Taylor Polynomial Error for sin(x)
The Taylor Series for sin(x) expanded around x=0 is
which is an infinite series.
If we approximate sin(x) with a finite portion of that series, we are using a Taylor Polynomial.
When we consider the error in that finite approximation, there are two factors to consider:
- the degree of the polynomial we will use (higher degree is more accurate, but takes longer to compute), and
- the specific x value where want to approximate sin(x); closer to the x=0 center is better, further out the approximation will get worse.
- If the series is alternating for the given x value, we can use the Alternating Series Error Bound of the first omitted term.
- If the series is not alternating for the given x value, we have to use Taylor's inequality, which is a little more complicated.