1. Irrationality proof

Prove that:

If a,b are relatively prime integers, and b is not a power of 2, then

$\frac{1}{\pi}\arccos\left (\frac{a}{b}\right )$

is irrational.

2. Re: Irrationality proof

we need the following result
for each natural number $n\geq 2$ there is a polynomial function of degree n with integer coefficients

$f(x)=2^{n-1}x^n+ a_{n-1}x^{n-1}+\text{...}+a_0$

such that

$\cos (n x)=f(\cos (x))$

3. Re: Irrationality proof

Yes, this is the way I obtained this result