I have the following trigonometric equation in $\theta$:
$0=G_{\omega}(1/r^2)({\csc^2}\theta){(r\cos\theta-x)}^2+(\cot\theta)(r\cos\theta-x)+r\sin\theta-y.$
Is there an analytical solution for $\theta$ to this equation for known values of $G_w \in \mathbb{R}$, $x \in \mathbb{R}$, and $y \in \mathbb{R}$. By the way: $G_w = -(g/2){\omega}^{-2}$. The above equation describes the parabolic path of the free fall trajectory of a particle in $(x,y)$. It has the following initial position and velocity components: $x_0=r\omega\cos\theta$, $y_0=r\omega\sin\theta$, $\dot{x}_0 = -r\omega\sin\theta$, and $\dot{y}_0=rw\cos\theta$ (where $\theta=\omega t$; the particle is being thrown of the inner wall of a cylinder with angular velocity $\omega$ and radius $r$).
Numerical methods can be used, and have been used. But I was looking for an analytical solution.
Thanks.