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Re: Trigonometric simplification

• To: mathgroup at smc.vnet.net
• Subject: [mg69328] Re: Trigonometric simplification
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Thu, 7 Sep 2006 04:30:20 -0400 (EDT)
• Organization: The University of Western Australia
• References: <ecbnnc$r29$1@smc.vnet.net> <ecmgm2$99o$1@smc.vnet.net> <200609040847.EAA23460@smc.vnet.net> <edjgsd$lt0$1@smc.vnet.net>

In article <edjgsd$lt0$1 at smc.vnet.net>,
 "Carl K. Woll" <carlw at wolfram.com> wrote:

> Yes, using ChebyshevT was what I had recommended before, and is a bit 
> simpler for this particular problem. See:
> 
> http://forums.wolfram.com/mathgroup/archive/2006/Apr/msg00709.html
> 
> The reason that I now suggest using TrigExpand and ArcCos is that this 
> method seems to be a bit more general. For example, how would you convert
> 
> 2 + 3 Sin[a] + Sin[3a]
> 
> into a sum of powers of Sin[a]? 

For odd multiples,

  Sin[(k_ /; OddQ[k]) a_] :> (-1)^((k - 1)/2) ChebyshevT[k, Sin[a]]

Similarly, for even multiples,

 Sin[(k_ /; EvenQ[k]) a_]:> (-1)^(k/2 + 1) Cos[a] ChebyshevU[k-1, Sin[a]]

> Or, how would you convert the following mixture of sines and cosines 
> into a trigonometric polynomial in Cos[a]:
> 
> 2 + 3 Cos[a] Sin[a] + Cos[3a] Sin[3a]

It is _not_ a trigonometric polynomial in Cos[a]! Clearly it involves 
Sin[a]. Moreover, using

 TrigExpand[2 + 3 Cos[a] Sin[a] + Cos[3a] Sin[3a] /. a -> ArcCos[x]] /.
   x -> Cos[a] // Simplify

leads to

 2+4 Cos[a] Sqrt[Sin[a]^2]+Cos[3a] Sqrt[Sin[a]^2]+Cos[5a] Sqrt[Sin[a]^2]

with unnecessary Sqrts. The "standard" decomposition is

 2 + 3 Cos[a] Sin[a] + Cos[3a] Sin[3a] /.
  {Cos[n_Integer a_] :> ChebyshevT[n, Cos[a]],
   Sin[n_Integer a_] :> Sin[a] ChebyshevU[n-1, Cos[a]]}

> I think both techniques are good to have in one's toolbox.

Agreed. However, for trig polynomials, one only needs a small set of 
transformations involving Chebyshev polynomials, and these always lead 
to a valid representation.

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul

• References:
  • Re: Trigonometric simplification
    • From: Paul Abbott <paul@physics.uwa.edu.au>