Circular Curve Solving Employing Non-Algebraic Methods

Question

How would I calculate the radius of a circular curve given only the tangent and arc lengths? An example being tangent = 260.28' and arc = 479.97'. The radius = 500.00'. I normally use Tangent = R tan(Δ/2) and Arc = (ΔπR)/180°. As you can see, both formulas use the central angle, or Δ, and the radius, or R. There's no geometric connection or relationship between the tangent length and arc length of any given circular curve. There must be an iterative approach for solving this. How would I know where to start an iteration, a best guess, for a problem like this? I assume from that point on an iteration would only approach the correct solution, not calculate an exact solution. After learning how to do this, I would like to apply it to a calculator program that I've written to solve circular curves. I only have a basic knowledge of calculus, but I've successfully passed precalculus algebra and analytic trigonometry in college.