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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
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