十三角形
Trigonometric constants expressed in real radicalsより
cos  2 π 13 = 13 − 1 + 104 − 20 13 − 12 i 39 3 + 104 − 20 13 + 12 i 39 3 12 {\displaystyle \cos {\frac {2\pi }{13}}={\frac {{\sqrt {13}}-1+{\sqrt[{3}]{104-20{\sqrt {13}}-12i{\sqrt {39}}}}+{\sqrt[{3}]{104-20{\sqrt {13}}+12i{\sqrt {39}}}}}{12}}}
また、以下の関係が成り立つ。
2 cos  2 π 13 + 2 cos  10 π 13 = − 2 + − 260 − 156 3 i 3 ω + − 260 + 156 3 i 3 ω 2 6 = 1 3 ( − 1 + 13 ⋅ − 5 − 3 3 i 2 13 3 ω + 13 ⋅ − 5 + 3 3 i 2 13 3 ω 2 ) {\displaystyle 2\cos {\frac {2\pi }{13}}+2\cos {\frac {10\pi }{13}}={\frac {-2+{\sqrt[{3}]{-260-156{\sqrt {3}}i}}\omega +{\sqrt[{3}]{-260+156{\sqrt {3}}i}}\omega ^{2}}{6}}={\frac {1}{3}}\left(-1+{\sqrt {13}}\cdot {\sqrt[{3}]{\frac {-5-3{\sqrt {3}}i}{2{\sqrt {13}}}}}\omega +{\sqrt {13}}\cdot {\sqrt[{3}]{\frac {-5+3{\sqrt {3}}i}{2{\sqrt {13}}}}}\omega ^{2}\right)}