Joint Density of $\sin(2\pi U)$ and $\cos(2\pi U)$,$ U$ uniform $(0,1)$

Question

I need help with finding the joint density of $X$ and $Y$ where $$X=\sin(2\pi U),\quad Y=\cos(2\pi U)$$ where $U$ is uniformly distributed on (0,1).

Answer 1

Since $(X,Y)$ is almost surely on the unit circle whose Lebesgue measure is zero, the distribution of $(X,Y)$ has no density with respect to the Lebesgue measure on the plane $\mathbb R^2$.

To describe the distribution of $(X,Y)$ one usually says that $(X,Y)$ is uniformly distributed on the unit circle, which is another way of saying that, for every measurable function $a$ on $\mathbb R^2$, $$ E(a(X,Y))=\int_0^{2\pi}a(\cos t,\sin t)\frac{\mathrm dt}{2\pi}. $$