十九角形
2 cos  2 π 19 + 2 cos  16 π 19 + 2 cos  14 π 19 = − 1 + ω 2 133 + 57 3 i 2 3 + ω 133 − 57 3 i 2 3 3 = α 2 cos  4 π 19 + 2 cos  6 π 19 + 2 cos  10 π 19 = − 1 + 133 + 57 3 i 2 3 + 133 − 57 3 i 2 3 3 = β 2 cos  8 π 19 + 2 cos  18 π 19 + 2 cos  12 π 19 = − 1 + ω 133 + 57 3 i 2 3 + ω 2 133 − 57 3 i 2 3 3 = γ {\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{19}}+2\cos {\frac {16\pi }{19}}+2\cos {\frac {14\pi }{19}}=&{\frac {-1+\omega ^{2}{\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}+\omega {\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}}{3}}=\alpha \\2\cos {\frac {4\pi }{19}}+2\cos {\frac {6\pi }{19}}+2\cos {\frac {10\pi }{19}}=&{\frac {-1+{\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}+{\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}}{3}}=\beta \\2\cos {\frac {8\pi }{19}}+2\cos {\frac {18\pi }{19}}+2\cos {\frac {12\pi }{19}}=&{\frac {-1+\omega {\sqrt[{3}]{\frac {133+57{\sqrt {3}}i}{2}}}+\omega ^{2}{\sqrt[{3}]{\frac {133-57{\sqrt {3}}i}{2}}}}{3}}=\gamma \\\end{aligned}}}