I have a vector that depends on the coordinates of randomly drawn unit vectors in $\mathbb{R}^2$:
$\sigma =\begin{pmatrix}\frac{\cos(\theta_a)+\cos(\theta_b)}{1+\cos(\theta_a)\cos(\theta_b)}\\ \frac{\sin(\theta_a)+\sin(\theta_b)}{1+\sin(\theta_a)\sin(\theta_b)}\end{pmatrix} $.
Here, $\theta_a,\theta_b \in [0,2\pi)$, are drawn uniformly random from the unit circle, and are thus angles that parametrize unit vectors in $\mathbb{R}^2$. I am interested in figuring out a probability distribution for the coordinates of the $\sigma$-vector, but I am not well-versed in probability theory. I have been told that it is possible to somehow find a probability distribution for a quantity that depends on other distributions by somehow using the Jacobian matrix, but I have been unable to figure out how on my own.
My questions are thus:
Is there a method for finding a probability distribution for a quantity that depends on other distributions? If so, how? Does it have a name so I can learn about form other sources?
If the method exists, how would I use it on my concrete example?
The following link shows a picture of 10000 plotted $\sigma$ made from randomly drawn unit vectors. There is an interesting pattern which I would love to explain, and I assume a probability distribution would be the key!
We know that the PDF (probability density function) of the random variable $\theta$ is $$PDF(\theta)=\lim_{\Delta x\rightarrow0}\frac{P(\theta\in(x, x+\Delta x))}{\Delta x}\rightarrow$$ $$P(\theta\in(x, x+\Delta x))=PDF(\theta)\Delta x$$ If the random variable $\sigma$ is dependent of $\theta$ by $y=y(x)$ we must have
$$P(\theta\in(x, x+\Delta x))=P(\sigma\in(y, y+\Delta y))\rightarrow$$ $$PDF(\theta)\Delta x=PDF(\sigma)\Delta y\rightarrow$$ $$PDF(\sigma)=lim_{\Delta x\rightarrow0}\frac{\Delta x}{\Delta y}PDF(\theta)=\frac{dx}{dy}PDF(\theta)$$