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

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

```

• Prev by Date: Re: Why doesn't Mathematica solve this simple differential equation?
• Next by Date: Re: repost -Difficulties with xml
• Previous by thread: Re: Re: Trigonometric simplification
• Next by thread: difficulties with xml