A Geometric Derivation of Pi, Arcsine, and the Inverse Polynomial Expression for Sine

In this paper, we propose a geometric method to derive pi using recursive bisections of chords on a semicircle. We extend this approach to a generic arc, obtaining a formula for arcsin(x) with nested radicals. Finally, we show that this process can be inverted algebraically, finding an exact formula for sin(θ) using an iterative chain of polynomials.