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>