Suppose that we have an under-determined linear system subject to a linear constraint as:
\begin{align} & A(\theta)x = b\\ s.t. \quad & Cx > 0, \end{align}
where $x \in \mathbb{R}^{n}$, $b \in \mathbb{R}^{m}$ is a known constant vector, $C \in \mathbb{R}^{k\times{n}}$ is a known constant matrix, and $A(\theta)$ is a $\mathbb{R}^{m\times{n}}$ matrix ($m < n$) which is only dependent on a real variable $\theta \in \mathbb{R}$. My goal: find all feasible sets (or necessary and sufficient conditions) of the variable $\theta$ which guarantees the existence of the solution $x$ for the above constrained linear system.
For example: \begin{align} &x_{1} + x_{2} + {x_{3}}\sin{\theta} = 5 \\ &\frac{1}{3}x_{1} + \frac{1}{2}x_{2} - \frac{1}{7}{x_{3}}\cos{\theta} = 6, \\ s.t. \quad & x_{1} > 0, x_{2} > 0, x_{3} > 0, x_{2} > \frac{1}{2}x_{3}. \end{align}
The problem is a trivial task if we convert it to a linear programming configuration and start a numerical computation. However, I would like to ask that is there any method/theorem which enables us to analytically find all feasible sets (or necessary and sufficient conditions) of $\theta$ to ensure existence of the solution?
Many thanks to any suggestion!