A property of roots of the truncated series for $\sin(x)$

I refer you to paper 230 from Richard S Varga's collection of papers with the title 'Zeros of the partial sums of $\cos (z)$ and $\sin(z)$'.

An important step is to view the scaled (by degree) truncated taylor polynomial. If your polynomial $p(z)$ is of degree $n$ you would look at $p(nz)$. You can also google the 'Szegő Curve' which is given by $|ze^{1-z}|=1$ with $|z| \leq 1$. This is the curve that the scaled taylor polynomial of $e^z$ approaches as $n\rightarrow\infty$.

Otherwise the link to MO that Olivier posted is also very helpful.

And of course Olivier's suggestion to look at the original paper by Szegő, which contains a lot of this already as well (if you can get it and can understand german). Otherwise you should loook at varga's paper 221 - which is mostly about convergence of the taylor polynomials of $e^z$ to the Szegő Curve. So I referenced two papers, namely 230 and 221 - not the same one twice.

Here's an interesting result that is related to your question.

Let $P_{n}(x)$ be the $n$th order Taylor polynomial for $\sin(x)$ about $x=0.$ Thus,

$$P_{1}(x) = P_{2}(x) = x,$$

$$P_{3}(x) = P_{4}(x) = x - {\tiny \frac{1}{6}}x^{3}, \;\; \mbox{etc.}$$

On each compact interval, these polynomials converge uniformly to $\sin(x)$ as $n \rightarrow \infty,$ so it follows that the number of zeros of $P_{n}(x)$ approaches $\infty$ as $n \rightarrow \infty.$

Let $Z(n)$ be the number of real zeros, counting multiplicity, of $P_{n}(x)$. Then

$$\lim_{n \rightarrow \infty} \frac{Z(n)}{n} \; = \; \frac{2}{\pi e}$$

A proof is given in the following 2 page paper by Rothe, which is on the internet (.pdf file). The proof given in this paper should be accessible to a fairly strong high school calculus student.

Frantz Rothe, Oscillations of the Taylor polynomials for the sin function, Nieuw Archief voor Wiskunde (5) 1 (2000), 397-398.

http://www.nieuwarchief.nl/serie5/pdf/naw5-2000-01-4-397.pdf

This result can also be found in the following paper (not mentioned by Rothe):

Norman Miller, The Taylor series approximation curves for the sine and cosine, American Mathematical Monthly 44 #2 (February 1937), 96-97.