I have a system of 9 coupled trigonometric equations (a 3X3 Matrix). I am trying to find four unknown angles $\theta_1, \theta_2, \phi_1, \phi_2$ by solving any 4 out of 9 such equations. $\text{M}=\left( \begin{array}{ccc} \text{s1xs2x} \cos (\text{$\theta $1}) \cos (\text{$\theta $2}) \cos (\text{$\phi $1}) \cos (\text{$\phi $2})+\text{s1ys2y} \cos (\text{$\theta $1}) \cos (\text{$\theta $2}) \sin (\text{$\phi $1}) \sin (\text{$\phi $2})+\text{s1zs2z} \sin (\text{$\theta $1}) \sin (\text{$\theta $2}) & \text{s1ys2y} \cos (\text{$\theta $1}) \sin (\text{$\phi $1}) \cos (\text{$\phi $2})-\text{s1xs2x} \cos (\text{$\theta $1}) \cos (\text{$\phi $1}) \sin (\text{$\phi $2}) & \text{s1xs2x} \cos (\text{$\theta $1}) \sin (\text{$\theta $2}) \cos (\text{$\phi $1}) \cos (\text{$\phi $2})+\text{s1ys2y} \cos (\text{$\theta $1}) \sin (\text{$\theta $2}) \sin (\text{$\phi $1}) \sin (\text{$\phi $2})-\text{s1zs2z} \sin (\text{$\theta $1}) \cos (\text{$\theta $2}) \\ \text{s1ys2y} \cos (\text{$\theta $2}) \cos (\text{$\phi $1}) \sin (\text{$\phi $2})-\text{s1xs2x} \cos (\text{$\theta $2}) \sin (\text{$\phi $1}) \cos (\text{$\phi $2}) & \text{s1xs2x} \sin (\text{$\phi $1}) \sin (\text{$\phi $2})+\text{s1ys2y} \cos (\text{$\phi $1}) \cos (\text{$\phi $2}) & \text{s1ys2y} \sin (\text{$\theta $2}) \cos (\text{$\phi $1}) \sin (\text{$\phi $2})-\text{s1xs2x} \sin (\text{$\theta $2}) \sin (\text{$\phi $1}) \cos (\text{$\phi $2}) \\ \text{s1xs2x} \sin (\text{$\theta $1}) \cos (\text{$\theta $2}) \cos (\text{$\phi $1}) \cos (\text{$\phi $2})+\text{s1ys2y} \sin (\text{$\theta $1}) \cos (\text{$\theta $2}) \sin (\text{$\phi $1}) \sin (\text{$\phi $2})-\text{s1zs2z} \cos (\text{$\theta $1}) \sin (\text{$\theta $2}) & \text{s1ys2y} \sin (\text{$\theta $1}) \sin (\text{$\phi $1}) \cos (\text{$\phi $2})-\text{s1xs2x} \sin (\text{$\theta $1}) \cos (\text{$\phi $1}) \sin (\text{$\phi $2}) & \text{s1xs2x} \sin (\text{$\theta $1}) \sin (\text{$\theta $2}) \cos (\text{$\phi $1}) \cos (\text{$\phi $2})+\text{s1ys2y} \sin (\text{$\theta $1}) \sin (\text{$\theta $2}) \sin (\text{$\phi $1}) \sin (\text{$\phi $2})+\text{s1zs2z} \cos (\text{$\theta $1}) \cos (\text{$\theta $2}) \\ \end{array} \right)$
I have the numerical value of the matrix elements $M=\left( \begin{array}{ccc} 144.57 & -2.21141 & 149.231 \\ -10.5118 & 8.78583 & 10.3137 \\ 28.3015 & 14.5596 & 16.4425 \\ \end{array} \right)$
Also, s1xs2x = 17.1482, s1ys2y = -17.1482, s1zs2z = 210.153
My code looks like this:
M = {{144.57, -2.21141, 149.231}, {-10.5118, 8.78583, 10.3137},
{28.3015, 14.5596, 16.4425}}
s1xs2x = 17.1482, s1ys2y = -17.1482, s1zs2z = 210.153
Sol = N@Reduce[{s1xs2x*Cos[Theta1]*Cos[Theta2]*Cos[Phi1]*Cos[Phi2] + s1zs2z*Sin[Theta1]*Sin[Theta2] + s1ys2y*Cos[Theta1]*Cos[Theta2]*Sin[Phi1]*Sin[Phi2] == M[[1,1]],
s1ys2y*Cos[Theta1]*Cos[Phi2]*Sin[Phi1] - s1xs2x*Cos[Theta1]*Cos[Phi1]*Sin[Phi2] == M[[1,2]],
(-s1zs2z)*Cos[Theta2]*Sin[Theta1] + s1xs2x*Cos[Theta1]*Cos[Phi1]*Cos[Phi2]*Sin[Theta2] + s1ys2y*Cos[Theta1]*Sin[Theta2]*Sin[Phi1]*Sin[Phi2] == M[[1,3]],
s1ys2y*Cos[Phi1]*Cos[Phi2] + s1xs2x*Sin[Phi1]*Sin[Phi2] == M[[2,2]], 0 < Theta1 < Pi, 0 <Theta2 < Pi, 0 < Phi2 < Pi, 0 < Phi2 < Pi},{Theta1, Theta2, Phi1, Phi2}]
HoweverThe set of values of the angles I should obtain are: Theta1 = 1.25869, Phi1 = 2.94731, Theta2 = 0.770075, Phi2 = 1.84307 However, it keeps on running forever. I am not sure where I'm going wrong. Any help shall be highly appreciated.
