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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
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