14.2 the trigonometric functions - cengage learning

TrigonometricFunctions

14.1 Radian Measure of Angles

14.2 The Trigonometric Functions

14.3 Graphs of Trigonometric Functions

14.4 Derivatives of Trigonometric Functions

14.5 Integrals of Trigonometric Functions

14100

90

80

70

60

50

40

30

20

10

t

T

Average 89.2≈ °

T = 72 + 18 sin(t − 8)12

π

Time (in hours)

Tem

pera

ture

(in

deg

rees

Fah

renh

eit)

4 8 12 16 20 24

Average Temperature

997

Example 10 on page 1042 shows how integration and a trigonometric

model can be used to find the averagetemperature during a four-hour period.

Yur

i Arc

urs/

ww

w.s

hutte

rsto

ck.c

omK

urha

n/w

ww

.shu

tters

tock

.com

9781133105060_1400.qxd 12/3/11 12:31 PM Page 997

1004 Chapter 14 ■ Trigonometric Functions

Analyzing Triangles In Exercises 33–40, solve the triangle for the indicated side and/or angle.

33. 34.

35. 36.

37. 38.

39. 40.

Finding the Area of an Equilateral Triangle InExercises 41–44, find the area of the equilateral trianglewith sides of length See Example 4.

41.

42.

43.

44.

45. Height A person 6 feet tall standing 16 feet from astreetlight casts a shadow 8 feet long (see figure). Whatis the height of the streetlight?

46. Length A guy wire is stretched from a broadcastingtower at a point 200 feet above the ground to an anchor 125 feet from the base (see figure). How long is the wire?

Arc Length In Exercises 47–50, use the followinginformation, as shown in the figure. For a circle of radius a central angle (in radians) intercepts an arc oflength given by

47. Using the Arc Length Formula Complete the tableusing the formula for arc length.

48. Distance A tractor tire that is 5 feet in diameter ispartially filled with a liquid ballast for additional traction. To check the air pressure, the tractor operatorrotates the tire until the valve stem is at the top so thatthe liquid will not enter the gauge. On a given occasion,the operator notes that the tire must be rotated tohave the stem in the proper position (see figure).

(a) What is the radius of the tractor tire?

(b) Find the radian measure of this rotation.

(c) How far must the tractor be moved to get the valvestem in the proper position?

s

80°

80�

r

θs = r

θ

s � r�.s�r,

θ

200 c

125

6

16 8

s � 12 cm

s � 5 ft

s � 8 m

s � 4 in.

s.

2.5 a

2.5

60°

2 s

2 3

θ

60°

3 2

4

h

5 5

40° θ

4

4s

θ60°

343

8 a

4

θ

60°

a

a288θ

45°

5c

30°

θ

35

r 8 ft 15 in. 85 cm

s 12 ft 96 in. 8642 mi

� 1.63�

44

2�

3

9781133105060_1401.qxd 12/3/11 12:32 PM Page 1004

Section 14.1 ■ Radian Measure of Angles 1005

49. Clock The minute hand on a clock is inches long(see figure). Through what distance does the tip of theminute hand move in 25 minutes?

Figure for 49 Figure for 50

50. Instrumentation The pointer of a voltmeter is 6 centimeters in length (see figure). Find the angle (inradians and degrees) through which the pointer rotateswhen it moves 2.5 centimeters on the scale.

51. Speed of Revolution A compact disc can have anangular speed of up to 3142 radians per minute.

(a) At this angular speed, how many revolutions perminute would the CD make?

(b) How long would it take the CD to make 10,000 revolutions?

Area of a Sector of a Circle In Exercises 53 and 54,use the following information. A sector of a circle is theregion bounded by two radii of the circle and their intercepted arc (see figure).

For a circle of radius the area of a sector of the circlewith central angle (in radians) is given by

53. Sprinkler System A sprinkler system on a farm isset to spray water over a distance of 70 feet and rotatesthrough an angle of Find the area of the region.

54. Windshield Wiper A car’s rear windshield wiperrotates The wiper mechanism has a total length of25 inches and wipes the windshield over a distance of14 inches. Find the area covered by the wiper.

True or False? In Exercises 55–58, determine whetherthe statement is true or false. If it is false, explain why orgive an example that shows it is false.

55. An angle whose measure is is obtuse.

56. is coterminal to

57. A right triangle can have one angle whose measure is

58. An angle whose measure is radians is a straight angle.�

89�.

325�.� � �35�

75�

125�.

120�.

A � 12 r2�.�

Ar,

rθ

6 cm

s

3 1 2 in.

312

52. HOW DO YOU SEE IT? Determine whichangles in the figure are coterminal angles withAngle Explain your reasoning.

A

D

B

C

A.

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14.2 The Trigonometric Functions■ Understand the definitions of the trigonometric functions.

■ Understand the trigonometric identities.

■ Evaluate trigonometric functions and solve right triangles.

■ Solve trigonometric equations.

The Trigonometric Functions

There are two common approaches to the study of trigonometry. In one case thetrigonometric functions are defined as ratios of two sides of a right triangle. In the othercase these functions are defined in terms of a point on the terminal side of an arbitraryangle. The first approach is the one generally used in surveying, navigation, and astronomy, where a typical problem involves a triangle, three of whose six parts (sidesand angles) are known and three of which are to be determined. The second approachis the one normally used in science and economics, where the periodic nature of thetrigonometric functions is emphasized. In the definitions below, the six trigonometricfunctions

sine, cosecant, cosine, secant, tangent, and cotangent

are defined from both viewpoints. These six functions are normally abbreviated sin, csc,cos, sec, tan, and cot, respectively.

Definitions of the Trigonometric Functions

Right Triangle Definition: (See Figure 14.11.)

The abbreviations opp, adj, and hyp represent the lengths of the three sides of aright triangle.

opp the length of the side opposite

adj the length of the side adjacent to

hyp the length of the hypotenuse

Circular Function Definition: Let be an angle in standard position with a point on the terminal ray of and (See Figure 14.12.)

cot � �xy

tan � �yx

sec � �rx

cos � �xr

csc � �ry

sin � �yr

r � �x2 � y2 � 0.��x, y��

�

��

��

cot � �adjopp

tan � �oppadj

sec � �hypadj

cos � �adjhyp

csc � �hypopp

sin � �opphyp

0 < � <�

2

In Exercise 72 on page 1015, you willuse trigonometric functions to find

the width of a river.

θ

Opposite

Adjacent

Hypotenuse

FIGURE 14.11

x

y r

x

(x, y)

θ

y

r = x2 + y2

FIGURE 14.12

Blaj Gabriel/www.shutterstock.com

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Trigonometric Identities

In the circular function definition of the six trigonometric functions, the value of isalways positive. From this, it follows that the signs of the trigonometric functions aredetermined from the signs of and as shown in Figure 14.13.

The trigonometric reciprocal identities below are also direct consequences of thedefinitions.

Furthermore, because

you can obtain the Pythagorean Identity Other trigonometric identities are listed below. In the list, is the lowercase Greek letter phi.

Although an angle can be measured in either degrees or radians, radian measure ispreferred in calculus. So, all angles in the remainder of this chapter are assumed to bemeasured in radians unless otherwise indicated. In other words, sin 3 means the sine of3 radians, and means the sine of 3 degrees.sin 3�

�sin2� � cos2� � 1.

� 1

�r2

r2

�x2 � y2

r2

sin2� � cos2� � �yr�

2

� �xr�

2

cot � �cos �sin �

�1

tan �sec � �

1cos �

csc � �1

sin �

tan � �sin �cos �

�1

cot �cos � �

1sec �

sin � �1

csc �

y,x

r

Section 14.2 ■ The Trigonometric Functions 1007

Trigonometric Identities

Pythagorean Identities

Reduction Formulas

Sum or Difference of Two Angles

Double Angle

Half Angle

cos2� �12 �1 � cos 2��

sin2� �12 �1 cos 2��

cos 2� � 2 cos2� 1 � 1 2 sin2�

sin 2� � 2 sin � cos �

tan�� ± �� tan � ± tan �

1 tan � tan �

cos�� ± �� cos � cos � sin � sin �

sin�� ± �� sin � cos � ± cos � sin �

tan � � tan�� cos � � cos�� sin � � sin��

tan�� tan�cos�� cos�sin�� sin�

cot2� � 1 � csc2�tan2� � 1 � sec2�sin2� � cos2� � 1

STUDY TIP

The symbol is used torepresent �sin��2.

sin2�

x

θniscos θtan θ

:::

Quadrant II Quadrant I

:::

θnatθsoc

sin θ

θniscos θtan θ

:::

Quadrant IVQuadrant III

:::

θnatθsoc

sin θ

y

FIGURE 14.13

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Evaluating Trigonometric Functions

There are two common methods of evaluating trigonometric functions: decimal approximations using a calculator and exact evaluations using trigonometric identitiesand formulas from geometry. The next three examples illustrate the second method.

Example 1 Evaluating Trigonometric Functions

Let be a point on the terminal side of as shown in Figure 14.14. Find the sine,cosine, and tangent of

SOLUTION Referring to Figure 14.14, you can see that and

So, the values of the sine, cosine, and tangent of are as shown.

Checkpoint 1

Let be a point on the terminal side of as shown in the figure. Find the sine, cosine, and tangent of

■

The sines, cosines, and tangents of several common angles are listed in the tablebelow. You should remember, or be able to derive, these values.

Trigonometric Values of Common Angles

1 2

1

2

( 3, 1)r

θx

y

�.�,��3, 1�

tan � �yx

� 43

cos � �xr

� 35

sin � �yr

�45

�

� 5.

� �25

� ��3�2 � 42

r � �x2 � y2

x � 3, y � 4,

�.�,�3, 4�


(degrees)� 0 30� 45� 60� 90� 180� 270�

(radians)� 0�

6�

4�

3�

2�

3�

2

sin � 012

�22

�32

1 0 1

cos � 1�32

�22

12

0 1 0

tan � 0�33

1 �3 Undefined 0 Undefined

(−3, 4)

1

2

3

4

r

θx

y

−3 −2 −1 1

FIGURE 14.14

STUDY TIP

Learning the table of values at the right is worth the effort because doing so willincrease both your efficiencyand your confidence. Here isa pattern for the sine functionthat may help you rememberthe values.

Reverse the order to getcosine values of the sameangles.

� 0 30� 45�

sin ��02

�12

�22

� 60� 90�

sin ��32

�42

9781133105060_1402.qxd 12/3/11 12:32 PM Page 1008

To extend the use of the values in the table on the preceding page, you can use theconcept of a reference angle, as shown in Figure 14.15, together with the appropriatequadrant sign. The reference angle for an angle is the smallest positive anglebetween the terminal side of and the -axis. For instance, the reference angle for is and the reference angle for is

FIGURE 14.15

To find the value of a trigonometric function of any angle first determine the functionvalue for the associated reference angle Then, depending on the quadrant in which

lies, prefix the appropriate sign to the function value.


Evaluate each trigonometric function.

a. b. c.

SOLUTION

a. Because the reference angle for is and the sine is positive in the secondquadrant, you can write

Reference angle

See Figure 14.16(a).

b. Because the reference angle for is and the tangent is negative in the fourthquadrant, you can write

Reference angle

See Figure 14.16(b).

c. Because the reference angle for is and the cosine is negative in the thirdquadrant, you can write

Reference angle

See Figure 14.16(c).

Checkpoint 2


a. b. c. ■tan 5�

3cos 135�sin

5�

6

� �32

.

cos 7�

6� cos

�

6

��67��6

� �33

.

tan 330� � tan 30�

30�330�

��22

.

sin 3�

4� sin

�

4

��43��4

cos 7�

6tan 330�sin

3�

4

��.

�,

θ

V

θ

Referenceangle

elgnaecnerefeR

elgnaecnerefeR

θ

IItnardauQ I ItnardauQ

Reference angle: θ:elgnaecnerefeR elgnaecnerefeRθ 2 θ:

Quadrant II

π π π

30�.210�45�135�x�

��


(b)

(c)

(a)

6

67

30°

330°

43

4

ππ

π

π

FIGURE 14.16

9781133105060_1402.qxd 12/3/11 12:32 PM Page 1009



a. b.

c. d.

e. f.

SOLUTION

a. By the reduction formula

b. By the reciprocal identity

c. By the difference formula

d. Because the reference angle for is 0,

e. Using the reciprocal identity

and the fact that you can conclude that is undefined.

f. Because the reference angle for is and the tangent is positive in the firstquadrant,

Checkpoint 3


a. b.

c. d.

e. f. ■cot 13�

4sec 0�

cos 2�cos 75�

csc 45�sin��

6�

tan 9�

4� tan

�

4� 1.

��49��4

cot 0�tan 0� � 0,

cot � �1

tan �

sin 2� � sin 0 � 0.

2�

��6 � �2

4.

��64

��24

� ��22 ��3

2 � � ��22 ��1

2� � �cos 45��cos 30�� sin 45��sin 30��

cos 15� � cos�45� 30��

cos�� cos� cos� � sin� sin�,

sec 60� �1

cos 60��

11�2

� 2.

sec � � 1�cos �,

sin��

3� � sin �

3�

�32

.

sin�� sin �,

tan 9�

4cot 0�

sin 2�cos 15�

sec 60�sin��

3�


TECH TUTOR

When using a calculator to evaluate trigonometric functions,remember to set the calculatorto the proper mode—eitherdegree mode or radian mode.Also, most calculators haveonly three trigonometric functions: sine, cosine, andtangent. To evaluate the otherthree functions, you shouldcombine these keys with thereciprocal key . Forinstance, you can evaluate

on most calculatorsusing the following keystrokesequence in radian mode.

7 ENTERx-1��COS

sec��7�

x-1

9781133105060_1402.qxd 12/3/11 12:32 PM 10

Example 4 Solving a Right Triangle

A surveyor is standing 50 feet from the base of a large tree, as shown in Figure 14.17.The surveyor measures the angle of elevation to the top of the tree as How tall isthe tree?

SOLUTION From Figure 14.17, you can see that

where and is the height of the tree. So, the height of the tree is

Checkpoint 4

Find the height of a building that casts a 75-foot shadow when the angle of elevationof the sun is ■

Example 5 Calculating Peripheral Vision

To measure the extent of your peripheral vision, stand 1 foot from the corner of a room,facing the corner. Have a friend move an object along the wall until you can just barelysee it. When the object is 2 feet from the corner, as shown in Figure 14.18, what is thetotal angle of your peripheral vision?

SOLUTION Let represent the total angle of your peripheral vision. As shown inFigure 14.19, you can model the physical situation with an isosceles right triangle whoselegs are feet and whose hypotenuse is 2 feet. In the triangle, the angle is given by

Using the inverse tangent function of a calculator in degree mode and the followingkeystrokes

3.414

you can determine that So, which impliesthat In other words, the total angle of your peripheral vision is about

FIGURE 14.18 FIGURE 14.19

Checkpoint 5

When the object in Example 5 is 4 feet from the corner, find the total angle of yourperipheral vision. ■

2

1 45°

α2

θ

x = 2 − 1

y = 2

2

1x

y

α

θ

212.6�.� 212.6�.��2 180� 73.7� � 106.3�,� 73.7�.

ENTER�TAN-1

tan � �yx

��2

�2 1 3.414.

��2

�

35�.

y � �x��tan 71.5�� 50��2.98868� 149.4 feet.

yx � 50

tan 71.5� �yx

71.5�.


x = 50

y

Angle ofelevation71.5°

FIGURE 14.17

Some occupations, such as that of amilitary pilot, require excellent vision,including good depth perception and

good peripheral vision.

Ivonne Wierink/www.shutterstock.com

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Solving Trigonometric Equations

An important part of the study of trigonometry is learning how to solve trigonometricequations. For example, consider the equation

You know that is one solution. Also, in Example 3(d), you saw that isanother solution. But these are not the only solutions. In fact, this equation has infinitelymany solutions. Any one of the values of shown below will work.

To simplify the situation, the search for solutions can be restricted to the interval

as shown in Example 6.

Example 6 Solving Trigonometric Equations

Solve for in each equation. Assume

a. b. c.

SOLUTION

a. To solve the equation first remember that

Because the sine is negative in the third and fourth quadrants, it follows that you areseeking values of in these quadrants that have a reference angle of The twoangles fitting these criteria are

and

as indicated in Figure 14.20.

b. To solve remember that and note that in the interval the only angles whose reference angles are 0 are 0, and Of these, 0 and have cosines of 1. The cosine of is So, the equation has two solutions:

and

c. Because and the tangent is positive in the first and third quadrants, itfollows that the two solutions are

and

Checkpoint 6

Solve for in each equation. Assume

a. b. c. ■sin � � 12

tan � � �3cos � ��22

0 � 2�.�

� � � ��

4�

5�

4.� �

�

4

tan ��4 � 1

� � 2�.� � 0

1.��2�2�.�,

�0, 2��,cos 0 � 1cos� � 1,

� � 2� �

3�

5�

3

� � � ��

3�

4�

3

��3.�

sin �

3�

�32

.

sin� � �3�2,

tan� � 1cos� � 1sin� � �32

0 � 2�.�

0 � 2�

. . . , 3�, 2�, �, 0, �, 2�, 3�, . . .

�

� � 2�� 0

sin � � 0.


x

x

y

e c n

3

3

3

θ3

4

53

θ

3

y

Quadrant III

Quadrant IV

Reference

angle:

Reference

angle: π

π

π

π

π

π

=

=

FIGURE 14.20

ALGEBRA TUTOR

For more examples of the algebra involved in solvingtrigonometric equations, seethe Chapter 14 Algebra Tutor,on pages 1046 and 1047.

xy

9781133105060_1402.qxd 12/3/11 12:32 PM Page 1012

Example 7 Solving a Trigonometric Equation

Solve the equation for

SOLUTION You can use the double-angle identity to rewrite theoriginal equation, as shown.

Next, set each factor equal to zero. For you have which hassolutions of

and

For you have which has a solution of

So, for the three solutions are

and

Checkpoint 7

Solve the equation for

■0 � 2�sin 2� � sin � � 0,

�.

5�

6.

�

2,� �

�

6,

0 � 2�,

� ��

2.

sin � � 1,sin � 1 � 0,

� �5�

6.� �

�

6

sin � �12,2 sin � 1 � 0,

0 � �2 sin � 1��sin � 1� 0 � 2 sin2� 3 sin � � 1

1 2 sin2� � 2 3 sin �

cos 2� � 2 3 sin �

cos 2� � 1 2 sin2�

0 � 2�cos 2� � 2 3 sin �,

�.


STUDY TIP

In Example 7, note that the expression is a quadratic inand as such can be factored. For instance, when you let the

quadratic factors as �2x 1��x 1�.2x2 3x � 1 �x � sin �,sin �,

2 sin2� 3 sin � � 1

SUMMARIZE (Section 14.2)

1. State the circular function definition of the six trigonometric functions (page1006). For an example of evaluating trigonometric functions, see Example 1.

2. Explain what is meant by a reference angle (page 1009). For an example ofevaluating trigonometric functions using reference angles, see Example 2.

3. State the reduction formulas (page 1007). For an example of evaluatingtrigonometric functions using a reduction formula, see Example 3(a).

4. Describe a real-life example of how a trigonometric function can be used tofind the height of a tree (page 1011, Example 4).

5. State the double-angle identities (page 1007). For an example of solving atrigonometric equation using a double-angle identity, see Example 7.

Andresr/www.shutterstock.com

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The following warm-up exercises involve skills that were covered in a previous course or in earlier sections. You will use these skills in the exercise set for this section. For additional help, reviewSections 1.1, 1.3, and 14.1.

In Exercises 1–4, convert the angle to radian measure.

1. 2. 3. 4.

In Exercises 5–8, solve for

5. 6. 7. 8.

In Exercises 9–12, solve for

9. 10. 11. 12.2�

12�t 4� �

�

22�

365�t 10� �

�

42�

12�t 2� �

�

42�

24�t 4� �

�

2

t.

x2 5x � 6x2 2x � 32x2 � x � 0x2 x � 0

x.

390�225�300�315�

Exercises 14.2 See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

Evaluating Trigonometric Functions In Exercises1–6, determine all six trigonometric functions of theangle See Example 1.

1. 2.

3. 4.

5. 6.

Finding Trigonometric Functions In Exercises 7–12,sketch a right triangle corresponding to the trigonometricfunction of the angle and find the other five trigonometricfunctions of

7. 8.

9. 10.

11. 12.

Determining a Quadrant In Exercises 13–18, determinethe quadrant in which lies.

13. 14.

15. 16.

17. 18.

Evaluating Trigonometric Functions In Exercises19–32, evaluate the six trigonometric functions of theangle without using a calculator. See Examples 2 and 3.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.

Evaluating Trigonometric Functions InExercises 33–42, use a calculator to evaluate thetrigonometric function to four decimal places.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42. tan 4.5sin�0.65�cot 210�tan 240�

cos 250�cos�110��

tan 10�

9tan

�

9

csc 10�sin 10�

17�

310�

3

510�750�

210�300�

4�

3225�

150��

2

5�

4�

4

2�

360�

cos � > 0, tan � < 0csc � > 0, tan � < 0

cot � < 0, cos � > 0sin � > 0, sec � > 0

sin � > 0, cos � < 0sin � < 0, cos � > 0

�

csc � � 4.25tan � � 3

cos � �57sec � � 2

cot � � 5sin � �13

�.�

x

(−2, −2)

θ

y

x

θ

(−2, 4)

y

x

θ

(1, −1)

y

x

θ

(−12, −5)

y

x

θ

(8, −15)

y

x

(4, 3)

θ

y

�.

SKILLS WARM UP 14.2

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Solving a Right Triangle In Exercises 43–48, solve foror as indicated. See Example 4.

43. Solve for 44. Solve for



Solving Trigonometric Equations In Exercises 49–60,solve the equation for Assume SeeExample 6.

49. 50.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

Solving Trigonometric Equations In Exercises 61–70,solve the equation for Assume For some ofthe equations, you should use the trigonometric identitieslisted in this section. Use the trace feature of a graphingutility to verify your results. See Example 7.

61. 62.

63. 64.

65.

66.

67. 68.

69. 70.

71. Length A 20-foot ladder leaning against the side of ahouse makes a angle with the ground (see figure).How far up the side of the house does the ladder reach?

72. Width of a River A biologist wants to know thewidth of a river in order to set instruments to study the pollutants in the water. From point A, the biologistwalks downstream 100 feet and sights to point C. Fromthis sighting it is determined that (see figure).How wide is the river?

73. Distance An airplane flying at an altitude of 6 milesis on a flight path that passes directly over an observer(see figure). Let be the angle of elevation from theobserver to the plane. Find the distance from theobserver to the plane when (a) (b) and (c)

74. Skateboard Ramp A skateboard ramp with a heightof 4 feet has an angle of elevation of (see figure).How long is the skateboard ramp?

4 ftc

18°

18�

6 mid

Not drawn to scale

θ

� � 90�.� � 60�,� � 30�,

d�

w

100 ft

= 50°θ

A

C

� � 50�

w

20 ft

75°

75�

cos �

2 cos � � 1cos2� � sin � � 1

sec � csc � � 2 csc �sin � � cos �

cos 2� � 3 cos � � 2 � 0

sin 2� cos � � 0

2 cos2� cos � � 1tan2� tan � � 0

tan2� � 32 sin2� � 1

0 � 2�.�.

tan � ��33

sin � ��32

cot � � �3sin � � �22

sec � � 2sec � � 2

cot � � 1csc � �2�3

3

cos � � �22

tan � � �3

cos � � 12sin � �

12

0 � 2�.�.

x

50

20°r

10

40°

x.r.

20r

45°

25

x

60°

r.x.

x

10

60°001

y

30°

x.y.

ry,x,

9781133105060_1402.qxd 12/3/11 12:32 PM Page 1015


75. Empire State Building You are standing 45 metersfrom the base of the Empire State Building. You estimate that the angle of elevation to the top of the 86thfloor is The total height of the building is another123 meters above the 86th floor.

(a) What is the approximate height of the building?

(b) One of your friends is on the 86th floor. What is thedistance between you and your friend?

76. Height A 25-meter line is used to tether a helium-filledballoon. Because of a breeze, the line makes an angle ofapproximately with the ground.

(a) Draw the right triangle that gives a visual representation of the problem. Show the known sidelengths and angles of the triangle and use a variableto indicate the height of the balloon.

(b) Use a trigonometric function to write an equationinvolving the unknown quantity.

(c) What is the height of the balloon?

77. Loading Ramp A ramp feet in length rises to aloading platform that is feet off the ground (see figure).Find the angle (in degrees) that the ramp makes with theground.

78. Height The height of a building is 180 feet. Find theangle of elevation (in degrees) to the top of the buildingfrom a point 100 feet from the base of the building (seefigure).

79. Height of a Mountain In traveling across flat land,you notice a mountain directly in front of you. Its angle of elevation (to the peak) is After you drive13 miles closer to the mountain, the angle of elevationis Approximate the height of the mountain.

81. Medicine The temperature (in degrees Fahrenheit)of a patient hours after arriving at the emergency roomof a hospital at 10:00 P.M. is given by

Find the patient’s temperature at (a) 10:00 P.M.,(b) 4:00 A.M., and (c) 10:00 A.M. (d) At what time doyou expect the patient’s temperature to return to normal?Explain your reasoning.

82. Sales A company that produces a window and door insulating kit forecasts monthly sales over the next 2 years to be

where is measured in thousands of units and is the time in months, with corresponding to January 2011. Find the monthly sales for (a) February2011, (b) February 2012, (c) September 2011, and (d) September 2012.

Graphing Functions In Exercises 83 and 84,use a graphing utility or a spreadsheet to completethe table. Then graph the function.

83. 84.


85. 86.

87.

88. Because it can be said that the sine ofa negative angle is a negative number.

sin�t� � sin t,

sin2 45� cos2 45� � 1

sin 60�

sin 30�� sin 2�sin 10� csc 10� � 1

f�x� �12

�5 x� � 3 cos �x5

f�x� �25

x � 2 sin �x5

t � 1tS

S � 23.1 � 0.442t � 4.3 sin �t6

0 t 18.T�t� � 98.6 � 4 cos �t36

,

tT

Not d r awn to scal e 13 mi

3.5° 9°

9�.

3.5�.

180 ft

θ

100 ft

1 2 1

2

17 ft 3 ft

θ

312

1712

75�

82�.

80. HOW DO YOU SEE IT? Consider an angle in standard position with centimeters,as shown in the figure. Describe the changes inthe values of and as increases from to

12 cm

θ

y

x

(x, y)

90�.0��tan �cos �,sin �,y,x,

r � 12

x 0 2 4 6 8 10

f �x�

9781133105060_1402.qxd 12/3/11 12:32 PM Page 1016

Section 14.3 ■ Graphs of Trigonometric Functions 1017

14.3 Graphs of Trigonometric Functions■ Sketch graphs of trigonometric functions.

■ Evaluate limits of trigonometric functions.

■ Use trigonometric functions to model real-life situations.

Graphs of Trigonometric Functions

When you are sketching the graph of a trigonometric function, it is common to use (rather than ) as the independent variable. For instance, you can sketch the graph of

by constructing a table of values, plotting the resulting points, and connecting themwith a smooth curve, as shown in Figure 14.21. Some examples of values are shown inthe table below.

In Figure 14.21, note that the maximum value of is 1 and the minimum valueis The amplitude of the sine function (or the cosine function) is defined to be half of the difference between its maximum and minimum values. So, the amplitude of

is 1.

FIGURE 14.21

The periodic nature of the sine function becomes evident when you observe that asincreases beyond the graph repeats itself over and over, continuously oscillating

about the -axis. The period of the function is the distance (on the -axis) between successive cycles, as shown in Figure 14.22. So, the period of is

FIGURE 14.22

f(x) = sin x

2π

2π 2π3

2π 3

2π 5

2πππ

−1

1

−− −

Period: 2 π

x

y

2�.f�x� � sin xxx

2�,x

1

−1

x

Amplitude = 1 f(x) = sin x

y

2 6 π

4 π

3 π

2 π

3 π 2 π3

4 π 3

2 π 5

3 π 7

4 π

6 11π5

6 π 7

6 π 5

4 π 4

3 ππ

f�x� � sin x

�1.sin x

f�x� � sin x

�x

x 0�

6�

4�

3�

22�

33�

45�

6�

sin x 0.00 0.50 0.71 0.87 1.00 0.87 0.71 0.50 0.00

In Exercise 73 on page 1025, you will use a trigonometric function

to model the air flow of a person’s respiratory cycle.

Benjamin Thorn/www.shutterstock.com

9781133105060_1403.qxd 12/3/11 12:33 PM Page 1017

Figure 14.23 shows the graphs of at least one cycle of all six trigonometric functions.

Familiarity with the graphs of the six basic trigonometric functions allows you tosketch graphs of more general functions such as

and

Note that the function oscillates between and and so has an amplitude of

Amplitude of

Furthermore, because when and when it follows thatthe function has a period of

Period of

When graphing general functions such as

or

note how the constants and affect the graph. You already know that is theamplitude and is the period of the graph. The constants and determine the horizontal shift and vertical shift of the graph, respectively. Two examples are shown inFigure 14.24. In the first graph, notice that relative to the graph of the graphof is shifted units to the right, stretched vertically by a factor of 2, and shifted upone unit. In the second graph, notice that relative to the graph of the graphof is shifted units to the right, stretched horizontally by a factor of stretchedvertically by a factor of 3, and shifted down two units.

12,��2g

y � cos x,��2f

y � sin x,

dcbadc,b,a,

g�x� � a cos�b�x � c�� df�x� � a sin�b�x � c�� d

y � a sin bx2�

�b�.

y � a sin bxx � 2��b,bx � 2�x � 0bx � 0

y � a sin bx�a�.

a�ay � a sin bx

y � a cos bx.y � a sin bx


ππ

5

4

1

2

3

−3

6

4

5

3

2

−1

4

2

1

3

4

2

3

−1

−3

−2

6

4

5

3

2

1

−1

4

2

1

3

x

x

x x

x x

y y y

y = sin x y = cos x

y = tan x

1sin x

y y y

π

π 2 π

π 2 π

π 2 π

π 2 π

y = csc x = 1cos x

y = sec x = 1tan x

y = cot x =

Domain: all reals Range: [−1, 1] Period: 2 π

Domain: all x ≠ nRange: (−∞, −1] � [1, ∞) Period: 2π

π

Domain: all reals Range: [−1, 1] Period: 2 π

Domain: all x ≠ + n

Range: (−∞, ∞)Period: π

2

ππDomain: all x ≠ nRange: (−∞, ∞) Period: π

πDomain: all x ≠ + n

Range: (−∞, −1] � [1, ∞) Period: 2 π

2

2 π

Graphs of the Six Trigonometric FunctionsFIGURE 14.23

−6

3

−2� 2�

y = cos x

g(x) = 3 cos[ [ − 2x − 2 )) π2

FIGURE 14.24

−2

3

� 2�

−3 y = sin x f(x) = 2 sin )) π

2 + 1x −

9781133105060_1403.qxd 12/3/11 12:33 PM Page 1018

Example 1 Graphing a Trigonometric Function

Sketch the graph of

SOLUTION The graph of has the characteristics below.

Amplitude: 4

Period:

Three cycles of the graph are shown in Figure 14.25, starting with the point

Checkpoint 1

Sketch the graph of ■


Sketch the graph of

SOLUTION The graph of has the characteristics below.

Amplitude: 3

Period:

Almost three cycles of the graph are shown in Figure 14.26, starting with the maximum point

Checkpoint 2

Sketch the graph of ■


Sketch the graph of

SOLUTION The graph of this function has a period of The vertical asymptotes of this tangent function occur at

Several cycles of the graph are shown in Figure 14.27, starting with the vertical asymptote

Checkpoint 3

Sketch the graph of ■g�x� � tan 4x.

x � ��6.

Period ��

3

x � . . . , ��

6,

�

6,

�

2,

5�

6, . . . .

��3.

f�x� � �2 tan 3x.

g�x� � 2 sin 4x.

�0, 3�.

2�

2� �

f�x� � 3 cos 2x

f�x� � 3 cos 2x.

g�x� � 2 cos x.

�0, 0�.

2�

f�x� � 4 sin x

f�x� � 4 sin x.


−3

−2

−1

1

2

3

x

Amplitude = 3

Period =

(0, 3)

y

2 π π

2 3 π

2 5 π2 π

f(x) = 3 cos 2xπ

FIGURE 14.26

y

x

3πPeriod =

f(x) = −2 tan 3x

6 π

6 π

2 π

6 5 π

6 7 π−

−3

−2

−4

FIGURE 14.27

−4 −3 −2 −1

4 3 2 1

y

x

Amplitude = 4

Period = 2

f(x) = 4 sin x

2 3 π

2 5 π

2 7 π

2 9 π

2 11 π

(0, 0)

π

FIGURE 14.25

9781133105060_1403.qxd 12/3/11 12:33 PM Page 1019

Limits of Trigonometric Functions

The sine and cosine functions are continuous over the entire real number line. So, youcan use direct substitution to evaluate a limit such as

When direct substitution with a trigonometric limit yields an indeterminate form, such as

Indeterminate form

you can rely on technology to help evaluate the limit. The next example examines thelimit of a function that you will encounter again in Section 14.4.

Example 4 Evaluating a Trigonometric Limit

Use a calculator to evaluate the function

at several -values near Then use the result to estimate

Use a graphing utility (set in radian mode) to confirm your result.

SOLUTION The table shows several values of the function at -values near zero. (Note that the function is undefined when )

From the table, it appears that the limit is 1. That is,

Figure 14.28 shows the graph of

From this graph, it appears that gets closer and closer to 1 as approaches zero (from either side).

Checkpoint 4

Use a calculator to evaluate the function

at several -values near Then use the result to estimate

■limx→0

1 � cos x

x.

x � 0.x

f�x� �1 � cos x

x

xf�x�

f�x� �sin x

x.

limx→0

sin x

x� 1.

x � 0.x

limx→0

sin x

x.

x � 0.x

f�x� �sin x

x

00

limx→0

sin x � sin 0 � 0.


x �0.20 �0.15 �0.10 �0.05 0.05 0.10 0.15 0.20

sin xx

0.9933 0.9963 0.9983 0.9996 0.9996 0.9983 0.9963 0.9933

1.5

−1.5

2 �−2 �

f(x) = sin xx

FIGURE 14.28

Andresr/www.shutterstock.com

9781133105060_1403.qxd 12/3/11 12:33 PM Page 1020

Applications

There are many examples of periodic phenomena in both business and biology. Manybusinesses have cyclical sales patterns, and plant growth is affected by the day-nightcycle. The next example describes the cyclical pattern followed by many types of predator-prey populations, such as coyotes and rabbits.

Example 5 Modeling Predator-Prey Cycles

The population of a predator at time (in months) is modeled by

and the population of its primary food source (its prey) is modeled by

Graph both models on the same set of axes and explain the oscillations in the size ofeach population.

SOLUTION Each function has a period of 24 months. The predator population has anamplitude of 3000 and oscillates about the line The prey population has an amplitude of 5000 and oscillates about the line The graphs of the twomodels are shown in Figure 14.29. The cycles of this predator-prey population areexplained by the diagram below.

FIGURE 14.29

Checkpoint 5

Repeat Example 5 for the following models.

Predator population

Prey population ■t � 0p � 18,000 � 6000 sin 2� t12

,

t � 0P � 12,000 � 2500 sin 2� t12

,

20,000

15,000

10,000

5,000

6 12 18 24 30 36 42 48 54 60 66 72 78 t

Amplitude = 5000

Amplitude = 3000

P = 10,000 + 3000 sinPredator Prey

Period = 24 months

Popu

latio

n

Time (in months)

Predator-Prey Cycles

2 t24π

p = 15,000 + 5000 cos 2 t24π

Preypopulation

increase

Predatorpopulationdecrease

Preypopulationdecrease

Predatorpopulation

increase

y � 15,000.y � 10,000.

t � 0.p � 15,000 � 5000 cos 2�t24

,

p

t � 0P � 10,000 � 3000 sin 2�t24

,

tP


9781133105060_1403.qxd 12/3/11 12:33 PM Page 1021

Example 6 Modeling Biorhythms

A theory that attempts to explain the ups and downs of everyday life states that eachperson has three cycles, which begin at birth. These three cycles can be modeled by sinewaves

Physical (23 days):

Emotional (28 days):

Intellectual (33 days):

where is the number of days since birth. Describe the biorhythms during the month ofSeptember 2011, for a person who was born on July 20, 1991.

SOLUTION Figure 14.30 shows the person’s biorhythms during the month ofSeptember 2011. Note that September 1, 2011 was the 7348th day of the person’s life.

Checkpoint 6

Use a graphing utility to describe the biorhythms of the person in Example 6 duringthe month of January 2011. Assume that January 1, 2011 is the 7105th day of the person’s life. ■

t

t � 0I � sin 2� t33

,

t � 0E � sin 2� t28

,

t � 0P � sin 2� t23

,


t2 4 6 8 10 20 24 26 30

23-day cycle

28-day cycle September 2011

33-day cycle

“Good” day

“Bad” day

t = 7348 July 20, 1991

7355 7369

Physical cycle Emotional cycle Intellectual cycle

FIGURE 14.30


1. Describe the graph of (page 1018). For an example of graphing asine function, see Example 1.

2. Describe the graph of (page 1018). For an example of graphing acosine function, see Example 2.

3. Describe the graph of (page 1018). For an example of graphing atangent function, see Example 3.

4. Describe a real-life example of how the graphs of trigonometric functionscan be used to analyze biorhythms (page 1022, Example 6).

y � tan x

y � cos x

y � sin x

Martin Novak/www.shutterstock.com

9781133105060_1403.qxd 12/3/11 12:33 PM Page 1022


The following warm-up exercises involve skills that were covered in earlier sections. You will usethese skills in the exercise set for this section. For additional help, review Sections 7.1 and 14.2.

In Exercises 1 and 2, find the limit.

1. 2.

In Exercises 3–10, evaluate the trigonometric function without using a calculator.

3. 4. 5. 6.

7. 8. 9. 10.

In Exercises 11–18, use a calculator to evaluate the trigonometric function to fourdecimal places.

11. 12. 13. 14.

15. 16. 17. 18. tan 140�tan 327�cos 72�sin 103�

cos 310�sin 275�sin 220�cos 15�

sin 4�

3cos

5�

3cos

5�

6sin

11�

6

cot 2�

3tan

5�

4sin �cos

�

2

limx→3

�x3 � 2x2 � 1�limx→2

�x2 � 4x � 2�

Exercises 14.3

Finding the Period and Amplitude In Exercises 1–14,find the period and amplitude of the trigonometric function.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

Finding the Period In Exercises 15–20, find the periodof the trigonometric function.

15. 16.

17. 18.

19. 20. y � 5 tan 2�x

3y � cot

�x6

y � csc 4xy � 3 sec 5x

y � 7 tan 2�xy � 3 tan x

y �23

cos �x10

y � 3 sin 4�x

y � 5 cos x4

y �12

sin 2x3

x

4π

1

−1

y

−1

−2

2

x

4π

y

y �13 sin 8xy � �

32 sin 6x

−3−2−1

123

x

y

π 2π−1

−2

2

x

2π

y

y � �cos 2x3

y � �sin 3x

8 6 4 2

−3 −2 −1

1 2 3

x

y

−2

−1

2

1

8 6 4 2 x

y

y �52

cos �x2

y �12

cos �x

−3−2−1

123

x

y

4 π

−3 −2 −1

1 2 3

x

y

π 2π 4π

y � �2 sin x3

y �32

cos x2

3 2 1

−1 −2 −3

x

y

π

−3 −2 −1

3 2 1

x

y

2 π

y � 3 cos 3xy � 2 sin 2x

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

SKILLS WARM UP 14.3

9781133105060_1403.qxd 12/3/11 12:33 PM Page 1023


Matching In Exercises 21–26, match the trigonometricfunction with the correct graph and give the period of thefunction. [The graphs are labeled (a)–(f).]

(a) (b)

(c) (d)

(e) (f)

21. 22.

23. 24.

25. 26.

Graphing Trigonometric Functions In Exercises27–40, sketch the graph of the trigonometric function byhand. Use a graphing utility to verify your sketch. SeeExamples 1, 2, and 3.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

Graphing Trigonometric Functions In Exercises41–50, use a graphing utility to graph thetrigonometric function.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

Evaluating Trigonometric Limits In Exercises51–60, complete the table using a spreadsheet ora graphing utility (set in radian mode) to estimate

Use a graphing utility to graph the function

and confirm your result. See Example 4.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

Graphical Reasoning In Exercises 61–64, find and for such that the graph of matchesthe figure.

61. 62.

63. 64.

−1

−2

−5

1

f

y

x π π−

−2 −

4

6

8

10

f

π π

y

x

−4

−6

2

4

xππ

y

−

f

1

−1

4

−2

y

x

f

2 π

ff�x � a cos x � dda

f�x� �1 � cos2 x

2x f�x� �

sin2 xx

y �cos x tan x

4x f�x� �

tan 2xx

f�x� �2 sin�x�4�

x f�x� �

3�1 � cos x�x

f�x� �1 � cos 2x

xy �

tan 3xtan 4x

f�x� �sin 2xsin 3x

f�x� �sin 4x

2x

limx→0

f �x.

y � �tan xy � csc 2�x

y � sec �xy � 2 sec 2x

y � cot 2x3

y � csc 2x3

y � 3 tan �xy � cot 2x

y � 10 cos �x6

y � �sin 2�x

3

y � �tan 3xy � 2 sec 4x

y � �csc x3

y �12

tan �x2

y � 2 cot xy � 2 tan x

y �32

sin �x4

y � cos 2�x

y � �3 cos 4xy � �2 sin 6x

y �32

cos 2x3

y � 2 cos �x3

y � 4 sin x3

y � sin x2

y � tan x2

y � 2 csc x2

y � �sec xy � cot �x2

y �12 csc 2xy � sec 2x

x

y

−1 1 2 3 4

−2

−3

1

2

33

2

1

−1

−2

−3

x

y

4 π

4 5π

1

x

y

− − 2 π

2 π ππ

−1

x

y

π

2

4

x

y

π 3π

1

2

x

y

π

x �0.1 �0.01 �0.001 0.001 0.01 0.1

f �x�

9781133105060_1403.qxd 12/3/11 12:33 PM Page 1024


Phase Shift In Exercises 65–68, match the functionwith the correct graph. [The graphs are labeled (a)–(d).]

(a) (b)

(c) (d)

65. 66.

67. 68.

69. Biology: Predator-Prey Cycle The population of a predator at time (in months) is modeled by

and the population of its prey is modeled by

(a) Use a graphing utility to graph both models in thesame viewing window.

(b) Explain the oscillations in the size of each population.

70. Biology: Predator-Prey Cycle The population of a predator at time (in months) is modeled by

and the population of its prey is modeled by

(a) Use a graphing utility to graph both models in thesame viewing window.

(b) Explain the oscillations in the size of each population.

71. Biorhythms For a person born on July 20, 1991, usethe biorhythm cycles given in Example 6 to calculatethis person’s three energy levels on December 31, 2015.Assume this is the 8930th day of the person’s life.

72. Biorhythms Use your birthday and the biorhythmcycles given in Example 6 to calculate your three energylevels on December 31, 2015.

73. Health For a person at rest, the velocity (in litersper second) of air flow into and out of the lungs duringa respiratory cycle is given by

where is the time (in seconds). Inhalation occurs whenand exhalation occurs when

(a) Find the time for one full respiratory cycle.

(b) Find the number of cycles per minute.

(c) Use a graphing utility to graph the velocity function.

74. Health After a person exercises for a few minutes,the velocity (in liters per second) of air flow into andout of the lungs during a respiratory cycle is given by

where is the time (in seconds). Inhalation occurs whenand exhalation occurs when

(a) Find the time for one full respiratory cycle.

(b) Find the number of cycles per minute.

(c) Use a graphing utility to graph the velocity function.

75. Music When tuning a piano, a technician strikes a tuning fork for the A above middle C and sets up wave motion that can be approximated by

where is the time (in seconds).

(a) What is the period of this function?

(b) What is the frequency of this note

(c) Use a graphing utility to graph this function.

76. Health The function

approximates the blood pressure (in millimeters ofmercury) at time (in seconds) for a person at rest.

(a) Find the period of the function.

(b) Find the number of heartbeats per minute.

(c) Use a graphing utility to graph the pressure function.

tP

P � 100 � 20 cos�5�t�3�

� f � 1�p�?f

p

t

y � 0.001 sin 880�t

v < 0.v > 0,t

v � 1.75 sin �t2

v

v < 0.v > 0,t

v � 0.9 sin �t3

v

p � 9800 � 2750 cos 2�t24

.

p

P � 5700 � 1200 sin 2�t24

tP

p � 12,000 � 4000 cos 2�t24

.

p

P � 8000 � 2500 sin 2�t24

tP

y � sinx �3�

2 �y � sin�x � ��

y � sinx ��

2�y � sin x

y

x 2ππ

1

−1

y

x 2ππ

1

−1

y

x

2 3π

2 π

1

−1

y

x

2 3π

2 π

1

−1

9781133105060_1403.qxd 12/3/11 12:33 PM Page 1025


77. Construction Workers The number (in thousands)of construction workers employed in the United Statesduring 2010 can be modeled by

where is the time (in months), with correspondingto January. (Source: U.S. Bureau of Labor Statistics)

(a) Use a graphing utility to graph

(b) Did the number of construction workers exceed 5.5 million in 2010? If so, during which month(s)?

78. Sales The snowmobile sales (in units) at a dealershipare modeled by

where is the time (in months), with correspondingto January.


(b) Will the sales exceed 75 units during any month? Ifso, during which month(s)?

79. Physics Use the graphs below to answer each question.

(a) Which graph (A or B) has a longer wavelength, orperiod?

(b) Which graph (A or B) has a greater amplitude?

(c) The frequency of a graph is the number of oscillationsor cycles that occur during a given period of time.Which graph (A or B) has a greater frequency?

(d) Based on the definition of frequency in part (c), howare frequency and period related?

(Source: Adapted from Shipman/Wilson/Todd, AnIntroduction to Physical Science, Eleventh Edition)

80. Think About It Consider the function given byon the interval

(a) Sketch the graph of

for and 3.

(b) How does the value of affect the graph?

(c) How many complete cycles occur between 0 and for each value of

81. Think About It Consider the functions given byand on the interval

(a) Graph and in the same coordinate plane.

(b) Approximate the interval in which

(c) Describe the behavior of each of the functions as approaches How is the behavior of related tothe behavior of as approaches


83. The amplitude of is

84. The period of is

85.

86. One solution of is 5�

4.tan x � 1

limx→0

sin 5x

3x�

53

3�

2.f�x� � 5 cot�

4x3 �

�3.f�x� � �3 cos 2x

�?xfg�.

x

f > g.

gf

�0, ��.g�x� � 0.5 csc xf�x� � 2 sin x

b?2�

b

b �12, 2,

y � cos bx

�0, 2��.y � cos bx

A

B

Wave amplitude

Wave amplitude

One wavelength

One wavelength

Particle displacement

Velocity

Velocity

Particle displacement

One wavelength

S.

t � 1t

S � 58.3 � 32.5 cos �t6

S

W.

t � 1t

W � 5488 � 347.6 sin�0.45t � 4.153�

W

82. HOW DO YOU SEE IT? The normal monthlyhigh temperatures for Erie, Pennsylvania areapproximated by

and the normal monthly low temperatures for Erie,Pennsylvania are approximated by

where is the time (in months), with corresponding to January. (Source: NationalClimatic Data Center)

(a) During what part of the year is the differencebetween the normal high and low temperaturesgreatest? When is it smallest?

(b) The sun is the farthest north in the sky around June 21, but the graph shows the highest temperatures at a later date. Approximate the lag time of the temperatures relative to the position of the sun.

Meteorology

t

908070605040302010

2 431 65 87 9 10 11 12

H(t)

L(t)

Month (1 ↔ January)

Tem

pera

ture

(i

n de

gree

s Fa

hren

heit)

t � 1t

L�t� � 41.80 � 17.13 cos �t6

� 13.39 sin �t6

H�t� � 56.94 � 20.86 cos �t6

� 11.58 sin �t6

9781133105060_1403.qxd 12/3/11 12:33 PM Page 1026

■ Quiz Yourself 1027

QUIZ YOURSELF

Take this quiz as you would take a quiz in class. When you are done, check yourwork against the answers given in the back of the book.

In Exercises 1–4, express the angle in radian measure as a multiple of Use a calculator to verify your result.

1. 2. 3. 4.

In Exercises 5–8, express the angle in degree measure. Use a calculator to verifyyour result.

5. 6. 7. 8.

In Exercises 9–14, evaluate the trigonometric function without using a calculator.

9. 10.

11. 12.

13. 14.

In Exercises 15–17, solve the equation for Assume

15.

16.

17.

In Exercises 18–20, find the indicated side and/or angle.

18. 19. 20.

21. A map maker needs to determine the distance across a small lake. The distancefrom point A to point B is 500 feet and the angle is (see figure). What is

In Exercises 22–24, (a) sketch the graph and (b) determine the period of the function.

22. 23. 24.

25. A company that produces snowboards forecasts monthly sales for 1 year to be

where is the sales (in thousands of dollars) and is the time (in months), withcorresponding to January.


(b) Use the graph to determine the months of maximum and minimum sales.

S.

t � 1tS

S � 53.5 � 40.5 cos � t6

y � tan �x3

y � �2 cos 4xy � �3 sin 3x4

d?35��d

θ

25°12

aθ

50°

16

a

θ

60°10

a

5

sin2 � � 3 cos2 �

cos2 � � 2 cos � � 1 � 0

tan � � 1 � 0

0 � 2�.�.

csc 3�

2sec��60��

cot 45�tan 5�

6

cos 240�sin��

4�

11�

12�

4�

34�

152�

3

35��80�105�15�

�.


A C

B

500 ft d

θ = 35°

Figure for 21

9781133105060_1403.qxd 12/3/11 12:33 PM Page 1027


14.4 Derivatives of Trigonometric Functions■ Find derivatives of trigonometric functions.

■ Find the relative extrema of trigonometric functions.

■ Use derivatives of trigonometric functions to answer questions about real-life situations.

Derivatives of Trigonometric Functions

In Example 4 and Checkpoint 4 in the preceding section, you looked at two importanttrigonometric limits:

and

These two limits are used in the development of the derivative of the sine function.

This differentiation rule is illustrated graphically in Figure 14.31. Note that the slope ofthe sine curve determines the value of the cosine curve. If is a function of then theChain Rule version of this differentiation rule is

The Chain Rule versions of the differentiation rules for all six trigonometric functionsare listed below. To help you remember these differentiation rules, note that eachtrigonometric function that begins with a “c” has a negative sign in its derivative.

ddx

�sin u� � cos ududx

.

x,u

� cos x

� �cos x��1� � �sin x��0�

� cos x � lim�x→0

sin �x

�x � � sin x � lim�x→0

1 � cos �x

�x � � lim

�x→0 ��cos x ��sin �x

�x � � �sin x��1 � cos �x�x �

� lim�x→0

� cos x sin �x � �sin x��1 � cos �x�

�x

� lim�x→0

sin x cos �x � cos x sin �x � sin x

�x

ddx

�sin x� � lim�x→0

sin�x � �x� � sin x

�x

lim�x→0

1 � cos �x

�x� 0.

lim�x→0

sin �x

�x� 1

Derivatives of the Six Basic Trigonometric Functions

ddx

�csc u� � �csc u cot u dudx

ddx

�sec u� � sec u tan u dudx

ddx

�cot u� � �csc2 u dudx

ddx

�tan u� � sec2 u dudx

ddx

�cos u� � �sin u dudx

ddx

�sin u� � cos u dudx

1

−1

−1

1

y

x

x

y

ddx

π

π 2 π

y increasing,y ′ positive

y increasing,y ′ positive

y decreasing,y ′ negative

y ′ = 0

y ′ = 0

y ′ = 1y ′ = 1

y ′ = −1

y = sin x

y ′ = cos x

[sin x] = cos x

2 π

2 π

FIGURE 14.31

In Exercise 71 on page 1035, you will use derivatives of trigonometricfunctions to find the times of day atwhich the maximum and minimum

growth rates of a plant occur.

Blaj Gabriel/www.shutterstock.com

9781133105060_1404.qxp 12/3/11 12:34 PM Page 1028

Example 1 Differentiating Trigonometric Functions

Differentiate each function.

a. b. c.

SOLUTION

a. Letting you obtain and the derivative is

b. Letting you can see that So, the derivative is

c. Letting you have which implies that

Example 2 Differentiating a Trigonometric Function

Differentiate the function

SOLUTION Letting you obtain

Apply Cosine Differentiation Rule.

Substitute for u.

Simplify.


Differentiate the function

SOLUTION Begin by rewriting the function.

Write original function.

Rewrite.

Apply Power Rule.

Apply Tangent Differentiation Rule.

Checkpoints 1, 2, and 3


a. b. c.

d. e. ■y � sin3 xy � 2 cos x3

y � tan x2

y � sin�x2 � 1�y � cos 4x

� 4 tan3 x sec2 x

f��x� � 4�tan x�3 ddx

�tan x�

� �tan x�4

f�x� � tan4 x

f�x� � tan4 x.

� �6x sin 3x2.

� ��sin 3x2��6x�

3x2 � �sin 3x2 ddx

�3x2�

f��x� � �sin u dudx

u � 3x2,

f�x� � cos 3x2.

dydx

� sec2 ududx

� sec2 3xddx

�3x� � �sec2 3x��3� � 3 sec2 3x.

u� � 3,u � 3x,

dydx

� �sin ududx

� �sin�x � 1� ddx

�x � 1� � �sin�x � 1��1� � �sin�x � 1�.

u� � 1.u � x � 1,

dydx

� cos u dudx

� cos 2x ddx

�2x� � �cos 2x��2� � 2 cos 2x.

u� � 2,u � 2x,

y � tan 3xy � cos�x � 1�y � sin 2x

Section 14.4 ■ Derivatives of Trigonometric Functions 1029

TECH TUTOR

When you use a symbolic differentiation utility to differentiatetrigonometric functions, youcan easily obtain results thatappear to be different fromthose you would obtain byhand. Try using a symbolicdifferentiation utility todifferentiate the function inExample 3. How does yourresult compare with thegiven solution?

9781133105060_1404.qxp 12/3/11 12:34 PM Page 1029


Differentiate

SOLUTION


Apply Cosecant Differentiation Rule.

Simplify.

Checkpoint 4


a. b. ■


Differentiate

SOLUTION Begin by rewriting the function in rational exponent form. Then apply theGeneral Power Rule to find the derivative.

Rewrite with rational exponent.

Apply General Power Rule.

Apply Sine Differentiation Rule.

Simplify.

Checkpoint 5


a. b. ■


Differentiate

SOLUTION Using the Product Rule, you can write


Apply Product Rule.

Simplify.

Checkpoint 6


a. b. ■y � t sin 2ty � x2 cos x

� x cos x � sin x.

dydx

� x ddx

�sin x� � sin x ddx

�x�

y � x sin x

y � x sin x.

f�x� � 3tan 3x f�x� � cos 2x

�2 cos 4tsin 4t

� �12��sin 4t��1�2 �4 cos 4t�

f��t� � �12��sin 4t��1�2

ddt

�sin 4t�

f�t� � �sin 4t�1�2

f�t� � sin 4t.

y � cot x2y � sec 4x

� �12

csc x2

cot x2

dydx

� �csc x2

cot x2

ddx�

x2

y � csc x2

y � csc x2

.


STUDY TIP

Notice that all of the differentiation rules that youlearned in earlier chapters in the text can be applied totrigonometric functions. Forinstance, Example 5 uses the General Power Rule and Example 6 uses theProduct Rule.

Benjamin Thorn/www.shutterstock.com

9781133105060_1404.qxp 12/3/11 12:34 PM Page 1030

Relative Extrema of Trigonometric Functions

Recall that the critical numbers of a function are the values for whichor is undefined.

Example 7 Finding Relative Extrema

Find the relative extrema of

in the interval

SOLUTION To find the relative extrema of the function, begin by finding its criticalnumbers. The derivative of is

By setting the derivative equal to zero, you obtain So, in the interval the critical numbers are and Using the First-Derivative Test, youcan conclude that yields a relative minimum and yields a relative maximum,as shown in Figure 14.32.

Checkpoint 7

Find the relative extrema of

in the interval ■

Example 8 Finding Relative Extrema

Find the relative extrema of in the interval

SOLUTION


Differentiate.

Set derivative equal to 0.

Identity:

Factor.

From this, you can see that the critical numbers occur when and when So, in the interval the critical numbers are

Using the First-Derivative Test, you can determine that and arerelative maxima, and and are relative minima, as shown inFigure 14.33.

Checkpoint 8

Find the relative extrema of on the interval ■�0, 2��.y �12 sin 2x � cos x

�11��6, �32��7��6, �3

2��3��2, �1��2, 3�

x ��

2,

3�

2,

7�

6,

11�

6.

�0, 2��,sin x � �12.

cos x � 0

0 � 2�cos x��1 � 2 sin x�sin 2x � 2 cos x sin x 0 � 2 cos x � 4 cos x sin x

0 � 2 cos x � 2 sin 2x

f��x� � 2 cos x � 2 sin 2x

f�x� � 2 sin x � cos 2x

�0, 2��.f�x� � 2 sin x � cos 2x

�0, 2��.

y �x2

� cos x

5��3��3x � 5��3.x � ��3

�0, 2��,cos x �12.

dydx

�12

� cos x.

y

�0, 2��.

y �x2

� sin x

f��x�f��x� � 0x-y � f�x�


−2

−1

1

2

3

4

x

Relativemaximum

minimumRelative

x2

y

π 2π3

4π3

5π

y = − sin x

FIGURE 14.32

xπ2

4

3

2

1

−1

−2

−3

y

f(x) = 2 sin x − cos 2x

Relative maxima

Relative minima

(0, −1)

, 3

, −1

π

( )

( )

( )

, − 6 π7

2 π3

2 π

3 2 ( ), −

6 π11 3

2

2 π

(2 , −1)

FIGURE 14.33

9781133105060_1404.qxp 12/3/11 12:34 PM Page 1031

Applications

Example 9 Modeling Seasonal Sales

A fertilizer manufacturer finds that the sales of one of its fertilizer brands follow a seasonal pattern that can be modeled by

where is the amount sold (in pounds) and is the time (in days), with corresponding to January 1. On which day of the year is the maximum amount of fertilizer sold?

SOLUTION The derivative of the model is

Setting this derivative equal to zero produces

Because cosine is zero at and you can find the critical numbers as shown.

The 151st day of the year is May 31 and the 334th day of the year is November 30.From Figure 14.34, you can see that, according to the model, the maximum sales occuron May 31.

FIGURE 14.34

Checkpoint 9

Using the model from Example 9, find the rate at which sales are changing when ■t � 59.

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

31 59 90 120 151 181 212 243 273 304 334 365

Fert

ilize

r so

ld (

in p

ound

s)

t

200,000

150,000

100,000

50,000

F

Maximum sales

Minimum sales

Seasonal Pattern for Fertilizer Sales

F = 100,000 1 + sin[ [2 (t − 60)365

π

t � 334 t � 151

t �3�365�

4� 60 t �

3654

� 60

t � 60 �3�365�

4 t � 60 �

3654

2��t � 60� ��365��3��

2 2��t � 60� �

365�

2

2��t � 60�

365�

3�

2 2��t � 60�

365�

�

2

3��2,��2

cos 2��t � 60�

365� 0.

dFdt

� 100,000� 2�

365� cos 2��t � 60�

365.

t � 1tF

t � 0F � 100,000�1 � sin 2��t � 60�

365 ,


TECH TUTOR

Because of the difficulty of solving sometrigonometric equations,it can be difficult to findthe critical numbers of atrigonometric function. Forinstance, consider the function

Setting the derivative of thisfunction equal to zero produces

This equation is difficult tosolve analytically. So, it isdifficult to find the relativeextrema of analytically.With a graphing utility,however, you can estimatethe relative extrema graphically using the zoomand trace features. You canalso try other approximationtechniques, such as Newton’sMethod, which is discussedin Section 15.8.

f

2 cos x � 3 sin 3x � 0.

f�x� � 2 sin x � cos 3x.

9781133105060_1404.qxp 12/3/11 12:34 PM Page 1032

Example 10 Modeling Temperature Change

The temperature (in degrees Fahrenheit) during a given 24-hour period can be modeled by

where is the time (in hours), with corresponding to midnight, as shown in Figure 14.35. Find the rate at which the temperature is changing at 6 A.M.

FIGURE 14.35

SOLUTION The rate of change of the temperature is given by the derivative

Because 6 A.M. corresponds to the rate of change at 6 A.M. is

Checkpoint 10

In Example 10, find the rate at which the temperature is changing at 8 P.M. ■

� 3.4 per hour.

�5�

4 �32 �

15�

12 cos ��2�

12 � �5�

4 cos ��

6�t � 6,

dTdt

�15�

12 cos

��t � 8�12

.

A.M. P.M.

T

t2 4 6 8 10 12 14 16 18 20 22 24

100

90

80

70

60

50

T = 70 + 15 sin (t − 8)12

π

Temperature Cycle over a 24-Hour Period

Rate of change

Tem

pera

ture

(i

n de

gree

s Fa

hren

heit)

Time (in hours)

t � 0t

t � 0T � 70 � 15 sin ��t � 8�

12,

T



1. State the Sine, Tangent, and Secant Differentiation Rules (page 1028). Forexamples of using these rules, see Examples 1, 3, 5, and 6.

2. State the Cosine, Cotangent, and Cosecant Differentiation Rules (page 1028).For examples of using these rules, see Examples 1, 2, and 4.

3. Explain how to find the relative extrema of a trigonometric function (page 1031). For examples of finding the relative extrema of trigonometric functions, see Examples 7 and 8.

4. Describe a real-life example of how the derivative of a trigonometric function can be used to analyze the rate of change of temperature (page 1033, Example 10).

S.M./www.shutterstock.com

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The following warm-up exercises involve skills that were covered in earlier sections. You will usethese skills in the exercise set for this section. For additional help, review Sections 7.4, 7.6, 7.7, 8.5, and 14.2.

In Exercises 1–4, find the derivative of the function.

1. 2.

3. 4.

In Exercises 5 and 6, find the relative extrema of the function.

5. 6.

In Exercises 7–10, solve the equation for x. Assume

7. 8. 9. 10. sin x2

� �22

cos x2

� 0cos x � �12

sin x �32

0 x 2�.

f�x� �13 x3 � 4x � 2f�x� � x2 � 4x � 1

g�x� �2x

x2 � 5f�x� � �x � 1��x2 � 2x � 3�

g�x� � �x3 � 4�4f�x� � 3x3 � 2x2 � 4x � 7


Differentiating Trigonometric Functions In Exercises1–28, find the derivative of the trigonometric function.See Examples 1, 2, 3, 4, 5, and 6.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

Differentiating Trigonometric Functions In Exercises29–40, find the derivative of the function and simplifyyour answer by using the trigonometric identities listed inSection 14.2.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

Finding an Equation of a Tangent Line In Exercises41–48, find an equation of the tangent line to the graphof the function at the given point.

Function Point

41.

42.

43.

44.

45.

46.

47.

48. ��

6, 22 �y � sin x

�3�

2, 0�y � ln�sin x � 2�

�3�

2, 0�y � sin x cos x

�3�

4, �1�y � cot x

��

2, 1�y � csc2 x

��, 0�y � sin 4x

��

3, 2�y � sec x

��

4, �1�y � tan x

y �12 ln�cos2 x�y � ln�sin2 x�

y �sec7 x

7�

sec5 x5

y �sin3 x

3�

sin5 x5

y � cot x � xy � tan x � x

y � 3 sin x � 2 sin3 xy � sin2 x � cos 2x

y �x2

�sin 2x

4y � cos2 x � sin2 x

y �14 sin2 2xy � cos2 x

y � e�x cos x2

y � e2x sin 2x

y � �sin4 2xy � 2 tan2 4x

y � x2 cos 1x

y � x sin 1x

y � csc2 x � sec 3xy � cos 3x � sin2 x

y � e�2x2 cot xy � ex2 sec x

f�x� �sin x

xg�t� �

cos tt

y � �x � 3� csc x f�t� � t2 cos t

y � 3sin 6xy � tan 2x

f�t� � cos 2t

y � cot�2x � 1�

y �12 csc 2xy � sec �x

y � cos4 xy � sin2 x

g�x� � 3 cos x4 f�t� � tan 5t

y � sin�3x � 1�y � tan 4x3

f�x� � cos 2xy � sin x3

SKILLS WARM UP 14.4

9781133105060_1404.qxp 12/3/11 12:34 PM Page 1034


Implicit Differentiation In Exercises 49 and 50, useimplicit differentiation to find and evaluate thederivative at the given point.

Function Point

49.

50.

Slope of a Tangent Line In Exercises 51–56, find theslope of the tangent line to the given sine function at theorigin. Compare this value with the number of completecycles in the interval

51. 52.

53. 54.

55. 56.

Finding Relative Extrema In Exercises 57–66, find therelative extrema of the trigonometric function in the interval Use a graphing utility to confirm yourresults. See Examples 7 and 8.

57. 58.

59. 60.

61. 62.

63. 64.

65. 66.

Finding Second Derivatives In Exercises 67–70, findthe second derivative of the trigonometric function.

67. 68.

69. 70.

71. Biology Plants do not grow at constant rates during anormal 24-hour period because their growth is affectedby sunlight. Suppose that the growth of a certain plantspecies in a controlled environment is given by themodel

where is the height of the plant (in inches) and is thetime (in days), with corresponding to midnight ofday 1. During what time of day is the rate of growth ofthis plant

(a) a maximum? (b) a minimum?

72. Meteorology The normal average daily temperaturein degrees Fahrenheit for a city is given by

where is the time (in days), with correspondingto January 1. Find the expected date of

(a) the warmest day. (b) the coldest day.

73. Construction Workers The numbers (in thousands) of construction workers employed in theUnited States during 2010 can be modeled by

where is the time (in months), with correspondingto January. Approximate the month in which the numberof construction workers employed was a maximum.What was the maximum number of construction workersemployed? (Source: U.S. Bureau of Labor Statistics)

74. Transportation Workers The numbers (in thousands) of scenic and sightseeing transportationworkers employed in the United States during 2010 canbe modeled by

where is the time (in months), with correspondingto January. Approximate the month in which the number of scenic and sightseeing transportation workersemployed was a maximum. What was the maximumnumber of scenic and sightseeing transportation workersemployed? (Source: U.S. Bureau of Labor Statistics)

75. Meteorology The number of hours of daylight inNew Orleans can be modeled by

where is the time (in months), with correspondingto January. Approximate the month in which NewOrleans has the maximum number of daylight hours.What is this maximum number of daylight hours?(Source: U.S. Naval Observatory)

tt � 1t

D � 12.12 � 1.87 cos� �t � 5.83�

6

D

tt � 1t

W � 27.8 � 8.25 sin�0.561t � 2.5913�

W

tt � 1t

W � 5488 � 347.6 sin�0.45t � 4.153�

W

t � 1t

T � 55 � 21 cos 2� �t � 32�

365

t � 0th

h � 0.2t � 0.03 sin 2�t

y � csc2 �xy � cos x2

y � sec xy � x sin x

y � sin2 x � cos xy � x � 2 sin x

y � e�x sin xy � ex cos x

y � 2 cos x � cos 2xy � 2 sin x � sin 2x

y � sec x2

y � cos2 x

y � �3 sin x3

y � cos �x2

�0, 2��.

1

−1

x

y

2 π π 3

2 π 2 π−1

1

x

y

2 π π

y � sin x2

y � sin x

1

−1

x

y

3 2 π

2 π 2 π

−1

1

x

y

2 π

y � sin 3x2

y � sin 2x

1

−12

x

y

π π 2

1

−1

x

y

π

y � sin 5x2

y � sin 5x4

[0, 2�].

�0, 0�tan�x � y� � x

��

2,

�

4�sin x � cos 2y � 1

dy/dx

9781133105060_1404.qxp 12/3/11 12:34 PM Page 1035


76. Tides Throughout the day, the depth of water (inmeters) at the end of a dock varies with the tides. Thedepth for one particular day can be modeled by

where is the time (in hours), with correspondingto midnight.

(a) Determine

(b) Evaluate for and and interpretyour results.

(c) Find the time(s) when the water depth is the greatestand the time(s) when the water depth is the least.

(d) What is the greatest depth? What is the least depth?Did you have to use calculus to determine thesedepths? Explain your reasoning.

77. Think About It Rewrite the trigonometric function interms of sine and/or cosine and then differentiate toprove the following differentiation rules.

(a)

(b)

(c)

(d)

Analyzing a Function In Exercises 79–84, usea graphing utility to (a) graph f and in the sameviewing window over the specified interval, (b) findthe critical numbers of f, (c) find the interval(s) onwhich is positive and the interval(s) on which isnegative, and (d) find the relative extrema in theinterval. Note the behavior of f in relation to thesign of .

Function Interval

79.

80.

81.

82.

83.

84.


85. If then

86. If then

87. If then

88. The minimum value of

is �1.

y � 3 sin x � 2

y� � 2x sin x.y � x sin2 x,

f��x� � 2�sin 2x��cos 2x�. f�x� � sin2�2x�,y� �

12�1 � cos x��1�2.y � �1 � cos x�1�2,

�0, 2��f�x� � ln x cos x

�0, 2��f�x� � 2x sin x

�0, ��f�x� � x sin x

�0, ��f�x� � sin x �13 sin 3x �

15 sin 5x

�0, 2��f�x� �x2 � 2sin x

� 5x

�0, 2��f�t� � t2 sin t

f�

f�f�

f�

ddx

�csc x� � �csc x cot x

ddx

�cot x� � �csc2 x

ddx

�sec x� � sec x tan x

ddx

�tan x� � sec2 x

t � 20,t � 4dD�dt

dD�dt.

t � 0t

0 t 24D � 3.5 � 1.5 cos �t6

,

D

78. HOW DO YOU SEE IT? The graph shows the height (in feet) above ground of a seat on a Ferris wheel at time (in seconds).

(a) What is the period of the model? What does the period tell you about the ride?

(b) Find the intervals on which the height is increasing and decreasing.

(c) Estimate the relative extrema in the interval[0, 60�.

5 10 15 20 25 30 35 40 45 50 55 60

25

50

75

100

125

t

h

Time (in seconds)

Hei

ght (

in f

eet)

Ferris Wheel

th

89. Project: Meteorology For a project analyzingthe mean monthly temperature and precipitation in Sioux City, Iowa, visit this text’s website at www.cengagebrain.com. (Source: NationalOceanic and Atmospheric Administration)

9781133105060_1404.qxp 12/3/11 12:34 PM Page 1036

Section 14.5 ■ Integrals of Trigonometric Functions 1037

14.5 Integrals of Trigonometric Functions ■ Learn the trigonometric integration rules that correspond directly to

differentiation rules.

■ Integrate the six basic trigonometric functions.

■ Use trigonometric integrals to solve real-life problems.

Trigonometric Integrals

For each trigonometric differentiation rule, there is a corresponding integration rule.For instance, corresponding to the differentiation rule

is the integration rule

and corresponding to the differentiation rule

is the integration rule

� sin u du � �cos u � C.

ddx


� cos u du � sin u � C

ddx


Integrals Involving Trigonometric Functions

Differentiation Rule Integration Rule

� csc u cot u du � �csc u � Cddx


� csc2 u du � �cot u � Cddx


� sec u tan u du � sec u � Cddx


� sec2 u du � tan u � Cddx


� sin u du � �cos u � Cddx


� cos u du � sin u � Cddx


STUDY TIP

Note that the list above gives you formulas for integrating only two of the six trigonometric functions: the sine function and the cosine function. The list doesnot show you how to integrate the other four trigonometric functions. Rules for integrating those functions are discussed later in this section.

In Exercise 61 on page 1045, you will use integrals of trigonometric

functions to find the cost of cooling a house.

David Gilder/www.shutterstock.com

9781133105060_1405.qxp 12/3/11 12:35 PM Page 1037

Example 1 Integrating a Trigonometric Function

Find

SOLUTION Let Then

Apply Constant Multiple Rule.

Substitute for x and dx.

Integrate.

Substitute for u.


Find

SOLUTION Let Then

Rewrite integrand.

Substitute for and

Integrate.

Substitute for u.

Checkpoints 1 and 2

Find (a) and (b) ■


Find

SOLUTION Let Then

Multiply and divide by 3.

Substitute for and .

Integrate.

Substitute for u.

Checkpoint 3

Find ■� sec2 5x dx.

�13

sec 3x � C

�13

sec u � C

3 dx3x �13� sec u tan u du

� sec 3x tan 3x dx �13� �sec 3x tan 3x�3 dx

du � 3 dx.u � 3x.

� sec 3x tan 3x dx.

�4x3 cos x4 dx.�5 sin x dx

� �cos x3 � C

� �cos u � C

3x2 dx.x3 � � sin u du

� 3x2 sin x3 dx � � �sin x3�3x2 dx

du � 3x2 dx.u � x3.

� 3x2 sin x3 dx.

� 2 sin x � C

� 2 sin u � C

� 2� cos u du

� 2 cos x dx � 2� cos x dx

du � dx.u � x.

� 2 cos x dx.


TECH TUTOR

If you have access to a symbolic integration utility, try using it to integrate the functions inExamples 1, 2, and 3. Doesyour utility give the sameresults that are given in theexamples?

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9781133105060_1405.qxp 12/3/11 12:35 PM Page 1038


Find

SOLUTION Let Then

Rewrite integrand.

Substitute for and

Integrate.

Substitute for u.

Checkpoint 4

Find ■

The next two examples use the General Power Rule for integration and the GeneralLog Rule for integration. Recall from Chapter 11 that these rules are

General Power Rule

and

General Log Rule

The key to using these two rules is identifying the proper substitution for

Example 5 Using the General Power Rule

Find

SOLUTION Let Then

Rewrite integrand.

Integrate.

Substitute for

Simplify.

Checkpoint 5

Find ■� cos3 2x sin 2x dx.

�112

sin3 4x � C

u. �14

�sin 4x�3

3� C

�14

u3

3� C

�14� u2 du

� sin2 4x cos 4x dx �14� �sin 4x�2 �4 cos 4x� dx

du�dx � 4 cos 4x.u � sin 4x.

� sin2 4x cos 4x dx.

u.

� du�dx

u dx � ln�u� � C.

n � �1� un dudx

dx �un�1

n � 1� C,

� 2 csc 2x cot 2x dx.

� tan ex � C

� tan u � C

ex dx.ex � � sec2 u du

� ex sec2 ex dx � � �sec2 ex�ex dx

du � ex dx.u � ex.

� ex sec2 ex dx.


Substitute for and4 cos 4x dx.

sin 4x

u2 du�dx

STUDY TIP

Remember to check youranswers to integration problems by differentiating.In Example 5, for instance,try differentiating the answer

You should obtain the originalintegrand, as shown.

� sin2 4x cos 4x

y� �112

3�sin 4x�2�cos 4x�4

y �112

sin3 4x � C.

9781133105060_1405.qxp 12/3/11 12:35 PM Page 1039

Example 6 Using the Log Rule

Find

SOLUTION Let Then

Rewrite integrand.

Substitute for and

Apply Log Rule.

Substitute for

Checkpoint 6

Find ■

Example 7 Evaluating a Definite Integral

Evaluate

SOLUTION

Checkpoint 7

Find ■

Example 8 Finding Area by Integration

Find the area of the region bounded by the -axis and the graph of

for

SOLUTION As indicated in Figure 14.36, this area is given by

So, the region has an area of 2 square units.

Checkpoint 8

Find the area of the region bounded by the graphs of

and

for ■0 � x ��

2.

y � 0y � cos x

Area � ��

0sin x dx � �cos x

�

0� ��1� � ��1� � 2.

0 � x � �.

y � sin x

x

��2

0 sin 2x dx.

��4

0 cos 2x dx � 1

2 sin 2x

��4

0�

12

� 0 �12

��4

0 cos 2x dx.

� cos xsin x

dx.

u. � �ln�cos x� � C

� �ln�u� � C

�sin x.cos x � �� du�dx

u dx

� sin xcos x

dx � �� sin xcos x

dx

du�dx � �sin x.u � cos x.

� sin xcos x

dx.


1

x

y = sin x

y

2 π π

FIGURE 14.36

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Integrals of the Six Basic Trigonometric Functions

At the beginning of this section, the integration rules for the sine and cosine functionswere listed. Now, using the result of Example 6, you have an integration rule for the tangent function. That rule is

Integration formulas for the other three trigonometric functions can be developed in asimilar way. For instance, to obtain the integration formula for the secant function, youcan integrate as shown.

Substitute:

Apply Log Rule.

Substitute for u.

The integrals of the six basic trigonometric functions are summarized below.


Find

SOLUTION Let Then

Rewrite integrand.

Substitute for and

Integrate.

Substitute for

Checkpoint 9

Find ■�sec 2x dx.

u. � �14

ln�cos 4x� � C

� �14

ln�cos u� � C

4 dx.4x �14� tan u du

� tan 4x dx �14� �tan 4x�4 dx

du � 4 dx.u � 4x.

� tan 4x dx.

� ln�sec x � tan x� � C

� ln�u� � C

u � sec x � tan x. � � du�dx

u dx

� � sec2 x � sec x tan x

sec x � tan x dx

� sec x dx � � sec x�sec x � tan x�

sec x � tan x dx

� tan x dx � � sin xcos x

dx � �ln�cos x� � C.


Integrals of the Six Basic Trigonometric Functions

� csc u du � ln�csc u � cot u� � C� cot u du � ln�sin u� � C

� sec u du � ln�sec u � tan u� � C� tan u du � �ln�cos u� � C

� cos u du � sin u � C� sin u du � �cos u � C

9781133105060_1405.qxp 12/3/11 12:35 PM Page 1041

Application

In the next example, recall from Section 11.4 that the average value of a function overan interval is given by

Example 10 Finding an Average Temperature

The temperature (in degrees Fahrenheit) during a 24-hour period can be modeled by

where is the time (in hours), with corresponding to midnight. Will the averagetemperature during the four-hour period from noon to 4 P.M. be greater than

SOLUTION To find the average temperature use the formula for the average valueof a function on an interval.

So, the average temperature is about as indicated in Figure 14.37. You can concludethat the average temperature from noon to 4 P.M. will be greater than

Checkpoint 10

Use the function in Example 10 to find the average temperature from 9 A.M. to noon. ■

85.89.2,

� 89.2

� 72 �54�

�14 �288 �

216�

�1472�16� � 18�12

� �12 � 72�12� � 18�12

� �12

�1472t � 18�12

� ��cos ��t � 8�

12 16

12

A �14�

16

12 72 � 18 sin

��t � 8�12 dt

A,

85?t � 0t

T � 72 � 18 sin ��t � 8�

12

T

1b � a

�b

a

f�x� dx.

�a, b�f


100 90 80 70 60 50 40

30 20 10

t

T

Average 89.2≈ °

T = 72 + 18 sin(t − 8)12

π

Time (in hours)

Tem

pera

ture

(in

deg

rees

Fah

renh

eit)

4 8 12 16 20 24

Average Temperature

FIGURE 14.37


1. For each trigonometric integration rule on page 1037, state the correspondingdifferentiation rule (page 1037). For examples of using these integrationrules, see Examples 1–8.

2. State the integration rules for the six basic trigonometric functions (page 1041). For examples of using these integration rules, see Examples 1,2, 5, 7, 8, and 9.

3. Describe a real-life example of how the integral of a trigonometric functioncan be used to find an average temperature (page 1042, Example 10).

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9781133105060_1405.qxp 12/3/11 12:35 PM Page 1042


The following warm-up exercises involve skills that were covered in earlier sections. You will usethese skills in the exercise set for this section. For additional help, review Sections 11.4 and 14.2.

In Exercises 1–8, evaluate the trigonometric function.

1. 2.

3. 4.

5. 6.

7. 8.

In Exercises 9–16, simplify the expression using the trigonometric identities.

9. 10.

11. 12.

13. 14.

15. 16.

In Exercises 17–20, evaluate the definite integral.

17. 18.

19. 20. �1

0 x�9 � x2� dx�2

0 x�4 � x2� dx

�1

�1 �1 � x2� dx�4

0 �x2 � 3x � 4� dx

cot x�sin2 x�cot x sec x

cot x cos ��

2� x sec x sin��

2� x

sin2 x�csc2 x � 1�cos2 x�sec2 x � 1�csc x cos xsin x sec x

cos �

2sec �

cot 5�

3tan

5�

6

cos��

6 sin��

3

sin 7�

6cos

5�

4


Integrating Trigonometric Functions In Exercises1–32, find the indefinite integral. See Examples 1, 2, 3, 4,5, 6, and 9.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32. � �1 � tan �2 d� �sin x � cos x�2 dx

� esec 4x sec 4x tan 4x dx� esin�x cos �x dx

� e�x tan e�x dx� ex sin ex dx

� sin x

cos2 x dx�

csc2 xcot3 x

dx

� csc3 5x cot 5x dx� sec6 x4

tan x4

dx

� sec x2

dx� csc 2x dx

� tan 5x dx� cot �x dx

� 1 � cos � sin

d� sin x

1 � cos x dx

� cos t

1 � sin t dt�

sec x tan xsec x � 1

dx

� csc2 xcot x

dx� sec2 xtan x

dx

� �cot x csc2 x dx� tan3 x sec2 x dx

� csc x3

cot x3

dx� sec 2x tan 2x dx

� csc2 4x dx� sec2 x5

dx

� 2x sin x2 dx� 4x3 cos x4 dx

� cos 6x dx� sin 2x dx

� 8 cos x dx� 4 sin x dx

SKILLS WARM UP 14.5

9781133105060_1405.qxp 12/3/11 12:35 PM Page 1043


Integration by Parts In Exercises 33–38, use integration by parts to find the indefinite integral.

33. 34.

35. 36.

37. 38.

Evaluating Definite Integrals In Exercises 39–46,evaluate the definite integral. See Example 7.

39. 40.

41. 42.

43.

44.

45.

46.

Finding Area by Integration In Exercises 47–52, findthe area of the region. See Example 8.

47. 48.

49. 50.

51. 52.

Finding the Area Bounded by Two Graphs InExercises 53–56, sketch the region bounded by thegraphs of the functions and find the area of the region.

53.

54.

55.

56.

57. Meteorology The average monthly precipitation (in inches), including rain, snow, and ice, forSacramento, California can be modeled by

where is the time (in months), with correspondingto January. Find the total annual precipitation forSacramento. (Source: National Climatic Data Center)

58. Meteorology The average monthly precipitation (in inches), including rain, snow, and ice, for Bismarck,North Dakota can be modeled by

where is the time (in months), with correspondingto January. Find the total annual precipitation forBismarck. (Source: National Climatic Data Center)

59. Consumer Trends Energy consumption in theUnited States is seasonal. The primary residential energy consumption (in trillion Btu) in the UnitedStates during 2009 can be modeled by

where is the time (in months), with correspondingto January. Find the average primary residential energyconsumption during

(a) the first quarter

(b) the fourth quarter

(c) the entire year

(Source: U. S. Energy Information Administration)

60. Construction Workers The number (in thousands)of construction workers employed in the United Statesduring 2010 can be modeled by

where is the time (in months), with correspondingto January. Find the average number of constructionworkers during

(a) the first quarter

(b) the second quarter

(c) the entire year

(Source: U.S. Bureau of Labor Statistics)

�0 � t � 12�.�3 � t � 6�.

�0 � t � 3�.

t � 1t

W � 5488 � 347.6 sin�0.45t � 4.153)

W

�0 � t � 12�.�9 � t � 12�.

�0 � t � 3�.

t � 1t

0 � t � 12Q � 936 � 737.3 cos�0.31t � 0.928�,

Q

t � 1t

0 � t � 12P � 1.07 sin�0.59t � 3.94� � 1.52,

P

t � 1t

0 � t � 12P � 2.47 sin�0.40t � 1.80� � 2.08,

P

y � sec2 x4

, y � 4 � x2, x � �1, x � 1

y � 2 sin x, y � tan x, x � 0, x ��

3

y � sin x, y � cos 2x, x � ��

2, x �

�

6

y � cos x, y � 2 � cos x, x � 0, x � 2�

x

y

3

2

1

2π π

1

x

y

2π π

y � 2 sin x � sin 2xy � sin x � cos 2x

x

y

2π π

0.5 1

1.5 4 3 2 1

x

y

2 π π

y �x2

� cos xy � x � sin x

x

y

−1

1

8 π

4 π

1

x

y

π 2π

y � tan xy � cos x4

��4

0 sec x tan x dx

�1

0 tan�1 � x� dx

��8

0 sin 2x cos 2x dx

��4

��12 csc 2x cot 2x dx

��2

0 �x � cos x� dx�2��3

��2 sec2

x2

dx

��6

0 sin 6x dx��4

0 cos

4x3

dx

� 4x csc2 x dx� t csc 3t cot 3t dt

�

3 sec tan d� 6x sec2 x dx

� x sin 5x dx� x cos 2x dx

9781133105060_1405.qxp 12/3/11 12:35 PM Page 1044


61. Cost The temperature (in degrees Fahrenheit) in ahouse is given by

where is the time (in hours), with correspondingto midnight. The hourly cost of cooling a house is $0.30per degree.

(a) Find the cost of cooling this house between 8 A.M.and 8 P.M., when the thermostat is set at F (see figure) by evaluating the integral

(b) Find the savings realized by resetting the thermostatto F (see figure) by evaluating the integral

62. Water Supply The flow rate (in thousands of gallons per hour) of water at a pumping station during aday can be modeled by

where is the time in hours, with correspondingto midnight.

(a) Find the average hourly flow rate from midnight tonoon

(b) Find the average hourly flow rate from noon to midnight

(c) Find the total volume of water pumped in one day.

63. Sales In Example 9 in Section 14.4, the sales of a seasonal product can be modeled by

where is the amount sold (in pounds) and is the time (in days), with corresponding to January 1.The manufacturer of this product wants to set up a manufacturing schedule to produce a uniform amounteach day. What should this amount be? (Assume thatthere are 200 production days during the year.)

Using Simpson’s Rule In Exercises 65 and 66,use the Simpson’s Rule program in Appendix I toapproximate the integral.

Integral n

65. 8

66. 20

True or False? In Exercises 67 and 68, determinewhether the statement is true or false. If it is false, explainwhy or give an example that shows it is false.

67.

68. 4� sin x cos x dx � 0

�b

a

sin x dx � �b�2�

a

sin x dx

��

0 �1 � cos2 x dx

��2

0 �x sin x dx

t � 1tF

t � 0F � 100,0001 � sin 2� �t � 60�

365 ,

�12 � t � 24�.

�0 � t � 12�.

t � 0t

0 � t � 24

R � 53 � 7 sin��t6

� 3.6 � 9 cos��t12

� 8.9 ,

R

Tem

pera

ture

(i

n de

gree

s Fa

hren

heit)

Time (in hours)

84 78 72 66 60

24 16 20 12 8 4

T

t

C � 0.3�18

10 72 � 12 sin

� �t � 8�12

� 78 dt.

78

Tem

pera

ture

(i

n de

gree

s Fa

hren

heit)

Time (in hours)

84 78 72 66 60

24 16 20 12 8 4

T

t

T = 72 + 12 sin (t − 8)12

π

C � 0.3�20

8 72 � 12 sin

� �t � 8�12

� 72 dt.

72C

t � 0t

T � 72 � 12 sin � �t � 8�

12

T

64. HOW DO YOU SEE IT? The graph shows the sales (in thousands of units) of a seasonalproduct, where is the time (in months), with

corresponding to January.

(a) Which is greater, the average monthly sales fromJanuary through March, or the average monthlysales from October through December? Explainyour reasoning.

(b) Estimate the average monthly sales for the entireyear. Explain your reasoning.

1 2 3 4 5 6 7 8 9 10 11 12

102030405060708090

t

S

Time (in months)

Sale

s (i

n th

ousa

nds

of u

nits

)

t � 1t

S

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Solving Trigonometric Equations

Solving a trigonometric equation requires the use of trigonometry, but it also requiresthe use of algebra. Some examples of solving trigonometric equations were presentedon pages 1012 and 1013. Here are several others.


Solve each trigonometric equation. Assume

a.

b.

c.

SOLUTION

a. Write original equation.

Subtract from each side.

Add sin to each side.

Combine like terms.

Divide each side by 2.

b. Write original equation.

Divide each side by 3.

Extract square roots.

c. Write original equation.

Subtract 2 cot x from each side.

Factor.

Setting each factor equal to zero, you obtain the solutions in the intervalas shown.

and

No solution is obtained from because are outside the range ofthe cosine function. So, the equation has two solutions

and

in the interval 0 � x � 2�.

x �3�

2x �

�

2

±�2cos x � ±�2

cos x � ±�2

cos2 x � 2 x ��

2,

3�

2

cos2 x � 2 � 0 cot x � 0

0 � x � 2�

cot x�cos2 x � 2� � 0

cot x cos2 x � 2 cot x � 0

cot x cos2 x � 2 cot x

0 � x � 2� x ��

6,

5�

6,

7�

6,

11�

6,

tan x � ±�33

tan2 x �13

3 tan2 x � 1

0 � x � 2� x �5�

4,

7�

4,

sin x � ��22

2 sin x � ��2

x sin x � sin x � ��2

�2 sin x � �sin x ��2

sin x � �2 � �sin x

cot x cos2 x � 2 cot x

3 tan2 x � 1

sin x � �2 � �sin x

0 � x � 2�.

ALGEBRA TUTOR xy

9781133105060_140R.qxp 12/3/11 12:36 PM Page 1046

■ Algebra Tutor 1047


Solve each trigonometric equation in the interval .

a.

b.

c.

SOLUTION

a. Write original equation.

Factor.

Set each factor equal to zero. The solutions in the interval are

and

b. Write original equation.

Pythagorean Identity

Multiply.

Combine like terms.

Multiply each side by

Factor.


and

c. Write original equation.

Double-Angle Identity

Remove parentheses.

Rewrite.

Factor.


and

t ��

6,

5�

6

t �3�

2. sin t �

12

sin t � �1 2 sin t � 1

sin t � 1 � 0 2 sin t � 1 � 0

�0, 2��

�2 sin t � 1��sin t � 1� � 0

2 sin2 t � sin t � 1 � 0

sin t � 1 � 2 sin2 t � 0

sin t � �1 � 2 sin2 t� � 0

sin t � cos 2t � 0

x ��

3,

5�

3

x � 0, 2�. cos x �12

cos x � 1 2 cos x � 1

cos x � 1 � 0 2 cos x � 1 � 0

�0, 2��

�2 cos x � 1��cos x � 1� � 0

�1. 2 cos2 x � 3 cos x � 1 � 0

�2 cos2 x � 3 cos x � 1 � 0

2 � 2 cos2 x � 3 cos x � 3 � 0

2�1 � cos2 x� � 3 cos x � 3 � 0

2 sin2 x � 3 cos x � 3 � 0

x ��

2. x �

7�

6,

11�

6

sin x � 1 sin x � �12

sin x � 1 � 0 2 sin x � 1 � 0

�0, 2��

�2 sin x � 1��sin x � 1� � 0

2 sin2 x � sin x � 1 � 0

sin t � cos 2t � 0

2 sin2 x � 3 cos x � 3 � 0

2 sin2 x � sin x � 1 � 0

�0, 2��

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After studying this chapter, you should have acquired the following skills. The exercise numbers are keyed to the Review Exercises that begin on page 1050.Answers to odd-numbered Review Exercises are given in the back of the text.*

Section 14.1 Review Exercises■ Find coterminal angles. 1–4

■ Convert from degree to radian measure and from radian to degree measure. 5–16

■ Use formulas relating to triangles. 17–20

■ Use formulas relating to triangles to solve real-life problems. 21, 22

Section 14.2■ Evaluate trigonometric functions. 23–32

Right Triangle Definition:

Circular Function Definition: Let be an angle in standard position with a point on the terminal ray of and

■ Use a calculator to approximate values of trigonometric functions. 33–40

■ Solve right triangles. 41–44

■ Solve trigonometric equations. 45–50

■ Use right triangles to solve real-life problems. 51, 52

Section 14.3■ Find the period and amplitude of trigonometric functions. 53–56

■ Sketch graphs of trigonometric functions. 57–64

■ Use trigonometric functions to model real-life situations. 65, 66

* A wide range of valuable study aids are available to help you master the material in this chapter. The Student Solutions Manual includes step-by-step solutions to all odd-numbered exercises to help you review and prepare. The student website at www.cengagebrain.com offers algebra help and a Graphing Technology Guide, which contains step-by-step commands and instructions for a wide variety of graphing calculators.

cot � �xy

tan � �yx

sec � �rx

cos � �xr

csc � �ry

sin � �yr

r � �x2 � y2 � 0.��x, y��

cot � �adjopp

tan � �oppadj

sec � �hypadj

cos � �adjhyp

csc � �hypopp

sin � �opphyp

0 < � <�

2

� radians � 180

SUMMARY AND STUDY STRATEGIES

9781133105060_140R.qxp 12/3/11 12:36 PM Page 1048

■ Summary and Study Strategies 1049

Section 14.4 Review Exercises■ Find derivatives of trigonometric functions. 67–84

■ Find the equations of tangent lines to graphs of trigonometric functions. 85–90

■ Find the relative extrema of trigonometric functions. 91–96

■ Use derivatives of trigonometric functions to answer questions about real-life situations. 97, 98

Section 14.5■ Find indefinite integrals of trigonometric functions. 99–110

■ Evaluate definite integrals of trigonometric functions. 111–118

■ Find the areas of regions in the plane. 119–122

■ Use trigonometric integrals to solve real-life problems. 123–126

Study Strategies■ Degree and Radian Modes When using a computer or calculator to evaluate or graph a trigonometric function,

be sure that you use the proper mode—radian mode or degree mode.

■ Checking the Form of an Answer Because of the abundance of trigonometric identities, solutions of problems in this chapter can take a variety of forms. For instance, the expressions and are equivalent. So, when you are checking your solutions with those given in the back of the text, remember that your solution mightbe correct, even if its form doesn’t agree precisely with that given in the text.

■ Using Technology Throughout this chapter, remember that technology can help you graph trigonometric functions,evaluate trigonometric functions, differentiate trigonometric functions, and integrate trigonometric functions. Consider,for instance, the difficulty of sketching the graph of the function below without using a graphing utility.

−3

3

−2 � 2 �

y = sin 2x + 2 sin x

ln�tan x� � C�ln�cot x� � C

� csc u du � ln�csc u � cot u� � C� cot u du � ln�sin u� � C

� sec u du � ln�sec u � tan u� � C� tan u du � �ln�cos u� � C

� csc u cot u du � �csc u � C� csc2 u du � �cot u � C

� sec u tan u du � sec u � C� sec2 u du � tan u � C

� sin u du � �cos u � C� cos u du � sin u � C

ddx


ddx


ddx


ddx


ddx


ddx


9781133105060_140R.qxp 12/3/11 12:36 PM Page 1049


Review Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

Finding Coterminal Angles In Exercises 1–4, determinetwo coterminal angles (one positive and one negative) foreach angle. Give the answers in degrees.

1. 2.

3. 4.

Converting from Degrees to Radians In Exercises5–12, express the angle in radian measure as a multipleof Use a calculator to verify your result.

5. 6.

7. 8.

9. 10.

11. 12.

Converting from Radians to Degrees In Exercises13–16, express the angle in degree measure. Use a calculator to verify your result.

13.

14.

15.

16.

Using Triangles In Exercises 17–20, solve the trianglefor the indicated side and/or angle.

17. 18.

19. 20.

21. Height A ladder of length 16 feet leans against theside of a house. The bottom of the ladder is 4.4 feetfrom the house (see figure). Find the height of the topof the ladder.

22. Length To stabilize a 75-foot tower for a radio antenna,a guy wire must be attached from the top of the tower toan anchor 50 feet from the base (see figure). How longis the wire?

Evaluating Trigonometric Functions In Exercises 23and 24, determine all six trigonometric functions of theangle

23. 24.

x

θ

(4, −2)

y

x

θ

(−3, 3)

y

�.

75 c

50

h16 ft

4.4 ft

h

b

12

35c

55 a

θ60°

1

c 3

60°

θ

b

4

8

θ

30°

�11�

6

�2�

3

5�

6

4�

3

320110

�540�480

�30�60

300340

�.

θ = −210°

θ = −405°

θ = 230°θ = 390°

9781133105060_140R.qxp 12/3/11 12:36 PM Page 1050

Evaluating Trigonometric Functions In Exercises25–32, evaluate the six trigonometric functions of theangle without using a calculator.

25. 26.

27. 28.

29. 30.

31. 32.

Evaluating Trigonometric Functions InExercises 33–40, use a calculator to evaluate thetrigonometric function to four decimal places.

33. 34.

35. 36.

37. 38.

39. 40.

Solving a Right Triangle In Exercises 41–44, solve forx, y, or r as indicated.



Solving Trigonometric Equations In Exercises 45–50,solve the equation for Assume For someof the equations, you should use the trigonometric identities listed in Section 14.2. Use the trace feature ofa graphing utility to verify your results.

45. 46.

47.

48.

49.

50.

51. Height The length of the shadow of a tree is 125 feetwhen the angle of elevation of the sun is (see figure). Approximate the height of the tree.

52. Length A guy wire runs from the ground to a celltower. The wire is attached to the cell tower 150 feetabove the ground. The angle formed between the wireand the ground is (see figure).

(a) How long is the guy wire?

(b) How far from the base of the tower is the guy wireanchored to the ground?

Finding the Period and Amplitude In Exercises53–56, find the period and amplitude of the trigonometricfunction.

53. 54.

55. 56.

−2−3

123

x

y

π 2π 4π−2−3

123

x

y

2π 2π

y � sin x2

y � �2 cos 3x2

−1−2−3

123

x2π π

y

−1−2−3

3

x

2π

y

y � cos 2xy � �2 sin 4x

= 43°

150 ft

θ

43

125 ft

h

33°

h33

2 sec2 � � tan2 � � 3 � 0

sec2 � � sec � � 2 � 0

cos3 � � cos �

2 sin2 � � 3 sin � � 1 � 0

2 cos2 � � 12 cos � � 1 � 0

0 � � � 2�.�.

r100

45°

x

25

20°

r.x.

30

y

30°

50

r

70°

y.r.

sin 224cos 105

cos�3�

7 sin��

9

csc 2�

9sec

12�

5

cot 216tan 33

5�

2�

11�

6

180�225

�4�

35�

3

240�45

■ Review Exercises 1051

9781133105060_140R.qxp 12/3/11 12:36 PM Page 1051


Graphing Trigonometric Functions In Exercises57–64, sketch the graph of the trigonometric function byhand.

57. 58.

59. 60.

61. 62.

63. 64.

65. Seasonal Sales The jet ski sales (in units) of acompany are modeled by



(b) Will the sales exceed 110 units during any month?If so, during which month(s)?

66. Seasonal Sales The bathing suit sales (inthousands of units) of a company are modeled by



(b) Will the sales exceed 42,000 units during anymonth? If so, during which month(s)?

Differentiating Trigonometric Functions In Exercises67–84, find the derivative of the trigonometric function.

67.

68.

69.

70.

71.

72.

73.

74.

75.

76.

77.

78.

79.

80.

81.

82.

83.

84.

Finding an Equation of a Tangent Line In Exercises85–90, find an equation of the tangent line to the graphof the function at the given point.

Function Point

85.

86.

87.

88.

89.

90.

Finding Relative Extrema In Exercises 91–96, findthe relative extrema of the trigonometric function in theinterval Use a graphing utility to confirm yourresults.

91.

92.

93.

94.

95.

96.

97. Seasonal Sales Refer to the model given inExercise 65. Approximate the month in which thesales of jet skis were a maximum. What was the maximum number of jet skis sold?

98. Seasonal Sales Refer to the model given inExercise 66.

(a) Approximate the month in which the sales ofbathing suits were a maximum. What was themaximum number of bathing suits sold?

(b) Approximate the month in which the sales ofbathings suits were a minimum. What was theminimum number of bathing suits sold?

t

t

t

f�x� �1

2 � sin x

f�x� � sin2 x � sin x

f�x� � sin x cos x

f�x� �x2

� cos x

y � cos 3x2

y � sin �x4

�0, 2��.

��, ��y � �x cos x

�

2,

12y �

12

sin2 x

�

2, 0y � sin 2x

3�

2, �1y � csc x

5�

8, 1y � tan 2x

�

4, 0y � cos 2x

y � esin x

y � ex cot x

y � sec2 2x

y � csc4 x

y � x cos x � sin x

y � sin2 x � x

y �sin�2x � 1�

x � 3

y �cos x

x2

y � csc 3x � cot 3x

y � �x tan x

y � 3�sec 5x

y � �cos 2�x

y � csc�3x � 4�

y � cot 3x2

5

y � sec 3x

y � tan 3x3

y � cos x4

y � sin 5�x

S.

t � 1t

S � 25 � 20 sin �t6

S

S.

t � 1t

S � 74 � 40 cos �t6

S

y � 3 csc 2xy � sec 2�x

y � 8 cos x4

y � 3 sin 2x5

y � cot x2

y �13

tan x

y � sin 2�xy � 2 cos 6x

9781133105060_140R.qxp 12/3/11 12:36 PM Page 1052

Integrating Trigonometric Functions In Exercises99–110, find the indefinite integral.

99.

100.

101.

102.

103.

104.

105.

106.

107.

108.

109.

110.

Evaluating Definite Integrals In Exercises 111–118,evaluate the definite integral.

111.

112.

113.

114.

115.

116.

117.

118.

Finding Area by Integration In Exercises 119–122,find the area of the region.

119. 120.

121. 122.

123. Meteorology The average monthly precipitation (in inches), including rain, snow, and ice, for DodgeCity, Kansas can be modeled by

where is the time (in months), with correspondingto January. Find the total annual precipitation for Dodge City. (Source: National Climatic DataCenter)

124. Health For a person at rest, the velocity (in litersper second) of air flow into and out of the lungs duringa respiratory cycle is given by

where is the time (in seconds). Find the volume inliters of air inhaled during one cycle by integrating this function over the interval

125. Sales The sales (in billions of dollars) for Lowe’sfor the years 2000 through 2009 can be modeled by

where is the year, with corresponding to 2000.Find the average sales for Lowe’s from 2000 through2009. (Source: Lowe’s Companies, Inc.)

126. Electricity The oscillating current in an electricalcircuit can be modeled by

where is measured in amperes and is measured in seconds. Find the average current for the time intervals(a) (b) and (c) 0 � t � 1

30.0 � t � 160,0 � t � 1

240,

tI

I � 2 sin�60�t� � cos�120�t�

t � 0t

S � 15.31 sin�0.37t � 1.27� � 33.66, 0 � t � 9

S

�0, 3�.

t

v � 0.9 sin �t3

v

t � 1t

P � 1.27 sin�0.58t � 2.05� � 1.98, 0 � t � 12

P

3

2

1

x

y

4 π

2

1

x

y

4 π

2 π

y � 2 cos x � cos 2xy � 2 sin x � cos 3x

1

x

y

4 π

2 π

x

y

1

6π

y � cot xy � sin 3x

��

0 2x sin x2 dx

�� 2

�� 2 �2x � cos x� dx

�� 3

� 6csc x cot x dx

�� 3

�� 3 4 sec x tan x dx

�� 2

� 6 csc2 x dx

�� 6

�� 6 sec2 x dx

��

��

12

�1 � cos 2x� dx

��

0 �1 � sin x� dx

� ecos 3x sin 3x dx

� sin3 x cos x dx

� csc 3x4

dx

� tan 3x dx

� cos xsin3 x

dx

� sec2 2xtan 2x

dx

� sec 8x tan 8x dx

� x csc2 x2 dx

� 2x sec2 x2 dx

� csc 5x cot 5x dx

� cos x4

dx

� sin 3x dx

■ Review Exercises 1053

9781133105060_140R.qxp 12/3/11 12:36 PM Page 1053


TEST YOURSELF

Take this test as you would take a test in class. When you are done, check yourwork against the answers given in the back of the book.

In Exercises 1–6, copy and complete the table. Use a calculator if necessary.

Function Function Value

1. sin

2. cos

3. tan

4. cot

5. sec

6. csc

7. A digital camera tripod has a height of 25 inches, and an angle of is formedbetween the vertical and the leg of length (see figure). What is

In Exercises 8–10, solve the equation for Assume

8. 9. 10.

In Exercises 11–13, sketch the graph of the trigonometric function by hand.

11. 12. 13.

In Exercises 14–16, (a) find the derivative of the trigonometric function and (b) findthe relative extrema of the trigonometric function in the interval

14. 15. 16.

In Exercises 17–19, find the indefinite integral.

17. 18. 19.

In Exercises 20–23, evaluate the definite integral.

20. 21.

22. 23.

24. The monthly sales (in thousands of dollars) of a company that produces insect repellent can be modeled by

where is the time (in months), with corresponding to January.

(a) Find the total sales during the year

(b) Find the average monthly sales from April through October �3 � t � 10�.�0 � t � 12�.

t � 1t

S � 20.3 � 17.2 cos �t6

S

�3� 8

� 4sin 4x cos 4x dx�5� 6

3� 4csc 2x cot 2x dx

��

0 sec2

x3

tan x3

dx�1 2

1 4 cos �x dx

� x csc x2 dx� sec2 x4

dx� sin 5x dx

y �1

3 � sin�x � ��y � secx ��

4y � cos x � cos2 x

�0, 2��.

y � cot �x5

y � 4 cos 3�xy � 3 sin 2x

csc � � �3 sec �cos2 � � sin2 � � 02 sin � ��2 � 0

0 � � � 2�.�.

�?�24

��5�

4�

��40

��

6�

��15

��

5�

��67.5

� �rad�� deg�


25 in.24°

Figure for 7

9781133105060_140R.qxp 12/3/11 12:36 PM Page 1054

14.2 the trigonometric functions - cengage learning

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