In this entry we compute several Taylor polynomials for the function $\sin x$ around $x=0$ and we produce graphs to compare the function with the corresponding Taylor polynomial. Recall that for a given function $y=f(x)$ (here we suppose $f$ is infinitely differentiable) and a point $x=a$, the Taylor polynomial of degree $n$ ($n\geq 0$) is given by:

where $f^{{(n)}}$ denotes the $n$th derivative of $f(x)$.

From now on we assume $f(x)=\sin x$ and $a=0$. Notice that the derivatives of $\sin x$ are cyclic:

Therefore, the Taylor polynomials are easy to compute. In fact:

Thus, the first Taylor polynomial is given by:

In the following graph one can compare the function $T_{1}(x)=x$ and $\sin x$.

The function $y=\sin x$ and the first Taylor polynomial.

Notice that $T_{2}(x)=T_{1}(x)$. More generally, $T_{{2n}}(x)=T_{{2n-1}}(x)$ so we will not compute any other even order Taylor polynomials. However, the third degree Taylor polynomial is given by the formula:

The function $y=\sin x$ and the third Taylor polynomial.

The Taylor polynomial of degree $5$ is given by:

The function $y=\sin x$ and the fifth Taylor polynomial.

Next, we compute some Taylor polynomials of higher degree. In particular, the Taylor polynomial of degree $15$ has the form:

The function $y=\sin x$ and the Taylor polynomial of degree $15$.

Finally, we produce a detailed view of the Taylor polynomial of degree $99$. In particular, notice that the graphs are very close until $x=34$ or so, but after that $T_{{99}}(x)$ behaves rather jittery and wildly.

A detail of the Taylor polynomial of degree $99$ (the interval $(34,39)$).