Trigometric Functions by Ben Green.

For real or complex z.

sin(z) = z – z^{3}/3! + z^{5}/5! – z^{7}/7! + …

cos(z) = 1 – z^{2}/2! + z^{4}/4! – z^{6}/6! + …

ez = 1 + z + z^{2}/2! + z^{3}/3! + z^{4}/4! + …

And the coup de grâce:

e^{i}^{q} = cos(q) + i sin(q).