正五十六角形
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2021/09/10 21:58 UTC 版)
正五十六角形においては、中心角と外角は6.428571…°で、内角は173.571428…°となる。一辺の長さが a の正五十六角形の面積 S は S = 14 a 2 cot π 56 a 2 {\displaystyle S=14a^{2}\cot {\frac {\pi }{56}}a^{2}} 関係式 x 1 = 2 cos 2 π 56 + 2 cos 50 π 56 + 2 cos 18 π 56 = 14 − 2 2 x 2 = 2 cos 10 π 56 + 2 cos 26 π 56 + 2 cos 22 π 56 = 14 + 2 2 x 3 = 2 cos 54 π 56 + 2 cos 6 π 56 + 2 cos 38 π 56 = − 14 + 2 2 x 4 = 2 cos 46 π 56 + 2 cos 30 π 56 + 2 cos 34 π 56 = − 14 − 2 2 {\displaystyle {\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{56}}+2\cos {\frac {50\pi }{56}}+2\cos {\frac {18\pi }{56}}={\frac {{\sqrt {14}}-{\sqrt {2}}}{2}}\\&x_{2}=2\cos {\frac {10\pi }{56}}+2\cos {\frac {26\pi }{56}}+2\cos {\frac {22\pi }{56}}={\frac {{\sqrt {14}}+{\sqrt {2}}}{2}}\\&x_{3}=2\cos {\frac {54\pi }{56}}+2\cos {\frac {6\pi }{56}}+2\cos {\frac {38\pi }{56}}={\frac {-{\sqrt {14}}+{\sqrt {2}}}{2}}\\&x_{4}=2\cos {\frac {46\pi }{56}}+2\cos {\frac {30\pi }{56}}+2\cos {\frac {34\pi }{56}}={\frac {-{\sqrt {14}}-{\sqrt {2}}}{2}}\\\end{aligned}}} 三次方程式の係数を求めると 2 cos 2 π 56 ⋅ 2 cos 50 π 56 + 2 cos 50 π 56 ⋅ 2 cos 18 π 56 + 2 cos 18 π 56 ⋅ 2 cos 2 π 56 = 2 cos 26 π 28 + 2 cos 22 π 28 + 2 cos 10 π 28 + 2 cos 6 π 7 + 2 cos 4 π 7 + 2 cos 2 π 7 = − 7 − 1 = 2 x 4 2 cos 2 π 56 ⋅ 2 cos 50 π 56 ⋅ 2 cos 18 π 56 = x 4 − 2 = x 4 + x 1 + x 4 = x 1 + 2 x 4 {\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{56}}\cdot 2\cos {\frac {50\pi }{56}}+2\cos {\frac {50\pi }{56}}\cdot 2\cos {\frac {18\pi }{56}}+2\cos {\frac {18\pi }{56}}\cdot 2\cos {\frac {2\pi }{56}}\\&=2\cos {\frac {26\pi }{28}}+2\cos {\frac {22\pi }{28}}+2\cos {\frac {10\pi }{28}}+2\cos {\frac {6\pi }{7}}+2\cos {\frac {4\pi }{7}}+2\cos {\frac {2\pi }{7}}=-{\sqrt {7}}-1={\sqrt {2}}x_{4}\\&2\cos {\frac {2\pi }{56}}\cdot 2\cos {\frac {50\pi }{56}}\cdot 2\cos {\frac {18\pi }{56}}=x_{4}-{\sqrt {2}}=x_{4}+x_{1}+x_{4}=x_{1}+2x_{4}\\\end{aligned}}} 解と係数の関係より u 3 − x 1 u 2 + 2 x 4 u − ( x 1 + 2 x 4 ) = 0 {\displaystyle u^{3}-x_{1}u^{2}+{\sqrt {2}}x_{4}u-(x_{1}+2x_{4})=0} 変数変換 u = v + x 1 / 3 {\displaystyle u=v+x_{1}/3} 整理すると v 3 − 7 + 2 7 3 v + ( 7 + 2 7 ) ( 5 2 + 14 ) 54 = 0 {\displaystyle v^{3}-{\frac {7+2{\sqrt {7}}}{3}}v+{\frac {(7+2{\sqrt {7}})(5{\sqrt {2}}+{\sqrt {14}})}{54}}=0} 三角関数、逆三角関数を用いた解は u 1 = 14 − 2 6 + 2 7 + 2 7 3 cos ( 1 3 arccos ( − 5 2 + 14 4 7 + 2 7 ) ) {\displaystyle u_{1}={\frac {{\sqrt {14}}-{\sqrt {2}}}{6}}+{\frac {2{\sqrt {7+2{\sqrt {7}}}}}{3}}\cos \left({\frac {1}{3}}\arccos \left(-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}\right)\right)} 平方根、立方根で表すと u 1 = 14 − 2 6 + 7 + 2 7 3 − 5 2 + 14 4 7 + 2 7 + i 48 + 12 7 4 7 + 2 7 3 + 7 + 2 7 3 − 5 2 + 14 4 7 + 2 7 − i 48 + 12 7 4 7 + 2 7 3 {\displaystyle u_{1}={\frac {{\sqrt {14}}-{\sqrt {2}}}{6}}+{\frac {\sqrt {7+2{\sqrt {7}}}}{3}}{\sqrt[{3}]{-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}+i{\frac {\sqrt {48+12{\sqrt {7}}}}{4{\sqrt {7+2{\sqrt {7}}}}}}}}+{\frac {\sqrt {7+2{\sqrt {7}}}}{3}}{\sqrt[{3}]{-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}-i{\frac {\sqrt {48+12{\sqrt {7}}}}{4{\sqrt {7+2{\sqrt {7}}}}}}}}} cos ( 2 π / 56 ) {\displaystyle \cos(2\pi /56)} を平方根と立方根で表すと cos 2 π 56 = 14 − 2 12 + 7 + 2 7 6 − 5 2 + 14 4 7 + 2 7 + i 48 + 12 7 4 7 + 2 7 3 + 7 + 2 7 6 − 5 2 + 14 4 7 + 2 7 − i 48 + 12 7 4 7 + 2 7 3 {\displaystyle \cos {\frac {2\pi }{56}}={\frac {{\sqrt {14}}-{\sqrt {2}}}{12}}+{\frac {\sqrt {7+2{\sqrt {7}}}}{6}}{\sqrt[{3}]{-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}+i{\frac {\sqrt {48+12{\sqrt {7}}}}{4{\sqrt {7+2{\sqrt {7}}}}}}}}+{\frac {\sqrt {7+2{\sqrt {7}}}}{6}}{\sqrt[{3}]{-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}-i{\frac {\sqrt {48+12{\sqrt {7}}}}{4{\sqrt {7+2{\sqrt {7}}}}}}}}} cos 2 π 56 = 14 − 2 12 + 1 6 − 49 2 + 17 14 4 + i 6048 + 2268 7 4 3 + 1 6 − 49 2 + 17 14 4 − i 6048 + 2268 7 4 3 {\displaystyle \cos {\frac {2\pi }{56}}={\frac {{\sqrt {14}}-{\sqrt {2}}}{12}}+{\frac {1}{6}}{\sqrt[{3}]{-{\frac {49{\sqrt {2}}+17{\sqrt {14}}}{4}}+i{\frac {\sqrt {6048+2268{\sqrt {7}}}}{4}}}}+{\frac {1}{6}}{\sqrt[{3}]{-{\frac {49{\sqrt {2}}+17{\sqrt {14}}}{4}}-i{\frac {\sqrt {6048+2268{\sqrt {7}}}}{4}}}}} cos 2 π 56 = 14 − 2 12 + 1 12 − 98 2 − 34 14 + i ⋅ 12 168 + 63 7 3 + 1 12 − 98 2 − 34 14 − i ⋅ 12 168 + 63 7 3 {\displaystyle \cos {\frac {2\pi }{56}}={\frac {{\sqrt {14}}-{\sqrt {2}}}{12}}+{\frac {1}{12}}{\sqrt[{3}]{-98{\sqrt {2}}-34{\sqrt {14}}+i\cdot 12{\sqrt {168+63{\sqrt {7}}}}}}+{\frac {1}{12}}{\sqrt[{3}]{-98{\sqrt {2}}-34{\sqrt {14}}-i\cdot 12{\sqrt {168+63{\sqrt {7}}}}}}}
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