Recently, I've asked a physics problem on physics.se . nevertheless, after stripping the physical basis that lead the equations, the problem can also be stated as a pure mathematical one: a set of nonlinear algebraic-trigonometric equations. So I will present them here for asking a mathematical solution. You can get insight into the variables and parameters from the original question's link.
Equations below are obtained by invoking statics of rigid bodies on a beam.
The UNKNOWNS are the followings: $T_{1f},T_{2f},\alpha,\theta,\phi$, where $T_{1f} , T_{2f}$ are the final stresses on the nylon and steel ropes, and the angles $\alpha,\theta,\phi$ are as stated in the figure, where $\theta$ and $\phi$ are the angles that the forces $T_{1f}$ and $T_{2f}$ make with the beam, and $\alpha$ is the angle that the beam makes with the horizontal.
The known parameters are the $W$ as the weight of the beam, initial beam length $L_b = 2$, and rope lengths $L_n=L_s=3$, as stated in the initial loading configuration.
$$\begin{align} F_x &= 0:\quad& T_{1f} \cos(\theta+\alpha) - T_{2f} \cos(\phi-\alpha) &= 0 \tag1\\ F_y &= 0:\quad& T_{1f} \sin(\theta+\alpha) + \,T_{2f} \sin(\phi-\alpha) &= W \tag2\\ \tau_1 &= 0:\quad& T_{2f} \sin(\theta) &= 0.5 W \cos(\alpha) \tag3 \\ \tau_2 &= 0:\quad& T_{1f} \sin(\phi) &= 0.5 W \cos(\alpha) \tag4 \end{align}$$
and a fifth equation comes from the final triangle geometry (cosine law), and elasticity of materials.
$$L_b^2 = L_{nf}^2 + L_{sf}^2 + 2 L_{nf} L_{sf} \cos(\theta+\phi) \tag5$$
where the final lengths of nylon and steel ropes are found from their final equilibrium stresses, $T_{1f}$ and $T_{2f}$, based on Young's elasticity as:
$$L_{nf} = L_n \left(1 + \frac{T_{1f}}{Y_n A_n} \right) \qquad\text{and}\qquad L_{sf} = L_s \left(1 + \frac{T_{2f}}{Y_s A_s} \right) $$
where cross-sectional areas of the ropes are $A_n = \pi R_n^2$ and $A_s = \pi R_s^2$, as given in the problem statement...
Any help in solving these five nonlinear equations for those five unknowns? Is there any theoretical tool that would provide insight into whether a solution can exist and/or can be reached if so?