I dont know if this counts as philosophy, but I already understand how to do trigonometry. I was just wandering, if someone asked me what means, what would I say?
Its the ratio of two certain sides of a right triangle, o and a?
My ancient greek is a bit rusty, but I believe that "sine" means "perpendicular".
If you drew a unit circle and drew in the radius, an angle is formed with the positive x axis. If you then drew in a line segment from the edge of the radius on the circumference to the x-axis, then this length is the sine of the angle, because it's perpendicular to the x axis.
Being a function, it relates two sets - in this case a set of numbers taken to be the angle and a set of numbers taken to be the ratio of o and a. For sin(x), you can answer with this interpretation: the function relates the angle x (for any real 'x'), formed by the radius and the x-axis, to a ratio (the ratio is as Prove It and you have already noted).