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Sine huh? People often associate the trig functions with pi, because, well, they are trig functions. But, the functions themselves have a lot to do with e (e being that interesting constant that shows up everywhere).
I mean, what is Sine? how do we actually compute it? On a right triangle, sure, there is a pretty simple geometric explanation. But what is the sine of, say, pi*i? What is the geometric interpretation of that?
Well, as it turns out, we can actually define sine in two very useful ways:
1) sin(z) = [e^(i*z) - e^(-i*z)]/(2*i)
This looks kind of wonky, but its actually a relatively simple equation. And it works for all numbers, be it imaginary, complex, or purely real.
Sin(pi/2)?
[e^(i*pi/2)-e^(-i*pi/2)]/(2*i) = i-(-i)/2i = 2i/2i = 1.
Yeah. It actually works. Weird, huh?
2) sin(z) = z - (z^3)/3! + (z^5)/5! - (z^7)7! .....
This is the Taylor series expansion for sine. Its an infinite polynomial, so you can't actually write the whole thing down or use it to explicitly compute anything. You can, however, use it to prove some pretty advanced things.
Some cool you can immediately do with it, however, is create a finite polynomial that approximates the sine function to whatever degree of accuracy you like (that's basically what a Taylor series is used for).
Here it is carried out to 5 terms:
And here it is with 10.