[jameshart]

well sine has to come from lengths of curves, doesn't it? Because the geometric definition of sine is a function from angles to numbers, and to get pi in there you need to introduce radians as a way to measure angles - otherwise I could argue that the roots of sin are 0, 180, 360, etc. - and none of them are particularly irrational.

[jordigh]

No. There's the analytic definition of sin that does not mention curves or lengths at all: the sine is the unique function that satisfies the following differential equation and initial value problem:

    s''(x) + s(x) = 0
    s(0)  = 0
    s'(0) = 1

It's a nifty way to define sine purely by its differential properties. Of course, it requires some work to show that differential equations have a solution and that this particular solution is sine (i.e. has the properties you want a sine to have), but once you do that work, it's pretty easy to prove things such as sin^2(x) + cos^2(x) = 1.

On the other hand, starting from the geometric definitions (and building the framework for that, such as arclength, which really requires calculus), it takes a longer route to get to the calculus of sine. Historically this was the route, but we have found shortcuts since then.

[retbull]

Dude I needed you when I was working through my Calc 2 stuff. I didn't ever see that definition as a route to use. Way nicer than memorizing the unit circle which eventually translated into "knowing" the answers to the derivatives and integrals.

[jameshart]

Interesting - I was sort of wondering if you could get sine just from simple harmonic motion, and that's basically what those equations do - the first is equivalent to saying s''(x) = -s(x), so acceleration is opposite to and linearly proportional to (well, equal to in this simple case) displacement. That's a good point.