Draw a line of length 1 which makes an angle of p with the positive x axis. The x and y coordinates of a, the point at the end of the line give cos(p) and sin(p).
Now think about what happens if you increase p by a tiny bit. a moves tangentially. (This works very similar to how the tangent to a curve gives the change in y for a change in x.) So the vector of cos'(p), sin'(p) is given by a vector starting from a, at a right angle to 0a, and pointing in the positive direction.
Since the point a moves through 2pi distance while p goes from 0 to 2pi (the definition of measuring angles in radians) the speed of the point is 1, and so the vector of derivatives has length 1.
You can check easily that this makes cos'(p) = -sin(p) and sin'(p) = cos(p).
(If I were trying to present this stuff in a maximally-elegant order without too much regard for what order human brains like to learn things in, the order of things would be: complex numbers, calculus, trigonometry. Then we define something that we might initially call e(t) to satisfy the differential equation de/dt = ie, and observe that having e and e' at right angles means that |e| remains constant, which means that |e'| also remains constant, so if we start with e(0)=1 then we have a point moving at unit speed around the unit circle, etc. Keep the linkage between the geometrical and formal points of view there at all times. But I suspect this wouldn't be great paedagogically for the majority of students.)
