Expressing $\phi$ and $\theta$ in terms of time difference of arrival

I will assume that the distance to the sender is much larger than the distance between any two receivers. Note that it is not possible to locate the sender exactly with only three receivers, but it is possible to estimate the direction to the sender. (In the planar example from the link there are two sensors at two different locations, so four sensors in total.) Let $v_1, v_2, v_3 \in \mathbb R^3$ be the locations of the receivers and the unit vector $n \in S^2$ the (approximate) direction to the sender. Then for two receivers $v_i, v_j$ the angle $\alpha$ between $n$ and $v_j - v_i$ satisfies $$\frac{\langle n, v_j-v_i \rangle}{\lVert v_j-v_i \rVert} = \cos(\alpha) = \frac{v \, (t_i - t_j)}{\lVert v_i - v_j \rVert}.$$ This leads to the equations $$\langle n, v_j-v_i \rangle = v \, (t_i - t_j)$$ for all pairs of indices $i, j$. Together with $\lVert n \rVert = 1$ this system has two different solutions in general, symmetric by reflection along a normal vector of the plane through $v_1, v_2, v_3$. Then write $n$ in polar coordinates.