PDF and CDF of $x$ and $y$ coordinates of a random vector.

Question

I have a random vector where the magnitude ($M$) and angle ($\Theta$) are independent and normally distributed

\begin{align} M &\sim N(\mu_1, \sigma_1^2) \\ \Theta &\sim N(\mu_2, \sigma_2^2) \end{align}

The Cartesian $x$ and $y$ coordinates for the vector are therefore also random variables:

\begin{align} X &= M\cos \Theta \\ Y &= M\sin \Theta \end{align}

How can I derive PDF and CDF of $X$ and $Y$?

The aim is to be able to work out the probability of a vector falling within certain $x$ and $y$ bounds.

$$P((x_1<X<x_2) \cap (y_1<Y<y_2))$$

Edit: as pointed out in the comments, $M$ cannot be negative, so should be modelled either as a $\chi_1$ distribution or a half-normal distribution.