I am computationally trying to solve the system of linear equations given below.
$ \frac{1}{1+tan(\theta/2)*tan(\phi/2)*tan(\psi/2)} \begin{bmatrix} tan(\psi/2)-tan(\theta/2)*tan(\phi/2) \\ tan(\phi/2)+tan(\theta/2)*tan(\psi/2) \\ tan(\theta/2)-tan(\phi/2)*tan(\psi/2) \end{bmatrix} = \begin{bmatrix} r_1 \\ r_2 \\ r_3 \end{bmatrix} $
I use the least square approximations to find the solution to this equation on MATLAB. However, no effective solution was found using this method. Considering that the system values for r1, r2, r3 and $\psi$ = 0 are known whereas $\theta$ and $\phi$ are the two unknown values. Is it possible to obtain exact solution for these two unknowns of the system for $r = [0,-0.101233861621737,0.365119069777688]$.
The values I obtain on solving this system is $\theta = 0.6950 $ and $\phi = -0.1785$, Even-though these are good approximate values, I was wondering if it's possible to obtain the exact values using a different approach.