五十角形
cos  2 π 50 = cos  π 25 = 1 2 ( cos  π 5 + i ⋅ sin  π 5 5 + cos  π 5 − i ⋅ sin  π 5 5 ) = 1 2 ( cos  2 π 10 + i ⋅ sin  2 π 10 5 + cos  2 π 10 − i ⋅ sin  2 π 10 5 ) = 1 2 ( 5 + 1 4 + i ⋅ 10 − 2 5 4 5 + 5 + 1 4 − i ⋅ 10 − 2 5 4 5 ) {\displaystyle {\begin{aligned}\cos {\frac {2\pi }{50}}=&\cos {\frac {\pi }{25}}={\frac {1}{2}}\left({\sqrt[{5}]{\cos {\frac {\pi }{5}}+i\cdot \sin {\frac {\pi }{5}}}}+{\sqrt[{5}]{\cos {\frac {\pi }{5}}-i\cdot \sin {\frac {\pi }{5}}}}\right)\\=&{\frac {1}{2}}\left({\sqrt[{5}]{\cos {\frac {2\pi }{10}}+i\cdot \sin {\frac {2\pi }{10}}}}+{\sqrt[{5}]{\cos {\frac {2\pi }{10}}-i\cdot \sin {\frac {2\pi }{10}}}}\right)\\=&{\frac {1}{2}}\left({\sqrt[{5}]{{\frac {{\sqrt {5}}+1}{4}}+i\cdot {\frac {\sqrt {10-2{\sqrt {5}}}}{4}}}}+{\sqrt[{5}]{{\frac {{\sqrt {5}}+1}{4}}-i\cdot {\frac {\sqrt {10-2{\sqrt {5}}}}{4}}}}\right)\\\end{aligned}}}