# Real zeros of sine Taylor polynomials

The more terms of Taylor series ${\displaystyle \sin x = x-\frac{x^3}{3!}+ \frac{x^5}{5!}- \cdots }$ we use, the more resemblance we see between the Taylor polynomial and the sine function itself. The first-degree polynomial matches one zero of the sine, and gets the slope right. The third-degree polynomial has three zeros in about the right places.

The fifth-degree polynomial will of course have … wait a moment.

Since all four critical points are in the window, there are no real zeros outside of our view. Adding the fifth-degree term not only fails to increase the number of zeros to five, it even drops it back to the level of ${T_1(x)=x}$. How odd.

Since the sine Taylor series converges uniformly on bounded intervals, for every ${ A }$ there exists ${ n }$ such that ${\max_{[-A,A]} |\sin x-T_n(x)|<1 }$. Then ${ T_n }$ will have the same sign as ${ \sin x }$ at the maxima and minima of the latter. Consequently, it will have about ${ 2A/\pi }$ zeros on the interval ${[-A,A] }$. Indeed, the intermediate value theorem guarantees that many; and the fact that ${T_n'(x) \approx \cos x }$ on ${ [-A,A]}$ will not allow for extraneous zeros within this interval.

Using the Taylor remainder estimate and Stirling's approximation, we find ${A\approx (n!)^{1/n} \approx n/e }$. Therefore, ${ T_n }$ will have about ${ 2n/(\pi e) }$ real zeros at about the right places. What happens when ${|x| }$ is too large for Taylor remainder estimate to be effective, we can't tell.

Let's just count the zeros, then. Sage online makes it very easy:

sineroots = [[2*n-1,len(sin(x).taylor(x,0,2*n-1).roots(ring=RR))] for n in range(1,51)]
scatter_plot(sineroots) 

The up-and-down pattern in the number of zeros makes for a neat scatter plot. How close is this data to the predicted number ${ 2n/(\pi e) }$? Pretty close.

scatter_plot(sineroots,facecolor='#eeee66') + plot(2*n/(pi*e),(n,1,100))

The slope of the blue line is ${ 2/(\pi e) \approx 0.2342 }$; the (ir)rationality of this number is unknown. Thus, just under a quarter of the zeros of ${ T_n }$ are expected to be real when ${ n }$ is large.

The actual number of real zeros tends to exceed the prediction (by only a few) because some Taylor polynomials have real zeros in the region where they no longer follow the function. For example, ${ T_{11} }$ does this:

Richard S. Varga and Amos J. Carpenter wrote a series of papers titled Zeros of the partial sums of ${ \cos z }$ and ${\sin z }$ in which they classify real zeros into Hurwitz (which follow the corresponding trigonometric function) and spurious. They give the precise count of the Hurwitz zeros: ${1+2\lfloor n/(\pi e)\rfloor }$ for the sine and ${2\lfloor n/(\pi e)+1/2\rfloor }$ for the cosine. The total number of real roots does not appear to admit such an explicit formula. It is the sequence A012264 in the OEIS.