The Sine and Cosine Series

There is a nice method of deriving the polynomial approximations to the sine and cosine functions based on the fact that integration preserves the inequality of functions: If for all x on the interval [a,b], then
We will also need the integrals

We begin with the inequality

We assume that x is positive. Integrating the functions on both sides of this inequality from 0 to x gives us

We again integrate both functions from 0 to x

which is equivalent to

We continue building such inequalities, yielding upper and lower bounds on the sine and cosine functions:

No matter how large x may be, we can always find an m so that is as small as we might desire. We can pin down both cos x and sin x to within an error of at most :