On the picture, we have a point on the circle $(a, b)$ parametrized using the arc length and distance to diameter. As you can see, $a = r\theta$. I want to write polar coordinates, $(r, \theta)$ in terms of $(a, b)$. As you can see, if we palce this circle to origin, this is quite like transformation from cartesian to polar coordinates but with one ceveat: $a$ is not $x$ coordinate value but arc length.
Probably the equation b/r = cos(a/r) cannot be solved for r with elementary methods/functions. And r must be a solution to this equation. You could transform to (b/a)*x = cos(x) with x=a/r.