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MathGroup Archive 2006

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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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