Quartic eqn solutions for Cos/Sin

  • Thread starter natski
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  • #1
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Hi all,

I am looking for a special case solution to the quartic equation x^4 + a x^3 + b x^2 + c x + d = 0 in the case where x = Cos[theta]. Are there are any special properties of the solutions? For example, I know there are numerous properties from Vieta's formula but none of these really help simplify things if x=cosine.

Natski
 

Answers and Replies

  • #2
CRGreathouse
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Depends on what you want to do, I suppose. The half-angle identity
[tex]\cos^2a=\frac{1+\cos(2a)}{2}[/tex]
could reduce the degree, but at the cost of introducing cosines of different arguments. This may have been what you were referring to as not useful, though; I'm not sure.
 
  • #3
Mentallic
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I don't see how it can be considered much of a special case if [itex]x=cos\theta[/itex]. All you're saying here is that [itex]-1\leq x\leq 1[/itex] which really doesn't help a great deal.
 
  • #4
disregardthat
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[tex]x=cos(\theta)[/tex] is algebraic if [tex]\theta[/tex] is a rational multiple of pi. I.e. it is the zero of a polynomial of integer coefficients.

It can be shown that if [tex]\theta[/tex] is not a rational multiple of pi, then [tex]x=cos(\theta)[/tex] is transcendental and thus not the solution of any polynomial of algebraic coefficients.
 
  • #5
CRGreathouse
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I don't see how it can be considered much of a special case if [itex]x=cos\theta[/itex]. All you're saying here is that [itex]-1\leq x\leq 1[/itex] which really doesn't help a great deal.
I don't know... if the equation was [itex]2ax^2-1=2a\cos^2(\theta)-1[/itex], I would consider [itex]\cos(2\theta)[/itex] a simplification.
 

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