for any rational number A/B, it is always possible to express sin(pi*A/B) in radicals (permitting square roots, cube roots, and higher power roots). cosine and tangent also. (permitting only square roots is not enough: famously, one cannot construct regular heptagons or nonagons with compass and straightedge.)
sin(pi/4) = sqrt(2)/2, sin(pi/3) = sqrt(3)/2, and sin(pi/6) = 1/2 are well known. that every rational multiple of pi is possible was news to me.
the very hand waving reason is because the cyclotomic equation, the polynomial that remains after factoring out the trivial solution z=1 from z^n - 1 = 0, has as its Galois group a cyclic group (because the words "cyclotomic" and "cyclic" are similar), and all cyclic groups are abelian, and all abelian groups are "solvable", meaning they can be solved in radicals.
the algorithm to solve cyclotomics was first described by Gauss (who did much more than just solve z^17 - 1 = 0 with square roots only), but implemented rarely. here is one implementation in Maple. there seems to be plenty of opportunities for optimization.
previously, continued fractions.
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