Rational Approximations of Sine and Cosine

In this paper, we use elementary methods to derive a rational function over the integers to approximate the trigonometric sine function on the interval. This formula can then be used to derive a quartic polynomial with a root close to, providing an interesting algebraic approximation to this value. A more accurate rational function over the reals is then computed using numerical optimization. This new formula, while more accurate, provides a worse approximation of in the corresponding quartic equation, showing the trade-offs in local vs. global approximation. This paper is accessible to undergraduates and illustrates a combination of mathematical constructions used in Algebra, Calculus and Numerical Optimization.