The formula for a Maclaurin series expansion is

The derivatives of the function sin x and their values at x = 0 are given by

            f(x) = sin x                              f(0) = 0

            f '(x) = cos x                           f '(0) = 1

            f ''(x) = - sin x                         f ''(0) = 0 

            f '''(x) = - cos x                        f '''(0) = -1

            f ''''(x) = sin x                         f ''''(0) = 0

            f '''''(x) = cos x                         f '''''(0) = 1

            .....................                           ...............

Substituting the values of f(0), f '(0), f ''(0), ... into 1) we have

or