Distance of rays from the center of a chord to the arc

Question

I have search the web for an answer. Here is what I know. I know the radius (500") of the arc. I know the arc height(48") from the center of the chord. Starting at the center of the cord find the length to the arc every 15 degrees. If the formula to find the answer does not input values provided more than likely I will not be able to understand.

Answer I am looking for

Answer 1

The point $C$ has coordinates $(0,452)$. In order to find the length of segment $CG$ as a function of the angle $\alpha$ one can first find the coordinates of the point $G$ as functions of $\alpha$. The point $G$ is one of the two points of intersection (the one with positive $x$-coordinate) of the two graphs:

\begin{equation} y=x\tan\alpha+452 \end{equation}

\begin{equation} x^2+y^2=500^2 \end{equation}

Substituting the first equation into the second gives

\begin{equation} x^2+(x\tan\alpha+452)^2=500 \end{equation}

which can be solved for the non-negative value of $x$ in terms of $\alpha$.

Then using the first of the two equations above, one can find $y$ as a function of $\alpha$.

Then one can derive the length of $CG$ as a function of $\alpha$ for $0\le\alpha\le\tfrac{\pi}{2}$. For $\tfrac{\pi}{2}<\alpha\le\pi$ the distance is the same as for the acute angle $\pi-\alpha$.