Covariance of coordinates at eight equally spaced angles

An angle $\theta$ is chosen uniformly from the eight values $0,\tfrac{\pi}{4},\tfrac{\pi}{2},\dots,\tfrac{7\pi}{4}$, and we set $X=\cos\theta$, $Y=\sin\theta$. Find $\operatorname{Cov}(X,Y)$.

$0$. By symmetry $\mathbb{E}[X]=\mathbb{E}[Y]=0$, and $\mathbb{E}[XY]=\tfrac12\mathbb{E}[\sin2\theta]=0$ because $\sin 2\theta$ sums to zero over these points. $X$ and $Y$ are uncorrelated yet clearly dependent, since $X^2+Y^2=1$.