When does a rational portion of a circle have a rational cosine?
If r is a rational number, cos(2πr) rational if and only if the denominator of r is 1, 2, 3, 4, or 6.
This means that the special values of cosine you learn in a trig class, with a simple argument and simple value, are the only ones possible. (Here simple argument means an angle with an integer number of degrees and simple value means a rational number.) And if you see a result such as cos(π/7) = 837/929, you know it can’t be exactly correct, though in this case it’s very close.