
> shelf Mathematica commands to directly solve their problem. But
mathematics
> is too vast a subject for that without having millions of commands (which
> would present a problem in itself). Often, very often, you are going to
have
> to write definitions, rules and small routines to use as tools in your
> specific application. One should view Mathematica more as a kit to build
> tools than as a ready-to-use problem solver.
>
> For you problem I would write the initial definition this way:
>
> m[a_, b_][t_] := a Sin[t] + b Cos[t]
>
> Then, checking with my favorite Mathematics Handbook, I would write the
> amplitude-phase angle conversion as:
>
> AmplitudePhaseSimplify[expr_] :=
>   expr /. (a_.)*Sin[t_] + (b_.)*Cos[t_] ->
>     Sqrt[a^2 + b^2]*Cos[t - ArcTan[a/b] -
>        If[a < 0, Pi, 0]]
>
> Since it has an If statement in it, it doesn't look too great with
symbolic
> expressions. But with number expressions it works nicely.
>
> m[3, 2][t]
> % // AmplitudePhaseSimplify
> 2 Cos[t] + 3 Sin[t]
> Sqrt[13]*Cos[t - ArcTan[3/2]]
>
> Of course, sometimes one later finds that Mathematica does have a direct
way
> to do it, and you might even get such a reply from MathGroup.
>
> David Park
> djmp at earthlink.net
> http://home.earthlink.net/~djmp/
>
>
> > From: Steven Warwick [mailto:warwick at jps.net]
To: mathgroup at smc.vnet.net
> >
> > So, A typical scenario for me is the combining of sinusoids like:
> >
> > m[t_]  = A Sin[t] + B Cos[t]
> >
> > ( A and B Real, although I don't know how to communicate this to
> > Mathematica
> > in an effective way)
> >
> > with the desired "simplified" output being of the form:
> >
> > C  Cos[t+th]
> >
> >
> > Trigreduce will not do this, as I've tried. Yes, C and th are more
> > algebraically complicated, but the overall expression is actually more
> > meaningful for me..
> >
> > I can solve for C and th using Solve, with creating 2 simultaneous
> > equations with t picked at 0, PI/2 to get the correct form, but that's
> > not the same as having a "reduce" capability.   Am I missing
> > something? is there a way to create preference for this form in
> > simplify?
> >
> > Thanks!
> >
>
>




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