九十角形
cos  2 π 90 = 1 2 ( cos  12 ∘ + i sin  12 ∘ 3 + cos  12 ∘ − i sin  12 ∘ 3 ) {\displaystyle \cos {\frac {2\pi }{90}}={\frac {1}{2}}\left({\sqrt[{3}]{\cos {12^{\circ }}+i\sin {12^{\circ }}}}+{\sqrt[{3}]{\cos {12^{\circ }}-i\sin {12^{\circ }}}}\right)} = 1 4 { [ 6 ( 5 + 5 ) + 5 − 1 ] + i [ 2 ( 5 + 5 ) − 3 ( 5 − 1 ) ] 3 {\displaystyle ={\frac {1}{4}}{\Bigg \{}{\sqrt[{3}]{\left[{\sqrt {6(5+{\sqrt {5}})}}+{\sqrt {5}}-1\right]+i\left[{\sqrt {2(5+{\sqrt {5}})}}-{\sqrt {3}}({\sqrt {5}}-1)\right]}}} + [ 6 ( 5 + 5 ) + 5 − 1 ] − i [ 2 ( 5 + 5 ) − 3 ( 5 − 1 ) ] 3 } {\displaystyle +{\sqrt[{3}]{\left[{\sqrt {6(5+{\sqrt {5}})}}+{\sqrt {5}}-1\right]-i\left[{\sqrt {2(5+{\sqrt {5}})}}-{\sqrt {3}}({\sqrt {5}}-1)\right]}}{\Bigg \}}}