七十二角形
三次方程式を解いて、整理すると cos  ( 2 π / 72 ) {\displaystyle \cos(2\pi /72)} が求められる。
2 cos  2 π 72 = cos  2 π 24 + i sin  2 π 24 3 + cos  2 π 24 − i sin  2 π 24 3 4 cos  2 π 72 = 8 cos  2 π 24 + i 8 sin  2 π 24 3 + 8 cos  2 π 24 − i 8 sin  2 π 24 3 4 cos  2 π 72 = 2 ( 6 + 2 ) + i ⋅ 2 ( 6 − 2 ) 3 + 2 ( 6 + 2 ) − i ⋅ 2 ( 6 − 2 ) 3 {\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{72}}=&{\sqrt[{3}]{\cos {\frac {2\pi }{24}}+i\sin {\frac {2\pi }{24}}}}+{\sqrt[{3}]{\cos {\frac {2\pi }{24}}-i\sin {\frac {2\pi }{24}}}}\\4\cos {\frac {2\pi }{72}}=&{\sqrt[{3}]{8\cos {\frac {2\pi }{24}}+i8\sin {\frac {2\pi }{24}}}}+{\sqrt[{3}]{8\cos {\frac {2\pi }{24}}-i8\sin {\frac {2\pi }{24}}}}\\4\cos {\frac {2\pi }{72}}=&{\sqrt[{3}]{2({\sqrt {6}}+{\sqrt {2}})+i\cdot 2({\sqrt {6}}-{\sqrt {2}})}}+{\sqrt[{3}]{2({\sqrt {6}}+{\sqrt {2}})-i\cdot 2({\sqrt {6}}-{\sqrt {2}})}}\\\end{aligned}}}
三次方程式の係数を求めると
2 cos  2 π 72 ⋅ 2 cos  50 π 72 + 2 cos  50 π 72 ⋅ 2 cos  46 π 72 + 2 cos  46 π 72 ⋅ 2 cos  2 π 72 = − 3 2 cos  2 π 72 ⋅ 2 cos  50 π 72 ⋅ 2 cos  46 π 72 = 2 cos  2 π 24 {\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{72}}\cdot 2\cos {\frac {50\pi }{72}}+2\cos {\frac {50\pi }{72}}\cdot 2\cos {\frac {46\pi }{72}}+2\cos {\frac {46\pi }{72}}\cdot 2\cos {\frac {2\pi }{72}}=-3\\&2\cos {\frac {2\pi }{72}}\cdot 2\cos {\frac {50\pi }{72}}\cdot 2\cos {\frac {46\pi }{72}}=2\cos {\frac {2\pi }{24}}\end{aligned}}}
解と係数の関係より
u 3 − 3 u − 2 cos  2 π 24 = 0 {\displaystyle u^{3}-3u-2\cos {\frac {2\pi }{24}}=0}