The Maclaurin series of $\sin x\,$.

Solution. We calculate the successive derivatives in left-hand column and their values at $x=0\,$ in the right-hand columns:

$$f(x)=\sin x,\qquad\qquad\qquad f(0)=0;\qquad\qquad$$

$$f^{\prime}(x)=\cos x\qquad\qquad\qquad f^{\prime}(0)=1;\qquad\qquad$$

$$f^{\prime\prime}(x)=-\sin x\quad\qquad\qquad f^{\prime\prime}(0)=0;\qquad\qquad$$

$$f^{\prime\prime\prime}(x)=-\cos x\quad\qquad\qquad f^{\prime\prime\prime}(0)=-1;\qquad\qquad$$

$$f^{(4)}(x)=\sin x\qquad\qquad\qquad f^{(4)}(0)=0;\qquad\qquad$$

and the pattern repeats with a period of four; hence only odd powers appear and the signs alternate. We feed the right-hand coefficients into the general Maclaurin formula to get