Abstract: It is well known that sawtooth wave, square wave, triangular wave, trapezoidal wave can be expressed as the superposition of a series of cosine or sine functions. Based on a formula related to the number theory, expansions of sine and cosine waves by means of sawtooth wave and square wave, triangular wave, trapezoidal wave are presented, and the relations between these waveforms are discussed as well. These relations play a basic and important role in applications.

Key words: square w ave, t riangular w ave, saw 2too th w ave, t rapezo idalw ave, sine w ave