One can use a feature of complex numbers, and the span of a finite set.
Consider the set of cyclotomic numbers, ie $C(n) = \operatorname{cis}(2\pi/n)^m$, where $\operatorname{cis}(x)=\cos(x)+i\sin(x)$. Such a set is closed to multiplication. The 'Z-span' of the set is the set of values $\sum(a_m \operatorname{cis}(2\pi/n)$, over n, is also closed to multiplication.
We now begin with the observation that a span of a finite set, closed to multiplication, can not include the fractions. This is proved by showing that if a rational number, not an integer, is in the set, so must all of its powers. (ie if $1/2$ is constructible by steps at multiples of $N°$, (eg a random walk of unit-size steps at exact degrees), so must all values of $1/2^a$).
Since this means that that the intersection of the cyclotomic numbers $\mathbb{C}_n$ and the rationals $\mathbb{F}$ can not include any fractions, and thus must give $\mathbb{Z}$.
The double-cosine of the half-angles, are given by $1-\operatorname{cis}(2\pi/n)$, and therefore we see that the only rational numbers that can occur in the sines and cosines, is $1/2$. The chord, and the supplement-chords are entirely free of rationals, and further, no product of such numbers can be rational.