3 ms·

Thanks for the comment. I did contemplate mentioning the definition of sin(x) = x - x^3/3! + ... but thought it would be too leading & tangential -- I wanted t

by kalid 17y ago

Thanks for the comment. I did contemplate mentioning the definition of sin(x) = x - x^3/3! + ... but thought it would be too leading & tangential -- I wanted to show visually that most curves can in fact be modeled by a straight line over a small distance, without the sledgehammer of showing sine's expansion.

RiderOfGiraffes 17y ago
Better would simply be to say that sin is opposite over hypotenuse, and when the angle is really, really small, the opposite is effectively the same as the arc length. Since the acr length is the angle in degrees, that means that sin(x) ~ x when x is very small. Then you can be more rigorous with the epsilon/delta, and say "you pick how close, and I'll get it closer by making x small enough." Similar arguments can show, for example, that acceleration in a circle is v^2/R. "Calculus" was, or course, done for decades, if not centuries, before "calculus proper" was (somewhat) formalised by Leibniz and Newton. This whole discussion should probably be taken to email if it's to be taken further. The question is, how can we have this sort of discussion, and produce something worth publishing back onto HN?
- kalid 17y ago
  Yep, definitely -- actually I have an article on radians which does just that (comparing height on the circle, sin, to distance traveled around, x). Always happy to discuss on email (kalid.azad@gmail.com) -- I'm not sure if there's something to produce out of it necessarily, but I enjoy the discussion / investigation of personal philosophies & teaching approaches.