If $V_1,V_2$ are known, how would I solve this system of equations: $$\begin{cases} V_1 = \dfrac{\sin(x)\sin(y)}{\cos(x)} \\V_2 = \dfrac{\cos(x)}{\sin(x) \cdot \cos(y)} \end{cases}$$ The final result should be in degrees or radians it doesn't matter,
Thank you for your help
Multiplying the two equations gives you $V_1V_2 = \tan y$. This gives you two possible values for the pair $(\sin y, \cos y)$ (corresponding to diametrically opposite points on the unit circle). Rearranging first equation gives you $\frac{V_1}{\sin y} = \tan x$, and from there we can solve for $x$.
For example, suppose $V_1 = \frac12$ and $V_2 = \frac2{\sqrt3}$. Then multiplying the two equations gives us $\tan y = V_1V_2 = \frac1{\sqrt3}$. From this we conclude that $y = \frac{\pi}6 + k\pi$ for $k \in \mathbb Z$. We have $\sin y = \frac12$ or $-\frac12$ depending on whether $k$ is even or odd. Plugging into the first equation gives us that $\tan x$ is either $1$ or $-1$. In the former case, we get $x = \frac{\pi}4 + k\pi$, and in the latter case, we get $x = -\frac{\pi}4 + k\pi$. In summary, our solutions are $(x,y) = \left( \frac{\pi}6+2k\pi, \frac{\pi}4 + \ell\pi \right), \left( \frac{7\pi}6+2k\pi, -\frac{\pi}4 + \ell\pi \right)$.
