Take independent Gaussians $\mathbf{g_1}, \mathbf{g_2} \thicksim N(0,1)$ and define correlated Gaussians $z_1 = \mathbf{g_1} + \mu_1$ and $z_2 = \rho_1 \mathbf{g_1} + \rho_2 \mathbf{g_2} + \mu_2$. Since this is an affine linear transform of the form, $$\begin{bmatrix}z_1\\ z_2\end{bmatrix} = \begin{bmatrix}\mu_1\\ \mu_2\end{bmatrix} + \begin{bmatrix}1 & 0\\\rho_1 & \rho_2\end{bmatrix}\begin{bmatrix}g_1\\g_2\end{bmatrix}$$ we can equivalently consider, $$\begin{bmatrix}z_1\\ z_2\end{bmatrix} \thicksim N\left(\begin{bmatrix}\mu_1\\ \mu_2\end{bmatrix}, \begin{bmatrix}1 & 0\\\rho_1 & \rho_2\end{bmatrix} I \begin{bmatrix}1 & 0\\\rho_1 & \rho_2\end{bmatrix}^T\right) = N\left(\begin{bmatrix}\mu_1\\ \mu_2\end{bmatrix}, \begin{bmatrix}1 & \rho_1\\\rho_1 & \rho_1^2 + \rho_2^2\end{bmatrix} \right)$$ Now consider the unit vector, $$(v_1,v_2) = \mathcal{N}(z_1, z_2) := \frac{(z_1,z_2)}{\sqrt{z_1^2 + z_2^2}}$$ I would like to understand the expected value of this vector, as a function of the parameters $\mu_1, \mu_2, \rho_1, \rho_2$. Does this vector follow any well-understood distribution? I was able to compute the 2nd-order moments $\mathbb E[v_1^2]$ and $\mathbb E[v_1v_2]$ by observing that it's a ratio of quadratic forms (see this question), but I was unable to find / come up with a similar notion for the 1st-order moments.
Relatedly, I found this question asking about the ratio of independent Gaussian random variables, which led me to Student's t-distribution. However, this seems to be different in a few crucial ways, like $z_1$ and $z_2$ being correlated Gaussians, rather than independent draw from a normal distribution. Also, if $z_1$ and $z_2$ were just both drawn from $N(0,1)$, this would be equivalent to considering a uniformly random vector from $S^1$ (e.g. here). Perhaps this could be viewed as a transformation of the such a vector $v \in S^1$? I'm thrown off by the normalization.
