If $$\cos{(a\pi)}=\dfrac{1}{3}$$ then show that ：$a$ is irrational
this result seems easy,but I can't prove it. Thank you...
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1See page 2 of http://arxiv.org/pdf/1006.2938.pdf , or the answer to http://math.stackexchange.com/questions/49297/the-only-two-rational-values-for-cosine-and-their-connection-to-the-kummer-rings , depending on the tools you want to use. – Greg Martin Jan 28 '14 at 07:45
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isn't enough for you to say that $\frac{arccos(\frac13)}{\pi}$ is not rational because $arccos(\frac13)$ is not a multiple of $\pi$? – Bman72 Jan 28 '14 at 07:58
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In this answer, it is shown that $\sin(a\pi)$ and $a$ are both rational if and only if $|\sin(a\pi)|\in\{0,\frac12,1\}$. Since $\cos(a\pi)=\sin((\frac12-a)\pi)$, the same is true for $\cos(a\pi)$ and $a$.
We can adjust the proof as follows:
Note that $e^{\pm ip\pi/q}$ is an algebraic integer since $$ \left(e^{\pm ip\pi/q}\right)^{2q}-1=0 $$ Since $2\cos(p\pi/q)=e^{ip\pi/q}+e^{-ip\pi/q}$ is the sum of two algebraic integers, it is an algebraic integer and is rational if and only if it is an integer. The only integers possible are $\{-2,-1,0,1,2\}$. Thus, the only possibilities for $\cos(p\pi/q)$ to be rational would be $\{-1,-\frac12,0,\frac12,1\}$.
