Re: Re: Trigonometric simplification

*To*: mathgroup at smc.vnet.net*Subject*: [mg69284] Re: [mg69267] Re: Trigonometric simplification*From*: "Carl K. Woll" <carlw at wolfram.com>*Date*: Tue, 5 Sep 2006 05:30:57 -0400 (EDT)*References*: <ecbnnc$r29$1@smc.vnet.net> 1115.HAA25039@sssssssssssss <ecmgm2$99o$1@smc.vnet.net> <200609040847.EAA23460@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Paul Abbott wrote: > In article <ecmgm2$99o$1 at smc.vnet.net>, > "Carl K. Woll" <carlw at wolfram.com> wrote: > >> carlos at colorado.edu wrote: >>> Here is a correction to my second post (Thanks to Paul Abbott for >>> noticing the factor of 2). The reason I need to get 2 + 4*Cos(a)^3 >>> is to match a published solution in a homework solutions manual. >>> >> A while back Maxim gave a nice method to convert Cos[n a] into Cos[a]^_.: >> >> d = 2 + 3*Cos[a] + Cos[3*a] >> >> First, replace a with ArcCos[x] and use TrigExpand: >> >> In[5]:= TrigExpand[d /. a -> ArcCos[x]] >> >> Out[5]= 4 x^3 + 2 >> >> Then, replace x with Cos[a]. All together we have: >> >> In[6]:= TrigExpand[d /. a -> ArcCos[x]] /. x -> Cos[a] >> >> Out[6]= 4 (Cos[a])^3 + 2 > > Personally, I prefer to use Cos[n a] == ChebyshevT[n, Cos[a]]. So > > 2 + 3 Cos[a] + Cos[3 a] /. Cos[n_ a] :> ChebyshevT[n, Cos[a]] > > yields > > 2 + 4 Cos[a]^3 > > Cheers, > Paul > 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]? 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] I think both techniques are good to have in one's toolbox. Carl Woll Wolfram Research > _______________________________________________________________________ > 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>