正三十九角形
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2021/09/10 22:01 UTC 版)
正三十九角形においては、中心角と外角は9.23…°で、内角は170.769…°となる。一辺の長さが a の正三十九角形の面積 S は S = 39 4 a 2 cot π 39 ≃ 120.77542 a 2 {\displaystyle S={\frac {39}{4}}a^{2}\cot {\frac {\pi }{39}}\simeq 120.77542a^{2}} cos ( 2 π / 39 ) {\displaystyle \cos(2\pi /39)} を平方根と立方根で表すと 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}}} 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}}} 関係式 2 cos 2 π 39 + 2 cos 32 π 39 + 2 cos 34 π 39 = 1 4 ( 1 − 13 − 6 ( 13 − 3 13 ) ) = x 1 2 cos 4 π 39 + 2 cos 14 π 39 + 2 cos 10 π 39 = 1 4 ( 1 + 13 + 6 ( 13 + 3 13 ) ) = x 2 2 cos 8 π 39 + 2 cos 28 π 39 + 2 cos 20 π 39 = 1 4 ( 1 − 13 + 6 ( 13 − 3 13 ) ) = x 3 2 cos 16 π 39 + 2 cos 22 π 39 + 2 cos 38 π 39 = 1 4 ( 1 + 13 − 6 ( 13 + 3 13 ) ) = x 4 {\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{39}}+2\cos {\frac {32\pi }{39}}+2\cos {\frac {34\pi }{39}}={\frac {1}{4}}\left(1-{\sqrt {13}}-{\sqrt {6\left(13-3{\sqrt {13}}\right)}}\right)=x_{1}\\2\cos {\frac {4\pi }{39}}+2\cos {\frac {14\pi }{39}}+2\cos {\frac {10\pi }{39}}={\frac {1}{4}}\left(1+{\sqrt {13}}+{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)=x_{2}\\2\cos {\frac {8\pi }{39}}+2\cos {\frac {28\pi }{39}}+2\cos {\frac {20\pi }{39}}={\frac {1}{4}}\left(1-{\sqrt {13}}+{\sqrt {6\left(13-3{\sqrt {13}}\right)}}\right)=x_{3}\\2\cos {\frac {16\pi }{39}}+2\cos {\frac {22\pi }{39}}+2\cos {\frac {38\pi }{39}}={\frac {1}{4}}\left(1+{\sqrt {13}}-{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)=x_{4}\\\end{aligned}}} さらに、以下のような関係式が得られる。 ( 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}}} 両辺の立方根を取ると 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}}} よって 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}}}
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