Carl Friedrich Gauss (1777–1855) derived the exact value of cos(2π/17)
This remarkable expression was part of his proof that a regular 17-gon is constructible with compass and straightedge; one of the great achievements in the history of mathematics.
cos(2π/17) = [-1 + √17 + √(34 - 2√17) + 2√17 + 3√17 - √(34 - 2√17) - 2√34 + 2√17] / 16