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.