Chapter 1: Trigonometry and Cyclic Functions

Trigonometry is a branch of mathematics concerned with functions that describe angles. Although knowledge of trigonometry is valuable in surveying and navigation, in control systems engineering its virtue lies in the fact that trigonometric functions can be used to describe the status of objects that exhibit repeatable behavior. This includes the motion of the planets, pendulums, a mass suspended on a spring, and perhaps most relevant here, the oscillation of process variables under control.

Units of Measurement

The most common unit of measurement for angles is the degree, which is 1/360 of a whole circle.

A lesser used unit is the radian. Although the radian is not ordinarily used in angular measurement, it should be understood because when differential equations, which occur in control systems engineering, are solved, the angles emerge in radians.

On the circumference of a circle, if an arc equal in length to the radius of the circle is marked off, then the arc will subtend, at the center of the circle, an angle of 1 radian. The angle ? (or POB) in Figure 1-1, illustrates this.

Figure 1-1: A radian defined.

In line with this definition of a radian, the relationship between radians and degrees can be worked out. The full circumference of the circle (length 2 ? r) subtends an angle of 360 at the center of the circle. An arc of length r will subtend an angle of