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Recall that the $12$th roots of unity are precisely the complex numbers $z$ satisfying
\[z^{12} = 1.\]Consider the complex numbers $s, t, u, v, w$ in the complex plane below:
[asy] size(250);import TrigMacros;real big = 4;for (int i = 1; i < big+1; ++i) { draw(Circle((0,0),i), gray+ linewidth(0.4));} for(int i=0;i<360;i+=15) { draw(rotate(i)*((-big,0)--(big,0)),gray+ linewidth(0.4));} rr_cartesian_axes(-big,big,-big,big,complexplane=true);pair SS, T, U, V, WW;SS = dir(15);T = 2*dir(45);U = dir(90);V = dir(120);WW = 2*dir(240);dot(Some of these complex numbers are $12$th roots of unity. Enter those complex numbers in any order, separated by commas.

 Mar 7, 2021
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The 12th roots of unity are s, u, v.

 Sep 13, 2021

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