三十九角形
cos  2 π 39 = cos  2 π 3 ⋅ 13 = 1 2 ⋅ ( cos  2 π 13 + i ⋅ sin  2 π 13 3 + cos  2 π 13 − i ⋅ sin  2 π 13 3 ) = 1 2 ⋅ cos  2 π 13 + i ⋅ sin  2 π 13 3 + 1 2 ⋅ cos  2 π 13 − i ⋅ sin  2 π 13 3 = 1 2 ⋅ 1 12 ( 13 − 1 + 104 − 20 13 − 12 i 39 3 + 104 − 20 13 + 12 i 39 3 ) + i ⋅ sin  2 π 13 3 + 1 2 ⋅ 1 12 ( 13 − 1 + 104 − 20 13 − 12 i 39 3 + 104 − 20 13 + 12 i 39 3 ) − i ⋅ sin  2 π 13 3 {\displaystyle {\begin{aligned}\cos {\frac {2\pi }{39}}=&\cos {\frac {2\pi }{3\cdot 13}}\\=&{\frac {1}{2}}\cdot \left({\sqrt[{3}]{\cos {\frac {2\pi }{13}}+i\cdot \sin {\frac {2\pi }{13}}}}+{\sqrt[{3}]{\cos {\frac {2\pi }{13}}-i\cdot \sin {\frac {2\pi }{13}}}}\right)\\=&{\frac {1}{2}}\cdot {\sqrt[{3}]{\cos {\frac {2\pi }{13}}+i\cdot \sin {\frac {2\pi }{13}}}}+{\frac {1}{2}}\cdot {\sqrt[{3}]{\cos {\frac {2\pi }{13}}-i\cdot \sin {\frac {2\pi }{13}}}}\\=&{\frac {1}{2}}\cdot {\sqrt[{3}]{{\frac {1}{12}}({\sqrt {13}}-1+{\sqrt[{3}]{104-20{\sqrt {13}}-12i{\sqrt {39}}}}+{\sqrt[{3}]{104-20{\sqrt {13}}+12i{\sqrt {39}}}})+i\cdot \sin {\frac {2\pi }{13}}}}\\&+{\frac {1}{2}}\cdot {\sqrt[{3}]{{\frac {1}{12}}({\sqrt {13}}-1+{\sqrt[{3}]{104-20{\sqrt {13}}-12i{\sqrt {39}}}}+{\sqrt[{3}]{104-20{\sqrt {13}}+12i{\sqrt {39}}}})-i\cdot \sin {\frac {2\pi }{13}}}}\end{aligned}}}