Prove that:
If a,b are relatively prime integers, and b is not a power of 2, then
$\displaystyle \frac{1}{\pi}\arccos\left (\frac{a}{b}\right )$
is irrational.
we need the following result
for each natural number $\displaystyle n\geq 2$ there is a polynomial function of degree n with integer coefficients
$\displaystyle f(x)=2^{n-1}x^n+ a_{n-1}x^{n-1}+\text{...}+a_0$
such that
$\displaystyle \cos (n x)=f(\cos (x))$