Given a range of the rational numbers, $x$, between $0$ and $2\pi$\, what is the set of rational numbers $ y = \cos(x) $?
I was inspired by the stackoverflow question Can $\cos(a)$ ever equal $0$ in floating point? (The irrational number $\frac{\pi}{2}$ does not translate well into a computer representation.)
I looked for rational cosines, and came up with the likes of $ 0, \frac{\pi}{3},\frac{\pi}{2}, \pi, \frac{3\pi}{2}$ Following this rabbit hole, I wondered if there were any rational (Floating Point) numbers (besides $0$) that yielded rational cosines.
One respondent opened a different question, on english.stackexchange.com, What is the upper bound on “several”? which involves the size of the set in question.