History and Beauty of Sine Function

Still remember when was the first time you learn about sine function? Still remember how the sine function was derived when you were in the secondary school? I first learnt about the sine function from the trigonometry chapter at 14 years old if I remember correctly. 10 years later, I almost forgot that the sine function introduction was from a right-angled triangle. Sine function of the angle A (not the right angle) in a right angular triangle is the length of the side opposite to the angle A divided by the length of hypotenuse, which is the longest side of a right-angle triangular. Cosine function of the angle A, on the other hand, is defined as the length of the adjacent side of the angle A divided by the length of hypotenuse. From the above explanation, it is not too difficult to think that cosine function is an extension from the sine function. Perhaps this is the reason why cosine function is named as co-sine function.


Everything looks so simple when I was beginning to learn the sine and cosine functions. Never could I have thought that these simple functions have so much influence on today’s mankind history. No kidding! What would happen if there was no sine function nowadays? The influence of the sine function is too much way beyond anyone imagination.


An example of the importance of Sine function is shown obviously in Fourier Series. Fourier Series was discovered by a French mathematician, Jean Baptiste Joseph Fourier (1768-1830, one of the French Revolution contributors) when he was studying and analyzing the heat flow in a metal rod. Therefore, the Fourier Series was named in honor of him. According to the Fourier Series, a periodic function can be represented by an infinite sum of sine or cosine functions that are harmonically related. For an instance, a square wave, which doesn’t seems to be any sinusoidal at all, can be represented by a Fourier Series. (If I have the time, I would like to prove it to you in graphics next time.) If you think that this is an easy statement, then you are totally wrong. Fourier Series as well as Fourier Transform which bears his name, are considered among the greatest discoveries in scientific and engineering discipline.


There are too many periodic motions or waveforms around us all the time. To name a few, signals transmitted by the cell phone base-station, television and radio stations are sinusoidal and periodic. The alternating current (a.c.) power sources generate voltages and currents are in sinusoidal form. Function or Signal Generators generate different kind of periodic waveforms in your laboratories. The generation and analyzing of the periodic motions or waveforms are made possible through Fourier Series. Anyway, the Fourier Series will not be possible if there was no Sine function before Joseph Fourier.

Isn’t it amazing? Why sinusoidal shape matters? Why not square, triangular, or circle? This is really intriguing.