Is there a general formula for $\\sin( {p \\over q} \\pi)$?

Question

Virtually everyone knows the basic values of the unit circle, $\\sin(\\pi) = 0; \\ \\ \\sin({\\pi \\over 2}) = 1; \\ \\ \\sin({\\pi \\over 3}) = {\\sqrt{3} \\over 2} \\\\$\nAnd other values can be calculated through various identities, like $\\sin({\\pi \\over 8}) =\\frac{1}{2} \\sqrt{2 - \\sqrt{2}}\\\\$\nDoes there exist a general formula for $\\sin({p\\over q} \\pi)$ for rational ${p \\over q}$ as an algebraic number?

Answer 1

Not that I am aware of.

Of course, sin x = ∑x^2k+1/(2k+1)! for k=0 to ∞, but I doubt that is what you want.

Also you might want to look up some math history to understand how the ancient Greeks calculated sines.

Start with Ptolemy's theorem!