Polynomial approximation was used throughout the video, and minimax was one of the final methods. Timestamp is sometime around 18:00. There seemed to be numerical issues with the approximation producing sin values above 1.0.
Yeah, turns out I missed that part of the video. So the author, in fact, knew that minimax approximation is a thing, however he only tried to do it by hand! It's a bit silly to try to optimize a polynomial for minimizing the maximum error and then complain when the result predictably does worse in a completely different metric. What he should have tried is taking some tool for minimax approximation, like Sollya (Remez-based, also offers other functionality), and then try too look for a minimax that would additionally satisfy his other constraints. Sollya allows a weighted approximation, so he could have tried constraining the polynomial to specifically be more accurate around zero, π/2 and π. NB: My WIP package will be even more flexible than Sollya's remez and offer more direct ways to specify such constraints, due to the completely different approach.
