How many equations required to solve (find $\theta$ , $\phi$) system of equations have form: $$v = \exp(j \cdot 2\pi \cdot (a \cdot \cos(\phi) \cdot \sin(\theta) + b \cdot \sin(\phi)))$$ given a, and b are integer numbers, and v is a complex number?
I have tried to use two equations, this is my solution: $$\sin(\phi) = \frac{a_1 \cdot \frac{\angle(v_2)+(2\pi m)}{2\pi} - a_2 \cdot \frac{\angle(v_1)+(2 \pi n)}{2\pi}}{a_1 \cdot b_2 - a_2 \cdot b_1} $$ where $\angle(v_1)$ is the phase of $v_1$, $$ \sin(\theta) = \frac{ \frac{\angle(v_1)+(2 \pi n)}{2\pi} - b_1 \cdot \sin(\phi)}{a_1 \cdot \cos(\phi)}, $$ so we have soulations from $m_1$ to $\frac{a_2+b_2}{\sqrt2}$, $n_1$ to $\frac{a_1+b_1}{\sqrt2}$, where $m_1$, and $n_2$ is minimum integer number to make solution exists.
Is there a way to reduce the solution range with an increasing number of equations?