Consider the unit circle. Start at the point (1,0). Move up along the circle an arc length of X - then sin(X) will be your y-coordinate and cos(X) will be your x-coordinate. For example, the diameter of this circle is 2, so the circumference is 2*Pi: this is why the period if 2*Pi. This is only true in radians though, which is why they are the natural way to define the functions.