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It is well known that if $x$ is a rational multiple of $\pi$ then $\cos x$, $\sin x$, etc, are algebraic numbers. What is known about the inverse problem?

That is, is there a set of conditions that if imposed on $\alpha$ imply that $\cos^{-1} \alpha$ is a rational multiple of $\pi$?

Another way of putting it is, given the complex number $z=a+ib$, is there a way of deciding whether there exists some $n$ such that $z^n=1$?

EDIT: 'factor' -> 'multiple'

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    I think you mean a rational multiple of $\pi$, not factor?2012-03-05
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    you're right. edited2012-03-05
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    A sylistic observation: Greek letters like $\alpha$ are traditionally used for angles when trigonometric functions are under discussions. The argument of $\cos^{-1}$ is best thought of as a _length_ or as a ratio of lengths, not truly an angle. So I would recommend a roman letter for the argument of $\cos^{-1}$, not a Greek one. I often wonder if confusion regarding inverse trig functions could be avoided if it was clear that, say, cosine eats angles and spits out lengths, while $\cos^{-1}$ does the opposite.2012-03-06

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