三十九角形
cos  2 π 39 = cos  ( 2 π 3 − 8 π 13 ) = cos  2 π 3 cos  8 π 13 + sin  2 π 3 sin  8 π 13 = − 1 2 cos  8 π 13 + 3 2 sin  8 π 13 = − 1 2 cos  8 π 13 + 3 2 1 + cos  16 π 13 2 = − 1 24 ⋅ ( 12 cos  8 π 13 ) + 3 24 72 + 72 cos  10 π 13 = − 1 24 ( 13 − 1 + ω 104 − 20 13 + 12 i 39 3 + ω 2 104 − 20 13 − 12 i 39 3 ) + 3 24 72 + 6 ⋅ ( 12 cos  10 π 13 ) = − 1 24 ( 13 − 1 + ω 104 − 20 13 + 12 i 39 3 + ω 2 104 − 20 13 − 12 i 39 3 ) + 3 24 72 + 6 ( − 13 − 1 + ω 2 104 + 20 13 + 12 i 39 3 + ω 104 + 20 13 − 12 i 39 3 ) {\displaystyle {\begin{aligned}\cos {\frac {2\pi }{39}}=&\cos \left({\frac {2\pi }{3}}-{\frac {8\pi }{13}}\right)\\=&\cos {\frac {2\pi }{3}}\cos {\frac {8\pi }{13}}+\sin {\frac {2\pi }{3}}\sin {\frac {8\pi }{13}}\\=&-{\frac {1}{2}}\cos {\frac {8\pi }{13}}+{\frac {\sqrt {3}}{2}}\sin {\frac {8\pi }{13}}\\=&-{\frac {1}{2}}\cos {\frac {8\pi }{13}}+{\frac {\sqrt {3}}{2}}{\sqrt {\frac {1+\cos {\frac {16\pi }{13}}}{2}}}\\=&-{\frac {1}{24}}\cdot \left(12\cos {\frac {8\pi }{13}}\right)+{\frac {\sqrt {3}}{24}}{\sqrt {72+72\cos {\frac {10\pi }{13}}}}\\=&-{\frac {1}{24}}\left({\sqrt {13}}-1+\omega {\sqrt[{3}]{104-20{\sqrt {13}}+12i{\sqrt {39}}}}+\omega ^{2}{\sqrt[{3}]{104-20{\sqrt {13}}-12i{\sqrt {39}}}}\right)\\&+{\frac {\sqrt {3}}{24}}{\sqrt {72+6\cdot (12\cos {\frac {10\pi }{13}})}}\\=&-{\frac {1}{24}}\left({\sqrt {13}}-1+\omega {\sqrt[{3}]{104-20{\sqrt {13}}+12i{\sqrt {39}}}}+\omega ^{2}{\sqrt[{3}]{104-20{\sqrt {13}}-12i{\sqrt {39}}}}\right)\\&+{\frac {\sqrt {3}}{24}}{\sqrt {72+6\left(-{\sqrt {13}}-1+\omega ^{2}{\sqrt[{3}]{104+20{\sqrt {13}}+12i{\sqrt {39}}}}+\omega {\sqrt[{3}]{104+20{\sqrt {13}}-12i{\sqrt {39}}}}\right)}}\end{aligned}}}
2 cos  2 π 39 + ω ⋅ 2 cos  32 π 39 + ω 2 ⋅ 2 cos  34 π 39 = 104 + 34 13 + 3 6 ( 13 − 3 13 ) + 15 6 ( 13 + 3 13 ) + 3 3 ( − 2 13 + 6 ( 13 − 3 13 ) + 6 ( 13 + 3 13 ) ) i 8 3 2 cos  2 π 39 + ω 2 ⋅ 2 cos  32 π 39 + ω ⋅ 2 cos  34 π 39 = 104 + 34 13 + 3 6 ( 13 − 3 13 ) + 15 6 ( 13 + 3 13 ) − 3 3 ( − 2 13 + 6 ( 13 − 3 13 ) + 6 ( 13 + 3 13 ) ) i 8 3 {\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{39}}+\omega \cdot 2\cos {\frac {32\pi }{39}}+\omega ^{2}\cdot 2\cos {\frac {34\pi }{39}}=&{\sqrt[{3}]{\tfrac {104+34{\sqrt {13}}+3{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+15{\sqrt {6\left(13+3{\sqrt {13}}\right)}}+3{\sqrt {3}}\left(-2{\sqrt {13}}+{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)i}{8}}}\\2\cos {\frac {2\pi }{39}}+\omega ^{2}\cdot 2\cos {\frac {32\pi }{39}}+\omega \cdot 2\cos {\frac {34\pi }{39}}=&{\sqrt[{3}]{\tfrac {104+34{\sqrt {13}}+3{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+15{\sqrt {6\left(13+3{\sqrt {13}}\right)}}-3{\sqrt {3}}\left(-2{\sqrt {13}}+{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)i}{8}}}\\\end{aligned}}}