This example derives the power series definition of the cosine function using a Taylor Series Expansion.

Start with the general form of the taylor series where . Note, the variable has been replaced with the variable , because cosine takes in an angle as input.

Choose an input point for cosine to expand the series around.

Observe that for the input angle of radians cosine returns , which makes it a likely point to choose to expand around.

Substitute into the formula.

Simplify.

Take the first, second, third and etc. derivatives of the function. Here we use the derivatives of sine and cosine.

The function and its derivatives are shown below.

Evaluate the functions at the point .

Substitute the values into the expression.

Remove the expressions that go to zero.

Simplify the numerators, flipping some of the signs.

Substitute the name of the function back into the expression .

This gives us the power series of the cosine function.