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RE: Wrestling with Mathematica on Trig Simplification


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

> From: Steven Warwick [mailto:warwick at]
To: mathgroup at
> 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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