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Re: Trig identity for linear combination of sines

  • To: mathgroup at smc.vnet.net
  • Subject: [mg100828] Re: Trig identity for linear combination of sines
  • From: Ray Koopman <koopman at sfu.ca>
  • Date: Mon, 15 Jun 2009 05:37:16 -0400 (EDT)
  • References: <h13vr9$8qh$1@smc.vnet.net>

On Jun 14, 4:07 pm, Chelly <chelly85... at gmail.com> wrote:
> Hi:
>
> I have a function which is a linear combination of sines, of the form
>
> y = a*sin(x+q) + b*sin(x+r) + c*sin(x+t)
>
> and I am looking for a trignometric identity
> that would reduce the above to the form
>
> y = A*sin(x+Z)
>
> I would also like to find out how to implement this rule in Mathematica.
>
> Thanks
> Chelly

Collect[TrigExpand[ a*Sin[x+q] + b*Sin[x+r] + c*Sin[x+t] ],
{Cos[x], Sin[x]} ] /. Sin[x]*A_ + Cos[x]*B_ ->
Sqrt[A^2 + B^2] * Sin[x + ArcTan[A,B]]

Sqrt[(a*Cos[q] + b*Cos[r] + c*Cos[t])^2 +
     (a*Sin[q] + b*Sin[r] + c*Sin[t])^2] *
Sin[x + ArcTan[a*Cos[q] + b*Cos[r] + c*Cos[t],
               a*Sin[q] + b*Sin[r] + c*Sin[t]]]


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