# 八十四角形

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## 正八十四角形

${\displaystyle S={\frac {84}{4}}a^{2}\cot {\frac {\pi }{84}}\simeq 561.23682a^{2}}$

{\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{84}}+2\cos {\frac {74\pi }{84}}+2\cos {\frac {50\pi }{84}}={\frac {{\sqrt {3}}-{\sqrt {7}}}{2}}=x_{1}\\2\cos {\frac {38\pi }{84}}+2\cos {\frac {62\pi }{84}}+2\cos {\frac {58\pi }{84}}={\frac {-{\sqrt {3}}-{\sqrt {7}}}{2}}=x_{2}\\2\cos {\frac {34\pi }{84}}+2\cos {\frac {82\pi }{84}}+2\cos {\frac {10\pi }{84}}={\frac {-{\sqrt {3}}+{\sqrt {7}}}{2}}=x_{3}\\2\cos {\frac {26\pi }{84}}+2\cos {\frac {46\pi }{84}}+2\cos {\frac {22\pi }{84}}={\frac {{\sqrt {3}}+{\sqrt {7}}}{2}}=x_{4}\\\end{aligned}}}

さらに、以下のような関係式が得られる。

{\displaystyle {\begin{aligned}\left(2\cos {\frac {2\pi }{84}}+\omega \cdot 2\cos {\frac {74\pi }{84}}+\omega ^{2}\cdot 2\cos {\frac {50\pi }{84}}\right)^{3}=&3x_{1}+2\cos {\frac {2\pi }{28}}+2\cos {\frac {6\pi }{28}}+2\cos {\frac {18\pi }{28}}+6(x_{4})+3\omega \left(2x_{1}+6\cos {\frac {10\pi }{12}}+2\cos {\frac {10\pi }{28}}+2\cos {\frac {26\pi }{28}}+2\cos {\frac {22\pi }{28}}\right)+3\omega ^{2}\left(2x_{1}+x_{4}+2\cos {\frac {2\pi }{28}}+2\cos {\frac {6\pi }{28}}+2\cos {\frac {18\pi }{28}}\right)\\=&{\frac {11{\sqrt {3}}+9{\sqrt {7}}-3{\sqrt {3}}(7{\sqrt {3}}+5{\sqrt {7}})i}{4}}\\\left(2\cos {\frac {2\pi }{84}}+\omega ^{2}\cdot 2\cos {\frac {74\pi }{84}}+\omega \cdot 2\cos {\frac {50\pi }{84}}\right)^{3}=&3x_{1}+2\cos {\frac {2\pi }{28}}+2\cos {\frac {6\pi }{28}}+2\cos {\frac {18\pi }{28}}+6(x_{4})+3\omega ^{2}\left(2x_{1}+6\cos {\frac {10\pi }{12}}+2\cos {\frac {10\pi }{28}}+2\cos {\frac {26\pi }{28}}+2\cos {\frac {22\pi }{28}}\right)+3\omega \left(2x_{1}+x_{4}+2\cos {\frac {2\pi }{28}}+2\cos {\frac {6\pi }{28}}+2\cos {\frac {18\pi }{28}}\right)\\=&{\frac {11{\sqrt {3}}+9{\sqrt {7}}+3{\sqrt {3}}(7{\sqrt {3}}+5{\sqrt {7}})i}{4}}\\\end{aligned}}}

{\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{84}}+\omega \cdot 2\cos {\frac {74\pi }{84}}+\omega ^{2}\cdot 2\cos {\frac {50\pi }{84}}=&{\sqrt[{3}]{\frac {11{\sqrt {3}}+9{\sqrt {7}}-3{\sqrt {3}}(7{\sqrt {3}}+5{\sqrt {7}})i}{4}}}\\2\cos {\frac {2\pi }{84}}+\omega ^{2}\cdot 2\cos {\frac {74\pi }{84}}+\omega \cdot 2\cos {\frac {50\pi }{84}}=&{\sqrt[{3}]{\frac {11{\sqrt {3}}+9{\sqrt {7}}+3{\sqrt {3}}(7{\sqrt {3}}+5{\sqrt {7}})i}{4}}}\\\end{aligned}}}

よって

{\displaystyle {\begin{aligned}\cos {\frac {2\pi }{84}}=&{\frac {1}{6}}\left({\frac {{\sqrt {3}}-{\sqrt {7}}}{2}}+{\sqrt[{3}]{\frac {11{\sqrt {3}}+9{\sqrt {7}}-3{\sqrt {3}}(7{\sqrt {3}}+5{\sqrt {7}})i}{4}}}+{\sqrt[{3}]{\frac {11{\sqrt {3}}+9{\sqrt {7}}+3{\sqrt {3}}(7{\sqrt {3}}+5{\sqrt {7}})i}{4}}}\right)\\\end{aligned}}}

## 脚注

 [脚注の使い方]