I have a non-linear equation $$-\sin(\eta) \sin(\delta) \sin(\theta) - \cos(\eta) \sin(\phi) \cos(\theta) + \sin(\eta) \cos(\delta) \cos(\theta) \cos(\phi)-\cos(\rho)=0$$
Suppose we have four $\delta$, i.e.,$\delta_1$,$\delta_2$,$\delta_3$,$\delta_4$.So that we have four equations to slove the problem, with 3 unknowns,i.e., $\theta$,$\phi$,$\rho$. Let $\eta$ is a known constant (we can use just 3 of the equations). I want to implement this on a hardware, so I can not solve that by Newton-Raphson (because of the matrix inverse in the method). What are the other solutions?