more on sine

x is the angle.

also the circumference of the unit circle!

theta = s/r

if r = 1, then theta = circumference. as angle gets smaller, it goes straight up.

limit x->0 f(x)/x

is the derivative at the point 0.

sine and cosine are really just the same function, delayed by 90 degrees.

thus, their derivatives should be the same, delayed by 90 degrees.

d/dx sine = cosine. (because cos is 90 degrees ahead).

d/dx cosine = -sin (because sin is 90 degrees behind).

(another intuitive argument).

ALSO: need to measure in RADIANS, where

theta = s/r.

using unit circle, we can get circumference from the angle! that's why it works so well. degrees won't do it!

SINE: only seeing projection of more complicated motion! only seeing the shadow of what is really happening.