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Re: Expanding powers of cosine

• To: mathgroup at smc.vnet.net
• Subject: [mg84206] Re: Expanding powers of cosine
• From: Januk <ggroup at sarj.ca>
• Date: Thu, 13 Dec 2007 14:49:50 -0500 (EST)
• References: <200712130102.UAA23849@smc.vnet.net> <fjqjfd$6o0$1@smc.vnet.net>

Be warned that this method isn't bullet-proof:

f = Cos[x]^3;
sf = First[GroebnerBasis[{f, 1 - Cos[x]^2 - Sin[x]^2}, {Sin[x]},
{Cos[x]}]];
FullSimplify[ sf / f ]

Gives:
Cos[x]


On Dec 13, 1:29 am, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
> On 13 Dec 2007, at 10:02, michael.p.crouc... at googlemail.com wrote:
>
>
>
>
>
> > Hi
>
> > I would like to express even powers of Cos[x] in terms of powers of
> > Sin[x] using the identity Sin[x]^2+Cos[x]^2 = 1.  For example
>
> > Cos[x]^4 = 1 - 2 Sin[x]^2 + Sin[x]^4
>
> > I could not get any of Mathematica's built in functions to do this for
> > me so I created my own rule:
>
> > expandCosn[z_] := Module[{s, res},
> >  s = Cos[x]^n_ :> (1 - Sin[x]^2) Cos[x]^(n - 2) ;
> >  res = z //. s;
> >  Expand[res]
> >  ]
>
> > which works fine:
>
> > In[14]:= expandCosn[Cos[x]^4]
>
> > Out[14]= 1 - 2 Sin[x]^2 + Sin[x]^4
>
> > My question is - have I missed something?  Is there an easier way to
> > do this?
>
> > Cheers,
> > Mike
>
> Here is one way. This is how to expand Cos[x]^24:
>
> First[GroebnerBasis[{Cos[x]^24, 1 - Cos[x]^2 - Sin[x]^2}, {Sin[x]},
>       {Cos[x]}]]
>
>   Sin[x]^24 - 12*Sin[x]^22 + 66*Sin[x]^20 -
>     220*Sin[x]^18 + 495*Sin[x]^16 - 792*Sin[x]^14 +
>     924*Sin[x]^12 - 792*Sin[x]^10 + 495*Sin[x]^8 -
>     220*Sin[x]^6 + 66*Sin[x]^4 - 12*Sin[x]^2 + 1
>
> Andrzej Kozlowski- Hide quoted text -
>
> - Show quoted text -

• Follow-Ups:
  • Re: Re: Expanding powers of cosine
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>

• References:
  • Expanding powers of cosine
    • From: michael.p.croucher@googlemail.com