THE SINE WAVE
PERIODIC MOVEMENT AND THE CIRCLE

Most of us are introduced to sine and cosine in Geometry class. We’re taught that sine and cosine are functions whose primary use is the determination of unknown lengths and angles when working with right triangles. I think it’s better to think of sine waves as expressions of pure periodic movement, and to focus on the deep relationship between sine waves and circles.1

We can describe the shape of a sine wave by spinning a line around in a circle. The vertical distance from the center of the circle to the tip of the line gives us the amplitude of the sine wave. The faster the line is spinning, the higher the frequency of the resulting sine wave. Figure 1 shows the generation of a sine wave via circular movement. You can change the amplitude and frequency of the resulting sine wave by adjusting the sliders at the bottom of the figure.

 Figure 1.  The Sine Wave Frequency Amplitude

The construction in Figure 1 is so important to our studies that we’ll give it a special name, the phasor.2 Any time you see the term phasor in this text you should think of a line spinning around in a circle. In later sections we will see how phasors can be combined to create arbitrarily complex wave shapes.

1. It’s worth watching Brad Osgood discuss the nature of sine and cosine in this lecture from Stanford University.

2. We’re being a bit fast and loose with the terminology here. "Phasor" can mean a number of different things depending upon who you’re speaking with. If you take umbrage at my naming and have a better idea, let me know.

THE COSINE WAVE
THE COUNTERPART TO SINE
 Figure 2.  The Cosine Wave

Cosine waves are generated in a similar fashion to sine waves, except that we trace the horizontal distance from the center of the circle to the tip of the line. In actuality, sine and cosine wave have exactly the same shape. One is just a rotated version of the other. If you don’t believe me, rotate your head 90 degrees to the right and you’ll find that the cosine wave has become a sine wave.

Sine and cosine are exceptionally interesting, and their relationship to one another is crucially important for the functioning of the Fourier Transform. For the time being, our main takeaway from this section should be that sine and cosine are periodic functions with a deep connection to movement about the unit circle. It’s this deep connection to the circle which gives sine and cosine a number of very astonishing properties which we'll investigate in detail in later sections. In the next section we’ll reacquaint ourselves with the unit-circle and some basic trigonometric properties before returning to the question of sampling.