# Finding probability distribution of quantity depending on other distributions

Finding probability distribution of quantity depending on other distributions
I have a vector that depends on the coordinates of randomly drawn unit vectors in ${\mathbb{R}}^{2}$:
$\sigma =\left(\begin{array}{c}\frac{\mathrm{cos}\left({\theta }_{a}\right)+\mathrm{cos}\left({\theta }_{b}\right)}{1+\mathrm{cos}\left({\theta }_{a}\right)\mathrm{cos}\left({\theta }_{b}\right)}\\ \frac{\mathrm{sin}\left({\theta }_{a}\right)+\mathrm{sin}\left({\theta }_{b}\right)}{1+\mathrm{sin}\left({\theta }_{a}\right)\mathrm{sin}\left({\theta }_{b}\right)}\end{array}\right)$
Here, ${\theta }_{a},{\theta }_{b}\in \left[0,2\pi \right)$, 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?
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Step 1
We know that the PDF (probability density function) of the random variable $\theta$ is $PDF\left(\theta \right)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{P\left(\theta \in \left(x,x+\mathrm{\Delta }x\right)\right)}{\mathrm{\Delta }x}\to$
$P\left(\theta \in \left(x,x+\mathrm{\Delta }x\right)\right)=PDF\left(\theta \right)\mathrm{\Delta }x$
Step 2
If the random variable $\sigma$ is dependent of $\theta$ by $y=y\left(x\right)$ we must have
$P\left(\theta \in \left(x,x+\mathrm{\Delta }x\right)\right)=P\left(\sigma \in \left(y,y+\mathrm{\Delta }y\right)\right)\to$
$PDF\left(\theta \right)\mathrm{\Delta }x=PDF\left(\sigma \right)\mathrm{\Delta }y\to$
$PDF\left(\sigma \right)=li{m}_{\mathrm{\Delta }x\to 0}\frac{\mathrm{\Delta }x}{\mathrm{\Delta }y}PDF\left(\theta \right)=\frac{dx}{dy}PDF\left(\theta \right)$