= Central angle between independent uniform sphere points
{title2=$f_\Theta(\theta)=\tfrac12\sin\theta,\quad 0<\theta<\pi$}
For two independent uniform points on the surface of a <sphere>, rotational invariance permits fixing the first at the north pole. The <surface area> of a latitude band of angular width $d\theta$ is $2\pi R^2\sin\theta\,d\theta$, yielding the displayed density for their central angle. Equivalently their unit position <vectors> have a <dot product> uniformly distributed on $[-1,1]$. The law is independent of sphere radius.
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