四十一角形
2 cos  2 π 41 = x 1 + x 1 2 − 4 ( x 2 + x 3 ) 2 + ( x 1 + x 1 2 − 4 ( x 2 + x 3 ) 2 ) 2 − 4 ( x 4 + x 4 2 − 4 ( x 5 + x 1 ) 2 ) 2 cos  2 π 41 = 1 8 ( x 1 + x 1 2 − 4 ( x 2 + x 3 ) + ( x 1 + x 1 2 − 4 ( x 2 + x 3 ) ) 2 − 8 ( x 4 + x 4 2 − 4 ( x 5 + x 1 ) ) ) cos  2 π 41 = 1 8 ( x 1 + 8 − x 2 − 2 x 3 + 2 x 4 + ( x 1 + 8 − x 2 − 2 x 3 + 2 x 4 ) 2 − 8 ( x 4 + 8 − x 5 − 2 x 1 + 2 x 2 ) ) cos  2 π 41 = 1 8 ( x 1 + 8 − x 2 − 2 x 3 + 2 x 4 + 16 + 2 x 2 + 4 x 4 + 2 x 1 ⋅ 8 − x 2 − 2 x 3 + 2 x 4 − 8 ( x 4 + 8 − x 5 − 2 x 1 + 2 x 2 ) ) cos  2 π 41 = 1 8 ( x 1 + 8 − x 2 − 2 x 3 + 2 x 4 + 16 + 2 x 2 − 4 x 4 + 2 46 − 6 x 1 + 14 x 2 + 5 x 3 + 12 x 4 − 8 9 − x 1 + 3 x 2 + x 3 + x 4 ) {\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{41}}={\frac {{\frac {x_{1}+{\sqrt {x_{1}^{2}-4(x_{2}+x_{3})}}}{2}}+{\sqrt {\left({\frac {x_{1}+{\sqrt {x_{1}^{2}-4(x_{2}+x_{3})}}}{2}}\right)^{2}-4\left({\frac {x_{4}+{\sqrt {x_{4}^{2}-4(x_{5}+x_{1})}}}{2}}\right)}}}{2}}\\&\cos {\frac {2\pi }{41}}={\frac {1}{8}}\left({x_{1}+{\sqrt {x_{1}^{2}-4(x_{2}+x_{3})}}}+{\sqrt {\left({x_{1}+{\sqrt {x_{1}^{2}-4(x_{2}+x_{3})}}}\right)^{2}-8\left({x_{4}+{\sqrt {x_{4}^{2}-4(x_{5}+x_{1})}}}\right)}}\right)\\&\cos {\frac {2\pi }{41}}={\frac {1}{8}}\left({x_{1}+{\sqrt {8-x_{2}-2x_{3}+2x_{4}}}}+{\sqrt {\left({x_{1}+{\sqrt {8-x_{2}-2x_{3}+2x_{4}}}}\right)^{2}-8\left({x_{4}+{\sqrt {8-x_{5}-2x_{1}+2x_{2}}}}\right)}}\right)\\&\cos {\frac {2\pi }{41}}={\frac {1}{8}}\left({x_{1}+{\sqrt {8-x_{2}-2x_{3}+2x_{4}}}}+{\sqrt {16+2x_{2}+4x_{4}+{2x_{1}\cdot {\sqrt {8-x_{2}-2x_{3}+2x_{4}}}}-8\left({x_{4}+{\sqrt {8-x_{5}-2x_{1}+2x_{2}}}}\right)}}\right)\\&\cos {\frac {2\pi }{41}}={\frac {1}{8}}\left({x_{1}+{\sqrt {8-x_{2}-2x_{3}+2x_{4}}}}+{\sqrt {16+2x_{2}-4x_{4}+{2{\sqrt {46-6x_{1}+14x_{2}+5x_{3}+12x_{4}}}}-8{\sqrt {9-x_{1}+3x_{2}+x_{3}+x_{4}}}}}\right)\\\end{aligned}}}