It took me a little while to think of a definition of the sine function that does mention pi, though it turned out to be the first one taught in (my) school: "the y coordinate after going t/2pi times counterclockwise around the unit circle starting at (1,0)". If I were to draw the curve, I'd use Euler's method or roll a circle, both of which use the derivative going between -1 and 1 instead of pi for the frame of reference.