TITLE: What, fundamentally, is the reason for the shape of a sin curve?
QUESTION [6 upvotes]: Say we have a metal bar in space aligned horizontally and we start rotating it counter-clockwise about its left end. Then, the sin of the angle from between the horizontal and the bar is the y coordinate of the far end divided by the length of the bar. So, I understand that sin/cos are related to projections of things when you rotate them. Is this the fundamental idea? Why do sin curves look the way they do? Why is the slope most negative when sin(x) = 0? I'm looking for an intuitive answer, not just that the derivative of sin is cos.

REPLY [3 votes]: I think the other answers fully answer the question, but this might be helpful.  My favorite real-life visualization of a sin curve comes from a telephone handset cable:

When stretched out on a flat surface, and viewed from above, the shape of the cable approximates a sin curve.  The equations for the center of the cable will be something like:
$$y=R\sin(Ax)$$
$$z=R\cos(Ax)$$
where R is the radius of the cable spiral, and A depends on how much the cable is stretched.