Angles and Applications

In this Chapter:

Introduction

Plane Angle

Measures of Angles

Arc Length

Lengths of Arcs on a Unit Circle

Area of a Sector

Angular Velocity

Introduction

Trigonometry, as the name implies, is concerned with the measurement of the parts sides, and angles of a triangle. Plane trigonometry, which is the topic of this book, is restricted to triangles lying in a plane. Trigonometry is based on certain ratios, called trigonometric functions to be defined in the next chapter. The early applications of the trigonometric functions were to surveying, navigation, and engraining. These functions also play an important role in the study of all sorts of vibratory phenomena sound, light, electricity, etc. As a consequence, a considerable portion of the subject master is concerned with a study of properties of and relations among the trigonometric functions.

Plane Angle

The plane angle XO, Figure 1-1, is formed by the two rays OX and OY. The point O is called the vertex and the half lines are called the sides of the angle.

An angle is called positive if the direction of rotation is counter clockwise and negative if the direction of motion is clockwise. The angle is positive in Figure 1-2(a) and (e). Negative in Figure 1-2 (b).

Measure of Angles

A degree is defined as the measure of the central angle subtended by an arc of a circle equal to 1/360 of the circumference of the circle.

A minute is 1/60 of a degree; a second is 1/60 of a minute, or 1/3600 of a degree.

When changing angles in decimals to minutes and seconds, the general rule is that angles in tenths will be changed to the nearest minute and all other angles will be rounded to the nearest hundredth and then changed to the nearest second. When changing angles in minutes and seconds to decimals, the results in minutes are rounded to tenths, and angles in seconds have the results to hundredths.