二十八角形
α = 2 cos  2 π 28 + 2 cos  6 π 28 + 2 cos  18 π 28 = 7 β = 2 cos  10 π 28 + 2 cos  26 π 28 + 2 cos  22 π 28 = − 7 {\displaystyle {\begin{aligned}&\alpha =2\cos {\frac {2\pi }{28}}+2\cos {\frac {6\pi }{28}}+2\cos {\frac {18\pi }{28}}={\sqrt {7}}\\&\beta =2\cos {\frac {10\pi }{28}}+2\cos {\frac {26\pi }{28}}+2\cos {\frac {22\pi }{28}}=-{\sqrt {7}}\\\end{aligned}}}
三次方程式の係数を求めると
2 cos  2 π 28 ⋅ 2 cos  6 π 28 + 2 cos  6 π 28 ⋅ 2 cos  18 π 28 + 2 cos  18 π 28 ⋅ 2 cos  2 π 28 = 0 2 cos  2 π 28 ⋅ 2 cos  6 π 28 ⋅ 2 cos  18 π 28 = − 7 {\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{28}}\cdot 2\cos {\frac {6\pi }{28}}+2\cos {\frac {6\pi }{28}}\cdot 2\cos {\frac {18\pi }{28}}+2\cos {\frac {18\pi }{28}}\cdot 2\cos {\frac {2\pi }{28}}=0\\&2\cos {\frac {2\pi }{28}}\cdot 2\cos {\frac {6\pi }{28}}\cdot 2\cos {\frac {18\pi }{28}}=-{\sqrt {7}}\\\end{aligned}}}