SCHAUM’S Outline® Trigonometry With Calculator-Based Solutions 5th Edition

CHAPTER 1
Angles and Applications

1.1 Introduction

Trigonometry is the branch of mathematics 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 engineering. 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 matter is concerned with a study of the properties of and relations among the trigonometric functions.

1.2 Plane Angle

The plane angle XOP, Fig. 1.1, is formed by the two rays OX and OP. The point O is called the vertex and the half lines are called the sides of the angle.

Fig. 1.1

More often, a plane angle is thought of as being generated by revolving a ray (in a plane) from the initial position OX to a terminal position OP. Then O is again the vertex, is called the initial side, and is called the terminal side of the angle.

An angle generated in this manner is called positive if the direction of rotation (indicated by a curved arrow) is counterclockwise and negative if the direction of rotation is clockwise. The angle is positive in Fig. 1.2(a) and (c) and negative in Fig. 1.2(b).

Fig. 1.2

1.3 Measures of Angles

When an arc of a circle is in the interior of an angle of the circle and the arc joins the points of intersection of the sides of the angle and the circle, the arc is said to subtend the angle.

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.

EXAMPLE 1.1

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 rounded to hundredths.

EXAMPLE 1.2

A radian (rad) is defined as the measure of the central angle subtended by an arc of a circle equal to the radius of the circle. (See Fig. 1.3.)

Fig. 1.3

The circumference of a circle = 2π(radius) and subtends an angle of 360°. Then 2π radians = 360°; therefore

and

where .

EXAMPLE 1.3

(See Probs. 1.1 and 1.2.)

1.4 Arc Length

On a circle of radius r, a central angle of θ radians, Fig. 1.4, intercepts an arc of length

that is, arc length = radius × central angle in radians.

(NOTE: s and r may be measured in any convenient unit of length, but they must be expressed in the same unit.)

Fig. 1.4

EXAMPLE 1.4

(a) On a circle of radius 30 in, the length of the arc intercepted by a central angle of rad is

(b) On the same circle a central angle of 50° intercepts an arc of length

(See Probs. 1.3–1.8.)

1.5 Lengths of Arcs on a Unit Circle

The correspondence between points on a real number line and the points on a unit circle, , with its center at the origin is shown in Fig. 1.5.

Fig. 1.5

The zero (0) on the number line is matched with the point (1, 0) as shown in Fig. 1.5(a). The positive real numbers are wrapped around the circle in a counterclockwise direction, Fig. 1.5(b), and the negative real numbers are wrapped around the circle in a clockwise direction, Fig. 1.5(c). Every point on the unit circle is matched with many real numbers, both positive and negative.

The radius of a unit circle has length 1. Therefore, the circumference of the circle, given by 2πr, is 2π. The distance halfway around is π and the distance 1/4 the way around is π/2. Each positive number is paired with the length of an arc s, and since , each real number is paired with an angle θ in radian measure. Likewise, each negative real number is paired with the negative of the length of an arc and, therefore, with a negative angle in radian measure. Figure 1.6(a) shows points corresponding to positive angles, and Fig. 1.6(b) shows points corresponding to negative angles.

Fig. 1.6

1.6 Area of a Sector

The area K of a sector of a circle (such as the shaded part of Fig. 1.7) with radius r and central angle θ radians is

that is, the area of a sector = ½× the radius × the radius × the central angle in radians.

(NOTE: K will be measured in the square unit of area that corresponds to the length unit used to measure r.)

Fig. 1.7

EXAMPLE 1.5 For a circle of radius 30 in, the area of a sector intercepted by a central angle of rad is

EXAMPLE 1.6 For a circle of radius 18 cm, the area of a sector intercepted by a central angle of 50° is

(NOTE: rad.)

(See Probs. 1.9 and 1.10.)

1.7 Linear and Angular Velocity

Consider an object traveling at a constant velocity along a circular arc of radius r. Let s be the length of the arc traveled in time t. Let 2 be the angle (in radian measure) corresponding to arc length s.

Linear velocity measures how fast the object travels. The linear velocity, v, of an object is computed by

Angular velocity measures how fast the angle changes. The angular velocity, ω (the lower-case Greek letter omega) of the object, is computed by

The relationship between the linear velocity v and the angular velocity ω for an object with radius r is

where ω is measured in radians per unit of time and v is distance per unit of time.

(NOTE: v and ω use the same unit of time and r and v use the same linear unit.)

EXAMPLE 1.7 A bicycle with 20-in wheels is traveling down a road at 15 mi/h. Find the angular velocity of the wheel in revolutions per minute.

Because the radius is 10 in and the angular velocity is to be in revolutions per minute (r/min), change the linear velocity 15 mi/h to units of in/min.