正四十二角形
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2021/09/10 21:59 UTC 版)
正四十二角形においては、中心角と外角は8.571…°で、内角は171.428…°となる。一辺の長さが a の正四十二角形の面積 S は S = 42 4 a 2 cot π 42 ≃ 140.11276 a 2 {\displaystyle S={\frac {42}{4}}a^{2}\cot {\frac {\pi }{42}}\simeq 140.11276a^{2}} cos ( 2 π / 42 ) {\displaystyle \cos(2\pi /42)} を平方根と立方根で表すことが可能である。 cos 2 π 42 = cos π 21 = 1 12 72 + 72 cos 2 π 21 = 1 12 72 + 72 ⋅ 1 + 21 + 154 − 30 21 + ( 42 3 − 18 7 ) i 3 + 154 − 30 21 + ( 18 7 − 42 3 ) i 3 12 = 1 12 72 + 6 ( 1 + 21 + 154 − 30 21 + ( 42 3 − 18 7 ) i 3 + 154 − 30 21 + ( 18 7 − 42 3 ) i 3 ) {\displaystyle {\begin{aligned}\cos {\frac {2\pi }{42}}=&\cos {\frac {\pi }{21}}\\=&{\frac {1}{12}}{\sqrt {72+72\cos {\frac {2\pi }{21}}}}\\=&{\frac {1}{12}}{\sqrt {72+72\cdot {\frac {1+{\sqrt {21}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(42{\sqrt {3}}-18{\sqrt {7}}\right)i}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(18{\sqrt {7}}-42{\sqrt {3}}\right)i}}}{12}}}}\\=&{\frac {1}{12}}{\sqrt {72+6\left({1+{\sqrt {21}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(42{\sqrt {3}}-18{\sqrt {7}}\right)i}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(18{\sqrt {7}}-42{\sqrt {3}}\right)i}}}\right)}}\end{aligned}}} cos 2 π 42 = cos 2 π 3 ⋅ 14 = 1 2 ⋅ ( cos 2 π 14 + i ⋅ sin 2 π 14 3 + cos 2 π 14 − i ⋅ sin 2 π 14 3 ) = 1 2 ⋅ 3 ( 20 + 2 28 − 84 i 3 3 + 2 28 + 84 i 3 3 ) 12 + i ⋅ 3 ( 28 − 2 28 − 84 i 3 3 − 2 28 + 84 i 3 3 ) 12 3 + 1 2 ⋅ 3 ( 20 + 2 28 − 84 i 3 3 + 2 28 + 84 i 3 3 ) 12 − i ⋅ 3 ( 28 − 2 28 − 84 i 3 3 − 2 28 + 84 i 3 3 ) 12 3 {\displaystyle {\begin{aligned}\cos {\frac {2\pi }{42}}=&\cos {\frac {2\pi }{3\cdot 14}}\\=&{\frac {1}{2}}\cdot \left({\sqrt[{3}]{\cos {\frac {2\pi }{14}}+i\cdot \sin {\frac {2\pi }{14}}}}+{\sqrt[{3}]{\cos {\frac {2\pi }{14}}-i\cdot \sin {\frac {2\pi }{14}}}}\right)\\=&{\frac {1}{2}}\cdot {\sqrt[{3}]{{\tfrac {\sqrt {3\left(20+2{\sqrt[{3}]{28-84i{\sqrt {3}}}}+2{\sqrt[{3}]{28+84i{\sqrt {3}}}}\right)}}{12}}+i\cdot {\tfrac {\sqrt {3\left(28-2{\sqrt[{3}]{28-84i{\sqrt {3}}}}-2{\sqrt[{3}]{28+84i{\sqrt {3}}}}\right)}}{12}}}}+{\frac {1}{2}}\cdot {\sqrt[{3}]{{\tfrac {\sqrt {3\left(20+2{\sqrt[{3}]{28-84i{\sqrt {3}}}}+2{\sqrt[{3}]{28+84i{\sqrt {3}}}}\right)}}{12}}-i\cdot {\tfrac {\sqrt {3\left(28-2{\sqrt[{3}]{28-84i{\sqrt {3}}}}-2{\sqrt[{3}]{28+84i{\sqrt {3}}}}\right)}}{12}}}}\end{aligned}}} 関係式 以下のように定義すると α = 2 cos 2 π 42 + 2 cos 10 π 42 + 2 cos 34 π 42 = − 1 + 21 2 β = 2 cos 22 π 42 + 2 cos 26 π 42 + 2 cos 38 π 42 = − 1 − 21 2 {\displaystyle {\begin{aligned}&\alpha =2\cos {\frac {2\pi }{42}}+2\cos {\frac {10\pi }{42}}+2\cos {\frac {34\pi }{42}}={\frac {-1+{\sqrt {21}}}{2}}\\&\beta =2\cos {\frac {22\pi }{42}}+2\cos {\frac {26\pi }{42}}+2\cos {\frac {38\pi }{42}}={\frac {-1-{\sqrt {21}}}{2}}\\\end{aligned}}} α , β {\displaystyle \alpha ,\beta } は以下の関係式より求められる。 α + β = − 1 ( α − β ) 2 = 21 {\displaystyle {\begin{aligned}&\alpha +\beta =-1\\&(\alpha -\beta )^{2}=21\\\end{aligned}}} 三次方程式の係数を求めると 2 cos 2 π 42 ⋅ 2 cos 10 π 42 + 2 cos 10 π 42 ⋅ 2 cos 34 π 42 + 2 cos 34 π 42 ⋅ 2 cos 2 π 42 = − α − 1 2 cos 2 π 42 ⋅ 2 cos 10 π 42 ⋅ 2 cos 34 π 42 = β − 2 {\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{42}}\cdot 2\cos {\frac {10\pi }{42}}+2\cos {\frac {10\pi }{42}}\cdot 2\cos {\frac {34\pi }{42}}+2\cos {\frac {34\pi }{42}}\cdot 2\cos {\frac {2\pi }{42}}=-\alpha -1\\&2\cos {\frac {2\pi }{42}}\cdot 2\cos {\frac {10\pi }{42}}\cdot 2\cos {\frac {34\pi }{42}}=\beta -2\\\end{aligned}}} 解と係数の関係より x 3 − α x 2 + ( − α − 1 ) x − ( β − 2 ) = 0 {\displaystyle x^{3}-\alpha x^{2}+(-\alpha -1)x-(\beta -2)=0} 変数変換、関係式より x = y + α / 3 , β = − 1 − α , α 2 = 5 − α , α 3 = 6 α − 5 {\displaystyle x=y+\alpha /3,\quad \beta =-1-\alpha ,\quad \alpha ^{2}=5-\alpha ,\quad \alpha ^{3}=6\alpha -5} 整理すると y 3 − 2 α + 8 3 x + 15 α + 46 27 = 0 {\displaystyle y^{3}-{\frac {2\alpha +8}{3}}x+{\frac {15\alpha +46}{27}}=0} 三角関数、逆三角関数を使用した解は x = α 3 + 2 2 α + 8 3 cos ( 1 3 arccos − ( 15 α + 46 ) 2 ( 2 α + 8 ) 3 2 ) {\displaystyle x={\frac {\alpha }{3}}+{\frac {2{\sqrt {2\alpha +8}}}{3}}\cos \left({\frac {1}{3}}\arccos {\frac {-(15\alpha +46)}{2({2\alpha +8})^{\frac {3}{2}}}}\right)} 平方根と立方根で表すと x = α 3 + 2 α + 8 3 − ( 15 α + 46 ) 2 ( 2 α + 8 ) 3 2 + i 189 ( α + 3 ) 2 ( 2 α + 8 ) 3 2 3 + 2 α + 8 3 − ( 15 α + 46 ) 2 ( 2 α + 8 ) 3 2 − i 189 ( α + 3 ) 2 ( 2 α + 8 ) 3 2 3 x = α 3 + 1 6 − 4 ( 15 α + 46 ) + i ⋅ 4 189 ( α + 3 ) 3 + 1 6 − 4 ( 15 α + 46 ) − i ⋅ 4 189 ( α + 3 ) 3 {\displaystyle {\begin{aligned}&x={\frac {\alpha }{3}}+{\frac {\sqrt {2\alpha +8}}{3}}{\sqrt[{3}]{{\frac {-(15\alpha +46)}{2({2\alpha +8})^{\frac {3}{2}}}}+i{\frac {\sqrt {189(\alpha +3)}}{2({2\alpha +8})^{\frac {3}{2}}}}}}+{\frac {\sqrt {2\alpha +8}}{3}}{\sqrt[{3}]{{\frac {-(15\alpha +46)}{2({2\alpha +8})^{\frac {3}{2}}}}-i{\frac {\sqrt {189(\alpha +3)}}{2({2\alpha +8})^{\frac {3}{2}}}}}}\\&x={\frac {\alpha }{3}}+{\frac {1}{6}}{\sqrt[{3}]{{-4(15\alpha +46)}+i\cdot 4{\sqrt {189(\alpha +3)}}}}+{\frac {1}{6}}{\sqrt[{3}]{-4(15\alpha +46)-i\cdot 4{\sqrt {189(\alpha +3)}}}}\end{aligned}}} αの値((-1+√21)/2)を代入して、整理すると cos 2 π 42 = − 1 + 21 + − 154 − 30 21 + ( 42 3 + 18 7 ) i 3 + − 154 − 30 21 − ( 42 3 + 18 7 ) i 3 12 {\displaystyle \cos {\frac {2\pi }{42}}={\frac {-1+{\sqrt {21}}+{\sqrt[{3}]{-154-30{\sqrt {21}}+\left(42{\sqrt {3}}+18{\sqrt {7}}\right)i}}+{\sqrt[{3}]{-154-30{\sqrt {21}}-\left(42{\sqrt {3}}+18{\sqrt {7}}\right)i}}}{12}}}
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