四十二角形
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}