# Precalculus with Limits A Graphing Approach ... 2014/08/19 آ x y âˆ’4 2 4 د€ 4 د€ 4...

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257

Chapter 4

x

y

−4

2

4

π 4

π 4

5π 4

3− x

y

−4

2

4

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5 x

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5

Trigonometric Functions

4.1 Radian and Degree Measure 4.2 Trigonometric Functions:

The Unit Circle 4.3 Right Triangle Trigonometry 4.4 Trigonometric Functions of

Any Angle 4.5 Graphs of Sine and Cosine

Functions 4.6 Graphs of Other

Trigonometric Functions 4.7 Inverse Trigonometric

Functions 4.8 Applications and Models

Selected Applications Trigonometric functions have many real-life applications. The applica- tions listed below represent a small sample of the applications in this chapter. ■ Sports, Exercise 95, page 267 ■ Electrical Circuits,

Exercise 73, page 275 ■ Machine Shop Calculations,

Exercise 83, page 287 ■ Meteorology, Exercise 109,

page 296 ■ Sales, Exercise 72, page 306 ■ Predator-Prey Model,

Exercise 59, page 317 ■ Photography, Exercise 83,

page 329 ■ Airplane Ascent, Exercises 29 and

30, page 339 ■ Harmonic Motion, Exercises

55–58, page 341

The six trigonometric functions can be defined from a right triangle perspective and as functions of real numbers. In Chapter 4, you will use both perspectives to graph trigonometric functions and solve application problems involving angles and trian- gles. You will also learn how to graph and evaluate inverse trigonometric functions.

Trigonometric functions are often used to model repeating patterns that occur in real life. For instance, a trigonometric function can be used to model the populations of two species that interact, one of which (the predator) hunts the other (the prey).

© Winfried Wisniewski/zefa/Corbis

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

4.1 Radian and Degree Measure

What you should learn � Describe angles.

� Use radian measure.

� Use degree measure and convert between degree and radian measure.

� Use angles to model and solve real-life problems.

Why you should learn it Radian measures of angles are involved in numerous aspects of our daily lives. For instance, in Exercise 95 on page 267, you are asked to determine the measure of the angle generated as a skater performs an axel jump.

Stephen Jaffe/AFP/Getty Images

Angles As derived from the Greek language, the word trigonometry means “measurement of triangles.” Initially, trigonometry dealt with relationships among the sides and angles of triangles and was used in the development of astronomy, navigation, and surveying. With the development of calculus and the physical sciences in the 17th century, a different perspective arose—one that viewed the classic trigonometric relationships as functions with the set of real numbers as their domains. Consequently, the applications of trigonometry expanded to include a vast number of physical phenomena involving rotations and vibrations, including sound waves, light rays, planetary orbits, vibrating strings, pendulums, and orbits of atomic particles.

The approach in this text incorporates both perspectives, starting with angles and their measure.

Figure 4.1 Figure 4.2

An angle is determined by rotating a ray (half-line) about its endpoint. The starting position of the ray is the initial side of the angle, and the position after rotation is the terminal side, as shown in Figure 4.1. The endpoint of the ray is the vertex of the angle. This perception of an angle fits a coordinate system in which the origin is the vertex and the initial side coincides with the positive x-axis. Such an angle is in standard position, as shown in Figure 4.2. Positive angles are generated by counterclockwise rotation, and negative angles by clockwise rotation, as shown in Figure 4.3. Angles are labeled with Greek letters such as (alpha), (beta), and (theta), as well as uppercase letters such as A, B, and C. In Figure 4.4, note that angles and have the same initial and ter- minal sides. Such angles are coterminal.

Figure 4.3 Figure 4.4

β

α x

y

β

α

x

yPositive angle (counterclockwise)

Negative angle (clockwise)

x

y

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x Initial side

Term inal side

y

Vertex

Initial side

Ter min

al s ide

258 Chapter 4 Trigonometric Functions

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Radian Measure The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. One way to measure angles is in radians. This type of measure is especially useful in calculus. To define a radian, you can use a central angle of a circle, one whose vertex is the center of the circle, as shown in Figure 4.5.

Section 4.1 Radian and Degree Measure 259

Definition of Radian

One radian (rad) is the measure of a central angle that intercepts an arc s equal in length to the radius r of the circle. See Figure 4.5. Algebraically this means that

where is measured in radians.�

� � s r

�

r

r s r=

θ x

y

Arc length radius when radian. Figure 4.5

� � 1�

r

r

r

r

r

r

1 radian2 radians

3 radians

4 radians 5 radians

6 radians

x

y

Figure 4.6

Figure 4.8

Because the circumference of a circle is units, it follows that a central angle of one full revolution (counterclockwise) corresponds to an arc length of

Moreover, because there are just over six radius lengths in a full circle, as shown in Figure 4.6. Because the units of measure for s and r are the same, the ratio has no units—it is simply a real number.

Because the radian measure of an angle of one full revolution is you can obtain the following.

revolution radians

revolution radians

revolution radians

These and other common angles are shown in Figure 4.7.

Figure 4.7

Recall that the four quadrants in a coordinate system are numbered I, II, III, and IV. Figure 4.8 shows which angles between 0 and lie in each of the four quadrants. Note that angles between 0 and are acute and that angles between

and are obtuse.���2 ��2

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Quadrant II

< <

Quadrant III

< < Quadrant IV

< <

Quadrant I

0 <

Two positive angles and are complementary (complements of each other) if their sum is Two positive angles are supplementary (supplements of each other) if their sum is See Figure 4.12.

Complementary angles Supplementary angles Figure 4.12

β α

β α

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Two angles are coterminal if they have the same initial and terminal sides. For instance, the angles 0 and are coterminal, as are the angles and

You can find an angle that is coterminal to a given angle by adding or subtracting (one revolution), as demonstrated in Example 1. A given angle has infinitely many coterminal angles. For instance, is coterminal with

where n is an integer. �

6 � 2n�,

� � ��6 �2�

�13��6. ��62�

260 Chapter 4 Trigonometric Functions

STUDY TIP

The phrase “the terminal side of lies in a quadrant” is often abbreviated by simply saying that “ lies in a quadrant.” The terminal sides of the “quadrant angles” 0, and do not lie within quadrants.

3��2�,��2,

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Example 1 Sketching and Finding Coterminal Angles

a. For the positive angle subtract to obtain a coterminal angle

See Figure 4.9.

b. For the positive angle subtract to obtain a coterminal angle

See Figure 4.10.

c. For the negative angle add to obtain a coterminal angle

See Figure 4.11.

Figure 4.9 Figure 4.10 Figure 4.11

Now try Exercise 11.

π

π3 2

0

π 2

= − πθ 2 3

π4 3=

πθ 3

π

π3 2

0

π 2

4

π5 4

−

= πθ 13

π

π3 2

0

π 2

6 π 6

� 2� 3

� 2� � 4� 3

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