If
$\Theta\sim U[0,2\pi]$
$R\sim U[0,0.1]$
$X=R\cdot cos(\Theta)$ and $Y=R\cdot sin(\Theta)$
How can I calculate $cov(X,Y)$ and check the independence between X and Y?
I succeed to find the density function of X and Y in form of; f(R,P) and f(R,Q) [P=sin($\Theta$) and Q=cos($\Theta$)]. But it's seems impossible to caculate the cov(X,Y) without the density function f(X,Y). There is another way?
Edit: I'm sorry that I forget to mention that R and $\theta$ are independent. and another correction $R\sim U[0,0.1]$
Update: I found the density functions of P=cos($\theta$) and Q=sin($\theta$). $f_P(p)=\frac{1}{\pi\sqrt{1-p^2}}\mathbb{1}_{[-1;1]}$(p). $f_Q(q)=\frac{1}{\pi\sqrt{1-q^2}}\mathbb{1}_{[-1;1]}$(q). so that mean the density for X=Rcos($\theta$) and Y=Rsin($\theta$). $f_{RP}{(r,p)}=\frac{10}{\pi\sqrt{1-p^2}}\mathbb{1}_{[-1;1]}(p)\mathbb{1}_{[0;0.1]}(r)$. $f_{RQ}{(r,q)}=\frac{10}{\pi\sqrt{1-p^2}}\mathbb{1}_{[-1;1]}(p)\mathbb{1}_{[0;0.1]}(r)$.
but it's not clear to me how I transfer this only to 2 variables X and Y. at mean $f_{RP}{(r,p)}=f_X(x)$ and $f_{RQ}{(r,q)}=f_Y(y)$
and of course, I need the join density function $f_{XY}{(x,y)}$ for check the cov(x,y). (as @tommik mention above but I don't understand how to get it). Did I do everything right? and if I get it right, in the end, the results are: X and Y are dependent and uncorrelated?
The expectations of $X$ and $Y$ are zero by antisymmetry, and again by antisymmetry the product $(X-\bar X)(Y-\bar Y)$ has a zero expectation (it is positive in two quadrants and negative in the other two).