I have the general parametric equation of an ellipse.
$$\begin{align*}x&=c_x+a\cos{t}\cos{\alpha}-b\sin{t}\sin{\alpha} \\ y&=c_y+a\cos{t}\sin{\alpha}+b\sin{t}\cos{\alpha}\end{align*}$$
I need to find the value of $t$ when $x$ and $y$ are known. I tried solving the equation and ended up with
$$t = \arccos{\frac{(x-c_x)\cos{\alpha} + (y-c_y)\sin{\alpha}}{a}}$$
The first thing that bothers me about this equation is that it doesn't depend on $b$. The major issue is, it seems that the value of $t$ I get is more and more inaccurate with increasing $\alpha$.
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You are almost there. Eliminating the $b$-terms (by multiplying the first of the given equations with $\cos\alpha$, the second with $\sin\alpha$) you got the first of the following two equations; eliminating in the same way the $a$-terms you get the second: $$\eqalign{ (x-c_x)\cos\alpha +(y-c_y)\sin\alpha&=a\cos t\ , \cr -(x-c_x)\sin\alpha+(y-c_y)\cos\alpha&=b\sin t\ .\cr}$$ As $\ t=\arg(\cos t,\sin t)$ (mod $2\pi$) we obtain $$t\ =\ \arg\Bigl({(x-c_x)\cos\alpha +(y-c_y)\sin\alpha\over a}\ ,\ {-(x-c_x)\sin\alpha+(y-c_y)\cos\alpha \over b}\Bigr)\ .$$