Sine is an elliptic curve. More to the point, any particular sin curve is the exact replica of the edge of a particular ellipse. Observe the sin curve for y=sin(x), 0<x<2*pi:
| 1 ,,oO```Oo,,
| ,oO` `Oo,
| ,/` `\,
| ,/` `\,
| ,/` `\,
| /` `\
| ,/ \,
| /` `\
| ,/ \, pi
| /` `\ |
| ,/ \, |
|,/` `\, 3*pi/2
|-------------------------|------------------------`\----------------------|--------------------/`
| pi/2 \, ,/
| `\ /`
| \, ,/
| `\ /`
| \, ,/
| `\, ,/`
| `\, ,/`
| `\, ,/`
| `Oo, ,oO`
| -1 ``Oo,,,oO``
Imagine that the straight horizontal line (above) is the path a circle of radius one follows as it rolls to the right. Then imagine that, instead of just a circle, it's a cylinder of radius one and length two. (The cylinder is rolling 'on top of' the entire picture above, as if you had it in your hand and were rolling it on your screen.) Now consider an ellipse on the surface of the cylinder: going from one 'corner' to the other. As the cylinder rolls, the ellipse will trace the sine curve above, and a circle at the midpoint of the cylinder will trace the straight line. Thus, whenever you are calculating anything with sin or cos, remember that you are actually using ellipses.