Quartic eqn solutions for Cos/Sin

[natski]

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

 

[CRGreathouse]

Depends on what you want to do, I suppose. The half-angle identity
$$\cos^2a=\frac{1+\cos(2a)}{2}$$
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.

 

[Mentallic]

I don't see how it can be considered much of a special case if $x=cos\theta$. All you're saying here is that $-1\leq x\leq 1$ which really doesn't help a great deal.

 

[disregardthat]

$$x=cos(\theta)$$ is algebraic if $$\theta$$ is a rational multiple of pi. I.e. it is the zero of a polynomial of integer coefficients.

It can be shown that if $$\theta$$ is not a rational multiple of pi, then $$x=cos(\theta)$$ is transcendental and thus not the solution of any polynomial of algebraic coefficients.

 

[CRGreathouse]

I don't see how it can be considered much of a special case if $x=cos\theta$. All you're saying here is that $-1\leq x\leq 1$ which really doesn't help a great deal.

I don't know... if the equation was $2ax^2-1=2a\cos^2(\theta)-1$, I would consider $\cos(2\theta)$ a simplification.