I have a circle of fixed radius $r$. I have a target that is $x$ units laterally separated from the center of the circle, and $y$ units vertically. I need to calculate the angle $θ$ which is the declination of the radius that connects that target to the upper tangent on the circle.
It feels like there should be a simpler formula than what I've shown in the diagram:
$$θ=180° - tan^{-1}\left(\frac{y}{x}\right) - cos^{-1}\left(\frac{r}{\sqrt{x^{2}+y^{2}}}\right)$$
Is there a way to simplify this calculation further?