三十九角形
cos  2 π 39 = 1 6 ( 1 − 13 − 6 ( 13 − 3 13 ) 4 + 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 + 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}\cos {\frac {2\pi }{39}}=&{\frac {1}{6}}\left({\tfrac {1-{\sqrt {13}}-{\sqrt {6\left(13-3{\sqrt {13}}\right)}}}{4}}+{\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}}}+{\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}}}\right)\\\end{aligned}}}
三十九角形(さんじゅうきゅうかくけい、さんじゅうきゅうかっけい、triacontaenneagon)は、多角形の一つで、39本の辺と39個の頂点を持つ図形である。内角の和は6660°、対角線の本数は702本である。
( 2 cos  2 π 39 + ω ⋅ 2 cos  32 π 39 + ω 2 ⋅ 2 cos  34 π 39 ) 3 = 3 x 1 + 2 cos  2 π 13 + 2 cos  8 π 13 + 2 cos  6 π 13 + 6 ( x 2 + 2 ) + 3 ω ( 2 x 1 + x 3 + 2 cos  4 π 13 + 2 cos  10 π 13 + 2 cos  12 π 13 ) + 3 ω 2 ( 2 x 1 + x 4 + 2 cos  4 π 13 + 2 cos  10 π 13 + 2 cos  12 π 13 ) = 3 x 1 + − 1 + 13 2 + 6 ( x 2 + 2 ) + 3 ω ( 2 x 1 + x 3 + − 1 − 13 2 ) + 3 ω 2 ( 2 x 1 + x 4 + − 1 − 13 2 ) = 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 ( 2 cos  2 π 39 + ω 2 ⋅ 2 cos  32 π 39 + ω ⋅ 2 cos  34 π 39 ) 3 = 3 x 1 + 2 cos  2 π 13 + 2 cos  8 π 13 + 2 cos  6 π 13 + 6 ( x 2 + 2 ) + 3 ω 2 ( 2 x 1 + x 3 + 2 cos  4 π 13 + 2 cos  10 π 13 + 2 cos  12 π 13 ) + 3 ω ( 2 x 1 + x 4 + 2 cos  4 π 13 + 2 cos  10 π 13 + 2 cos  12 π 13 ) = 3 x 1 + − 1 + 13 2 + 6 ( x 2 + 2 ) + 3 ω 2 ( 2 x 1 + x 3 + − 1 − 13 2 ) + 3 ω ( 2 x 1 + x 4 + − 1 − 13 2 ) = 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 {\displaystyle {\begin{aligned}\left(2\cos {\frac {2\pi }{39}}+\omega \cdot 2\cos {\frac {32\pi }{39}}+\omega ^{2}\cdot 2\cos {\frac {34\pi }{39}}\right)^{3}=&3x_{1}+2\cos {\frac {2\pi }{13}}+2\cos {\frac {8\pi }{13}}+2\cos {\frac {6\pi }{13}}+6(x_{2}+2)+3\omega \left(2x_{1}+x_{3}+2\cos {\frac {4\pi }{13}}+2\cos {\frac {10\pi }{13}}+2\cos {\frac {12\pi }{13}}\right)+3\omega ^{2}\left(2x_{1}+x_{4}+2\cos {\frac {4\pi }{13}}+2\cos {\frac {10\pi }{13}}+2\cos {\frac {12\pi }{13}}\right)\\=&3x_{1}+{\frac {-1+{\sqrt {13}}}{2}}+6(x_{2}+2)+3\omega \left(2x_{1}+x_{3}+{\frac {-1-{\sqrt {13}}}{2}}\right)+3\omega ^{2}\left(2x_{1}+x_{4}+{\frac {-1-{\sqrt {13}}}{2}}\right)\\=&{\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}}\\\left(2\cos {\frac {2\pi }{39}}+\omega ^{2}\cdot 2\cos {\frac {32\pi }{39}}+\omega \cdot 2\cos {\frac {34\pi }{39}}\right)^{3}=&3x_{1}+2\cos {\frac {2\pi }{13}}+2\cos {\frac {8\pi }{13}}+2\cos {\frac {6\pi }{13}}+6(x_{2}+2)+3\omega ^{2}\left(2x_{1}+x_{3}+2\cos {\frac {4\pi }{13}}+2\cos {\frac {10\pi }{13}}+2\cos {\frac {12\pi }{13}}\right)+3\omega \left(2x_{1}+x_{4}+2\cos {\frac {4\pi }{13}}+2\cos {\frac {10\pi }{13}}+2\cos {\frac {12\pi }{13}}\right)\\=&3x_{1}+{\frac {-1+{\sqrt {13}}}{2}}+6(x_{2}+2)+3\omega ^{2}\left(2x_{1}+x_{3}+{\frac {-1-{\sqrt {13}}}{2}}\right)+3\omega \left(2x_{1}+x_{4}+{\frac {-1-{\sqrt {13}}}{2}}\right)\\=&{\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}}}