What is the distribution of Cartesian coordinate x, y, z if the spherical coordinate r, $\theta$ and $\phi$ are all Gaussian distribution?

Question

Assuming that I know the spherical coordinate r, $\theta$, $\phi$ are all Gaussian distributions, what will be the distribution of the x,y,z in Cartesian coordinate if I transform r, $\theta$, $\phi$ to x,y,z?

I tried the code, and get the results for x, y,z. It looks that x and y are normal distributions, but z is an unknown distribution. Here is the Matlab code.

r = randn(1,10000) + 500; % generate random r with mean of 500, and variance of 1.  
phi = randn(1,10000)*10/180*pi + 45/180*pi; % generate random phi with mean of 45 degree and variance of 10 degree. 
theta = randn(1,10000)*10/180*pi + 60/180*pi; % generate random theta with mean of 60 degree and variance of 10 degree.

% convert to x,y,z; 
x = r .* cos(phi) .*cos(theta);
y = r .* sin(phi) .*cos(theta);
z = r .* sin(theta);

% plot the histogram of x,y,z;
figure; histogram(x); title('Hist of x');
figure; histogram(y); title('Hist of y');
figure; histogram(z); title('Hist of z');