[eh_why_not]

> Perhaps the author merely meant it as an illustration.

Yes I too was expecting oscillations (since I saw Chebyshev polynomials mentioned on the wiki page). I'm beginning to think that the word "might" in their statement "One way to calculate is Remez’s algorithm. The results might look like this:" is doing a lot of heavy lifting.

A couple of notes about your results:

- The interval only needs to be [0, pi/16] and not [0, pi/4]; adjusting that in your wolframalpha query yields a similar bound to what the author lists: 3.5 e-9. (Although same observation about the change of behavior at the bounds as you made).

- The polynomial you obtained actually looks pretty close to the Taylor one: the even coefficients are small, and rounding to 5 or 6 digits might yield a close one, maybe if we look for a degree 9 polynomial instead of degree 8. (I don't have an implementation of Remez handy right now to try it myself).

> Although of course the first few digits of the low-order matching terms will be very similar

Makes sense.