You phrase this as a disagreement, but in my mind this is complementary.
The point of the post is that you should define things in terms of what you care about, and then prove stuff about it. The sum:
Sum from n=0 to infinity of (-1)^n / (2n+1)! * x^(2n+1)
isn't something you should start with, it should be the punchline. Instead, you should start with angles (the thing you care about), then prove that they behave the same as that sum (what an incredible claim!).
The post proposed "angles of the acute corners of right triangles" as the starting point. You've argued very well that "angles in the unit circle relative to (1, 0)" is a better starting point. Pedagogically, I think it's a wonderful starting point:
"Let's talk about angles. Just angles, nothing else. When you look at angles of actual objects, the side lengths are all different, which complicates matters. Since all we care about is the angle itself, not the side lengths, let's make all side lengths equal to 1. Etc. etc."
The point of the post is that you should define things in terms of what you care about, and then prove stuff about it. The sum:
isn't something you should start with, it should be the punchline. Instead, you should start with angles (the thing you care about), then prove that they behave the same as that sum (what an incredible claim!).The post proposed "angles of the acute corners of right triangles" as the starting point. You've argued very well that "angles in the unit circle relative to (1, 0)" is a better starting point. Pedagogically, I think it's a wonderful starting point:
"Let's talk about angles. Just angles, nothing else. When you look at angles of actual objects, the side lengths are all different, which complicates matters. Since all we care about is the angle itself, not the side lengths, let's make all side lengths equal to 1. Etc. etc."