Spline Based Series for Sine and Arbitrarily Accurate Bounds for Sine, Cosine and Sine Integral

Abstract

Based on two point spline approximations of arbitrary order, a series of functions that define lower bounds for sin(x) and sin(x)/x, over the interval [0,Pi/2], with increasingly low relative errors and smaller relative errors than published results, are defined. Second, fourth and eighth order approximations have, respectively, maximum relative errors over the interval [0,Pi/2] of 3.31 x 10-4, 2.48 x 10-8, and 2.02 x 10-18. New series for the sine function, which have significantly better convergence that a Taylor series over the interval [0,Pi/2], are proposed. Applications include functions that are upper bounds for the sine function, upper and lower bounds for the cosine function and lower bounds for the sine integral function. These bounded functions can be made arbitrarily accurate.