Irrationality of $\frac{1}{\pi} \arctan(x)$ using complex roots of the unit

Question

I stumbled upon a problem where we have to show that $\frac{1}{5}(3+4i)$ can't be a complex root of the unit which is equivalent to show that $(3+4i)^n=5^n$ has no positive integer solutions, for which I used the fact that $3+4i$ is idempotent modulo $5$ (if you know other methods, feel free to comment them), and from this result, we can deduce that $\frac{1}{\pi}\arctan(\frac{4}{3})$ is irrational since it's the argument of $3+4i$. So I was wondering if this procedure can be generalised for some kind of complex numbers perhaps of the form $\frac{1}{c_n}(a_n+ib_n)$. I first thought that it's true for Pythagorean triples $(a_n,b_n,c_n)$ but it doesn't seem to work for $(6,8,10)$. I would like to know if there's a way to generalise this.