Maclaurin Series for sin x
This page shows how to derive the Maclaurin expansion for sin x. The first thing we need to do is to find out the values of the derivatives. Once we have their values we simply plug them in this general formula to find the series expansion.
The first term is simply sin x with x = 0. You can use your calculator if you are new to this, but sin 0 = 0.
If you differentiate sin x the result is cos x, and therefore cos 0 = 1.
If you differentiate cos x the result is -sin x, therefore -sin 0 = 0.
If you differentiate -sin x, then the result is -cos x, therefore -cos 0 = -1
If you differentiate -cos x the result is sin x, which is what we started out with four steps ago. Therefore, there is a cyclic pattern here obviously.
The pattern is therefore, 0, +1, 0, -1, 0... Obviously, this can go on infinitely.
In this stage, I have substituted the values into the general formula of the Maclaurin series. The first 5 terms are the ones I calculated, but the next four in the second row is just a simple repetition of the cycle. The terms with the zero will go obviously.
This is the Maclaurin series for sin x.