Solving a system of coupled trigonometric equations to find angles

Question

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}]

The 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.

Answer 1

FindInstance (nor NSolve) returns a solution in Reals. Changed one of "Phi2" to "Phi1" also in the inequalities. Also should not start user-defined variables with upper case letter as the name may conflict with built-in names but I left them below for now:

s1xs2x = 17.1482;
s1ys2y = -17.1482;
s1zs2z = 210.153;
m11 = 144.57;
m12 = -2.21141;
m13 = 149.231;
m22 = 8.78583;
FindInstance[{s1xs2x*Cos[Theta1]*Cos[Theta2]*Cos[Phi1]*Cos[Phi2] + 
    s1zs2z*Sin[Theta1]*Sin[Theta2] + 
    s1ys2y*Cos[Theta1]*Cos[Theta2]*Sin[Phi1]*Sin[Phi2] == m11, 
  s1ys2y*Cos[Theta1]*Cos[Phi2]*Sin[Phi1] - 
    s1xs2x*Cos[Theta1]*Cos[Phi1]*Sin[Phi2] ==m12,
  (-s1zs2z)*Cos[Theta2]*Sin[Theta1] + 
    s1xs2x*Cos[Theta1]*Cos[Phi1]*Cos[Phi2]*Sin[Theta2] + 
    s1ys2y*Cos[Theta1]*Sin[Theta2]*Sin[Phi1]*Sin[Phi2] == m13, 
  s1ys2y*Cos[Phi1]*Cos[Phi2] + s1xs2x*Sin[Phi1]*Sin[Phi2] == m22
  && 0 < Theta1 < Pi
  && 0 < Theta2 < Pi 
  && 0 < Phi1 < Pi 
  && 0 < Phi2 < Pi},
  {Theta1, Theta2, Phi1, Phi2}, Reals]

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