八十五角形
cos  2 π 85 = cos  ( π 5 − 3 π 17 ) = cos  π 5 cos  3 π 17 + sin  π 5 sin  3 π 17 = 5 + 1 4 cos  3 π 17 + 1 4 10 − 2 3 sin  3 π 17 = 5 + 1 4 ⋅ 1 16 ( + 1 + 17 + 34 + 68 + 68 − 2448 − 2720 − 6284288 ) + 1 4 10 − 2 5 ⋅ 1 8 ( 34 + 68 − 136 + 1088 − 272 − 39168 + 43520 − 1608777728 ) {\displaystyle {\begin{aligned}\cos {\frac {2\pi }{85}}=&\cos \left({\frac {\pi }{5}}-{\frac {3\pi }{17}}\right)\\=&\cos {\frac {\pi }{5}}\cos {\frac {3\pi }{17}}+\sin {\frac {\pi }{5}}\sin {\frac {3\pi }{17}}\\=&{\frac {{\sqrt {5}}+1}{4}}\cos {\frac {3\pi }{17}}+{\frac {1}{4}}{\sqrt {10-2{\sqrt {3}}}}\sin {\frac {3\pi }{17}}\\=&{\frac {{\sqrt {5}}+1}{4}}\cdot {\frac {1}{16}}\left(+1+{\sqrt {17}}+{\sqrt {34+{\sqrt {68}}}}+{\sqrt {68-{\sqrt {2448}}-{\sqrt {2720-{\sqrt {6284288}}}}}}\right)\\&+{\frac {1}{4}}{\sqrt {10-2{\sqrt {5}}}}\cdot {\frac {1}{8}}\left({\sqrt {34+{\sqrt {68}}-{\sqrt {136+{\sqrt {1088}}}}-{\sqrt {272-{\sqrt {39168}}+{\sqrt {43520-{\sqrt {1608777728}}}}}}}}\right)\end{aligned}}}