  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.
