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 

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.