I want to have a simple and intuitive explanation of why the sin and cos waves have such a simple repetitive values for their derivatives at a specific point. Their derivative values are also periodic in respect to the derivative order. For example, e^-x is also periodic, but its derivatives are never zero. Is there a good explanation of this without involving complex numbers? Or if not, is there an easy answer to why complex numbers adds their angles when multiplied? Because that kind of answers the same question. The simpler the proof the better!