四十五角形
2 cos  2 π 45 ⋅ 2 cos  32 π 45 + 2 cos  32 π 45 ⋅ 2 cos  28 π 45 + 2 cos  28 π 45 ⋅ 2 cos  2 π 45 = − 3 2 cos  2 π 45 ⋅ 2 cos  32 π 45 ⋅ 2 cos  28 π 45 = 2 cos  2 π 15 = 1 + 5 + 30 − 6 5 4 {\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{45}}\cdot 2\cos {\frac {32\pi }{45}}+2\cos {\frac {32\pi }{45}}\cdot 2\cos {\frac {28\pi }{45}}+2\cos {\frac {28\pi }{45}}\cdot 2\cos {\frac {2\pi }{45}}=-3\\&2\cos {\frac {2\pi }{45}}\cdot 2\cos {\frac {32\pi }{45}}\cdot 2\cos {\frac {28\pi }{45}}=2\cos {\frac {2\pi }{15}}={\frac {1+{\sqrt {5}}+{\sqrt {30-6{\sqrt {5}}}}}{4}}\end{aligned}}}
解と係数の関係より
u 3 − 3 u − 2 cos  2 π 15 = 0 {\displaystyle u^{3}-3u-2\cos {\frac {2\pi }{15}}=0}
三次方程式を解いて、整理すると cos  ( 2 π / 45 ) {\displaystyle \cos(2\pi /45)} が求められる。
2 cos  2 π 45 + 2 cos  32 π 45 + 2 cos  28 π 45 = 0 2 cos  4 π 45 + 2 cos  26 π 45 + 2 cos  34 π 45 = 0 2 cos  8 π 45 + 2 cos  38 π 45 + 2 cos  22 π 45 = 0 2 cos  16 π 45 + 2 cos  14 π 45 + 2 cos  44 π 45 = 0 {\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{45}}+2\cos {\frac {32\pi }{45}}+2\cos {\frac {28\pi }{45}}=0\\2\cos {\frac {4\pi }{45}}+2\cos {\frac {26\pi }{45}}+2\cos {\frac {34\pi }{45}}=0\\2\cos {\frac {8\pi }{45}}+2\cos {\frac {38\pi }{45}}+2\cos {\frac {22\pi }{45}}=0\\2\cos {\frac {16\pi }{45}}+2\cos {\frac {14\pi }{45}}+2\cos {\frac {44\pi }{45}}=0\\\end{aligned}}}
三次方程式の係数を求めると