Rotation and the Unit Circle

The unit circle is a convenient vehicle for working with rotation and other trigonometric activities. A unit circle is a circle that you graph with its center on the origin of a Cartesian plane and its radius set to 1. Figure 12.13 illustrates a unit circle. Such a representation of a circle proves useful when you seek a convenient way to relate measurements of angles to measurements of rotation. When you set the diameter of a circle to 1, if an angle measures, for example, (90°), then the radian measure of the angle also tells you the length of the arc that the angle designates on the perimeter of the circle. At the same time, as emphasized previously, even though you work with radians, you can always convert back to degrees if you have a need to do so.

If you employ a unit circle to map the cardinal and a few other coordinates and their related sine values, you see a pattern unfold. With each revolution, you move a distance of 2π radians. Regardless of how large the angle of rotation, it is still the case that the same set of values can be used to map the arc. When you factor 4π radians, you find that the terminal side of the angle rests on the x axis, as does the terminal angle of 6π or 8π radians. The same holds true for all the other angles that are multiples of 2π as well. As the arc rotates around the circle, it visits the same points over and over. While the length of the arc grows, factoring allows you to understand the arc in terms of a stable set of radian and degree values.

Figure 12.14 illustrates inner and outer rings surrounding a unit circle. The inner ring shows you the coordinate values of the points on the perimeter of the unit circle. The outer ring displays radian values associated with the plotted points.