Polynomial approximations to sine

Taylor polynomials are terrific local approximations but poor global approximations. Taylor polynomials are optimal in some sense near their center, but are seldom the best choice over a large interval.

This post will look at approximating sin(πx) over [−1, 1] with fifth degree polynomials.

First, this plot compares the approximation error for a fifth order polynomials based on Taylor series and a fifth order polynomial based on Euler’s series from the previous post.

Very near 0 the Taylor approximation is more accurate, but over an entire period of sine, the polynomial based on Euler’s series is much better.

But we can do even better by using a fifth order polynomial based on the first few terms of a Chebyshev series to approximate sine.

Chebyshev series may not have minimal sup-norm error, but there is a theorem that says they are never far from optimal. See the bottom of this post.