Trigonometric Functions

1022

14 TrigonometricFunctions

The amounts of precipitation during certain times of the year can be modeled by trigonometric functions. (See Section 14.5, Exercise 57.)

Trigonometric functions have many real-life applications. The applications listed below represent a sample of the applications in this chapter.

Sprinkler System, Exercise 53, page 1030 Empire State Building, Exercise 69, page 1041 Make a Decision: Construction Workers, Exercise 79,

page 1050 Consumer Trends: Energy Consumption, Exercise 53,

page 1069 Inventory: Petroleum in the U.S., Exercise 55, page 1069

ApplicationsSt

ockb

yte/

Get

ty Im

ages

14.1 Radian Measure ofAngles

14.2 The TrigonometricFunctions

14.3 Graphs ofTrigonometricFunctions

14.4 Derivatives ofTrigonometricFunctions

14.5 Integrals ofTrigonometricFunctions

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SECTION 14.1 Radian Measure of Angles 1023

Find coterminal angles.

Convert from degree to radian measure and from radian to degree measure.

Use formulas relating to triangles.

Angles and Degree Measure

As shown in Figure 14.1, an angle has three parts: an initial ray, a terminal ray,and a vertex. An angle is in standard position if its initial ray coincides with thepositive x-axis and its vertex is at the origin.

Figure 14.2 shows the degree measures of several common angles. Note that(the lowercase Greek letter theta) is used to represent an angle and its measure.

Angles whose measures are between and are acute, and angles whose measures are between and are obtuse. An angle whose measure is isa right angle, and an angle whose measure is is a straight angle.

FIGURE 14.2

Positive angles are measured counterclockwise beginning with the initial ray.Negative angles are measured clockwise. For instance, Figure 14.3 shows an anglewhose measure is

Merely knowing where an angles initial and terminal rays are located doesnot allow you to assign a measure to the angle. To measure an angle, you mustknow how the terminal ray was revolved. For example, Figure 14.3 shows that theangle measuring has the same terminal ray as the angle measuring Such angles are called coterminal.

315.45

45.

Acute angle:between 0 and 90

Obtuse angle:between 90 and 180

Right angle:quarter revolution

Straight angle:half revolution

= 30 = 90 = 135

= 180 = 360

Full revolution

1809018090

900

Section 14.1

Radian Measureof Angles

Initial ray

Term

inal ra

y

Vertex

FIGURE 14.1 Standard Positionof an Angle

= 45

= 315

FIGURE 14.3 Coterminal Angles

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1024 CHAPTER 14 Trigonometric Functions

Although it may seem strange to consider angle measures that are larger thansuch angles have very useful applications in trigonometry. An angle that is

larger than is one whose terminal ray has revolved more than one full revolution counterclockwise. Figure 14.4 shows two angles measuring more than

In a similar way, you can generate an angle whose measure is less thanby revolving a terminal ray more than one full revolution clockwise.

Example 1 Finding Coterminal Angles

For each angle, find a coterminal angle such that

a.

b.

c.

d.

SOLUTION

a. To find an angle coterminal to subtract as shown in Figure 14.5(a).

b. To find an angle that is coterminal to subtract as shown inFigure 14.5(b).

c. To find an angle coterminal to add as shown in Figure 14.5(c).

d. To find an angle that is coterminal to add as shown in Figure14.5(d).

(a) (b)

(c) (d)FIGURE 14.5

= 330

390

= 200

160

= 30750

= 90

450

390 2360 390 720 330

2360,390,

160 360 200

360,160,

750 2360 750 720 30

2360,750,

450 360 90

360,450,

390

160

750

450

0 < 360.

360360.

360360,

FIGURE 14.4

= 405

= 720

CHECKPOINT 1For each angle, find a coterminalangle such that

a.

b.

c.

d. 390

495

330

210

0 < 360.

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Radian Measure

A second way to measure angles is in terms of radians. To assign a radian measureto an angle consider to be the central angle of a circular sector of radius 1,as shown in Figure 14.6. The radian measure of is then defined to be the lengthof the arc of the sector. Recall that the circumference of a circle is given by

So, the circumference of a circle of radius 1 is simply and you can concludethat the radian measure of an angle measuring is In other words

or

Figure 14.7 gives the radian measures of several common angles.

FIGURE 14.7 Radian Measures of Several Common Angles

It is important for you to be able to convert back and forth between thedegree and radian measures of an angle. You should remember the conversionsfor the common angles shown in Figure 14.7. For other conversions, you can usethe conversion rule below.

30 = 6

90 = 180 =

360 = 2

2

45 = 4 60 = 3

180 radians.

360 2 radians

2.3602,

Circumference 2radius.

,


Angle Measure Conversion Rule

The degree measure and radian measure of an angle are related by theequation

Conversions between degrees and radians can be done as follows.

1. To convert degrees to radians, multiply degrees by

2. To convert radians to degrees, multiply radians by 180 radians.

radians180 .

180 radians.

The arclength of thesector is theradian measureof .

r = 1

FIGURE 14.6

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Example 2 Converting from Degrees to Radians

Convert each degree measure to radian measure.

a. b. c. d.

SOLUTION To convert from degree measure to radian measure, multiply thedegree measure by

a.

b.

c.

d.

Although it is common to list radian measure in multiples of this is notnecessary. For instance, if the degree measure of an angle is the radianmeasure is

Example 3 Converting from Radians to Degrees

Convert each radian measure to degree measure.

a. b. c. d.

SOLUTION To convert from radian measure to degree measure, multiply theradian measure by

a.

b.

c.

d. 92 radians 92 radians

180 degrees radians 810

116 radians

116 radians


74 radians

74 radians


2 radians

2 radians180 degrees radians 90

180 radians.

92

116

74

2

79.3 79.3 degrees radians180 degrees 1.384 radians.

79.3,,

270 270 degrees radians180 degrees 32 radians

540 540 degrees radians180 degrees 3 radians40 40 degrees radians180 degrees

29 radian

135 135 degrees radians180 degrees 34 radians

radians180.

27054040135

CHECKPOINT 3Convert each radian measure todegree measure.

a. b.

c. d.

34

32

76

53

Most calculators andgraphing utilities have

both degree and radian modes.You should learn how to useyour calculator to convert fromdegrees to radians, and viceversa. Use a calculator orgraphing utility to verify theresults of Examples 2 and 3.*

T E CHNO LOGY

*Specific calculator keystroke instructions for operations in this and other technology boxes can befound at college.hmco.com/info/larsonapplied.

CHECKPOINT 2Convert each degree measure toradian measure.

a.

b.c.

d. 15024045225

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Example 4 Finding the Area of a Triangle

Find the area of an equilateral triangle with one-foot sides.

SOLUTION To use the formula you must first find the height of the triangle, as shown in Figure 14.10. To do this, apply the Pythagorean Theorem tothe shaded portion of the triangle.

Pythagorean Theorem

Simplify.

Solve for h.

So, the area of the triangle is

A 12bh 12 132 34 square foot.

h 32

h2 34

h2 122 12

A 12 bh,


A Summary of Rules About Triangles

1. The sum of the angles of a triangle is

2. The sum of the two acute angles of a right triangle is

3. Pythagorean Theorem The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse, as shown in Figure 14.8.

4. Similar Triangles If two triangles are similar (have the same angle measures), then the ratios of the corresponding sides are equal, as shownin Figure 14.9.

5. The area of a triangle is equal to one-half the base times the height. Thatis,

6. Each angle of an equilateral triangle measures

7. Each acute angle of an isosceles right triangle measures

8. The altitude of an equilateral triangle bisects its base.

45.

60.

A 12bh.

90.

180.a

b

c

FIGURE 14.8 a2 b2 c2

B

a

b

A

h

12

1

b

FIGURE 14.10

CHECKPOINT 4Find the area of an isosceles righttriangle with a hypotenuse of feet.

2

1. The measure of an angle is Is the angle obtuse or acute?

2. Is the angle whose measure is coterminal to an angle whose measureis

3. What is the measure of a right angle? What is the measure of a straightangle?

4. Name the three parts of an angle.

315?45

35.

C O N C E P T C H E C K

Triangles

FIGURE 14.9ab

AB

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In Exercises 14, determine two coterminal angles (onepositive and one negative) for each angle. Give theanswers in degrees.

1. (a) (b)

2. (a) (b)

3. (a) (b)

4. (a) (b)

In Exercises 58, determine two coterminal angles(one positive and one negative) for each angle. Givethe answers in radians.

5. (a) (b)

6. (a) (b)

67

116

32

9

= 230 = 420

= 740 = 300

= 390 = 120

= 41

= 45


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 1.2 and 1.3.

In Exercises 1 and 2, find the area of the triangle.

1. Base: 10 cm; height: 7 cm 2. Base: 4 in.; height: 6 in.

In Exercises 36, let and represent the lengths of the legs, and let represent the length of the hypotenuse, of a right triangle. Solve for themissing side length.

3. 4. 5. 6.

In Exercises 710, let and represent the side lengths of a triangle. Usethe information below to determine whether the figure is a right triangle,an isosceles triangle, or an equilateral triangle.

7. 8.9. 10. a 1, b 1, c 2a 12, b 16, c 20

a 3, b 3, c 4a 4, b 4, c 4

cb,a,

b 8, c 10a 8, c 17a 3, c 5a 5, b 12

cba

Skills Review 14.1

Exercises 14.1 See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

1053715_1401.qxp 11/5/08 11:49 AM Page 1028


7. (a) (b)

8. (a) (b)

In Exercises 920, express the angle in radian measureas a multiple of Use a calculator to verify your result.

9. 10.11. 12.13. 14.15. 16.17. 18.19. 20.

In Exercises 2130, express the angle in degree measure.Use a calculator to verify your result.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

In Exercises 3134, find the indicated measure of theangle. Express radian measure as a multiple of

Degree Measure Radian Measure31.

32.

33.

34.

In Exercises 3542, solve the triangle for the indicatedside and/or angle.

35. 36.

37. 38.

39. 40.

41. 42.

In Exercises 4346, find the area of the equilateraltriangle with sides of length

43. 44.45. 46.

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

6

16 8

s 12 cms 5 fts 8 ms 4 in.

s.

2.5 a

2.5

60

2s

2 3

60

2 1

2

h

5 5

40

4

4s

60

343

8a

4

60

a

a288

45

5c

30

35

712

144

9

270

.

83

196

32

94

74

12

973

54

52

405330315270240201203152102706030

.

458 9

8

215 4

9

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48. 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?

49. Arc Length Let represent the radius of a circle, thecentral angle (measured in radians), and the length of thearc intercepted by the angle (see figure). Use the relation-ship and a spreadsheet to complete the table.

50. Arc Length The minute hand on a clock is incheslong (see figure). Through what distance does the tip of theminute hand move in 25 minutes?

51. Distance A tractor tire that is 5 feet in diameter is partially filled with a liquid ballast for additional traction.To check the air pressure, the tractor operator rotates thetire until the valve stem is at the top so that the liquid willnot enter the gauge. On a given occasion, the operator notesthat the tire must be rotated to have the stem in theproper position (see figure).(a) Find the radian measure of this rotation.

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

52. Speed of Revolution A compact disc can have anangular speed up to 3142 radians per minute.(a) At this angular speed, how many revolutions per minute

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

revolutions?

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

For a circle of radius the area of a sector of the circle with central angle (measured in radians) is

given by

53. Sprinkler System A sprinkler system on a farm is set tospray water over a distance of 70 feet and rotates throughan angle of Find the area of the region.

54. Windshield Wiper A cars rear windshield wiperrotates The wiper mechanism has a total length of 25 inches and wipes the windshield over a distance of 14 inches. Find the area covered by the wiper.

True or False? In Exercises 5558, determinewhether the statement is true or false. If it is false,explain why or give 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. 3575

125.

120.

A 12r2.

Ar,

r

s

80

80

d

s

3 12 in.

312

r

s = r

sr

s

r

200c

125

r 8 ft 15 in. 85 cm

s 12 ft 96 in. 8642 mi

1.6 34

4 23

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SECTION 14.2 The Trigonometric Functions 1031

Recognize trigonometric functions.

Use 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 theother case these functions are defined in terms of a point on the terminal side of an arbitrary angle. The first approach is the one generally used in surveying,navigation, and astronomy, where a typical problem involves a triangle, three ofwhose six parts (sides and angles) are known and three of which are to be determined. The second approach is the one normally used in science and economics, where the periodic nature of the trigonometric functions is emphasized. In the definitions below, the six trigonometric functions are definedfrom both viewpoints.

Section 14.2

The TrigonometricFunctions

Opposite

Adjacent

Hypot

enuse

FIGURE 14.11

x

yr

x

(x, y)

yr = x2 + y2

FIGURE 14.12

Definitions of the Trigonometric Functions

Right Triangle Definition: (See Figure 14.11.)

Circular Function Definition: is any angle in standard position andis a point on the terminal ray of the angle. (See Figure 14.12.)

The full names of the trigonometric functions are sine, cosecant, cosine,secant, tangent, and cotangent.

cot x

ytan

yx

sec r

xcos

x

r

csc r

ysin y

r

x, y

cot adj.opp.tan

opp.adj.

sec hyp.adj.cos

adj.hyp.

csc hyp.opp.sin

opp.hyp.

0 < 0, tan < 0csc > 0, tan < 0cot < 0, cos > 0sin > 0, sec > 0sin > 0, cos < 0sin < 0, cos > 0

csc 4.25tan 3cos 57sec 2cot 5sin 13

.


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SECTION 14.2 The Trigonometric Functions 1041

67. Solve for r. 68. Solve for x.

69. Empire State Building You are standing 45 metersfrom the base of the Empire State Building. You estimatethat the angle of elevation to the top of the 86th floor is If the total height of the building is another 123 metersabove the 86th floor, what is the approximate height of thebuilding? One of your friends is on the 86th floor. What isthe distance between you and your friend?

70. Height A six-foot person walks from the base of abroadcasting tower directly toward the tip of the shadowcast by the tower. When the person is 132 feet from thetower and 3 feet from the tip of the shadow, the personsshadow starts to appear beyond the towers shadow.(a) Draw the right triangle that gives a visual representation

of the problem. Show the known quantities of the trian-gle and use a variable to indicate the height of the tower.

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

(c) What is the height of the tower?71. Length A 20-foot ladder leaning against the side of a

house makes a angle with the ground (see figure). Howfar up the side of the house does the ladder reach?

72. Width of a River A biologist wants to know the widthw of a river in order to set instruments to study the pollutants in the water. From point A the biologist walksdownstream 100 feet and sights to point C. From this sighting it is determined that (see figure). Howwide is the river?

73. Height of a Mountain In traveling across flat land,you notice a mountain directly in front of you. Its angle ofelevation (to the peak) is After you drive 13 milescloser to the mountain, the angle of elevation is Approximate the height of the mountain.

74. Distance From a 150-foot observation tower on thecoast, a Coast Guard officer sights a boat in difficulty. Theangle of depression of the boat is (see figure). How faris the boat from the shoreline?

75. Medicine The temperature T in degrees Fahrenheit of apatient t hours after arriving at the emergency room of ahospital at 10:00 P.M. is given by

Find the patients temperature at each time.(a) 10:00 P.M. (b) 4:00 A.M. (c) 10:00 A.M.At what time do you expect the patients temperature toreturn to normal? Explain your reasoning.

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

where S is measured in thousands of units and t is the timein months, with corresponding to January 2008. Usea graphing utility to estimate sales for each month.(a) February 2008 (b) February 2009(c) September 2008 (d) September 2009

In Exercises 77 and 78, use a graphing utility or aspreadsheet to complete the table. Then graph thefunction.

77. 78. f x 12 5 x 3 cos x

5f x 25 x 2 sin

x

5

t 1

S 23.1 0.442t 4.3 sin t6

0 t 18.Tt 98.6 4 cos t36,

3150 ft

Not drawn to scale

3

Not drawn to scale13 mi

3.5 9

9.3.5.

w

100 ft

= 50

A

C

50

20 ft

75

75

82.

x

3020

r 10

40

x 0 2 4 6 8 10

f x

1053715_1402.qxp 11/5/08 11:50 AM Page 1041

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 touse x (rather than ) as the independent variable. On the simplest level, you cansketch the graph of a function such as

by constructing a table of values, plotting the resulting points, and connectingthem with a smooth curve, as shown in Figure 14.21. Some examples of valuesare shown in the 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 tobe half of the difference between its maximum and minimum values. So, theamplitude of is 1.

The periodic nature of the sine function becomes evident when you observethat as x increases beyond the graph repeats itself over and over, continuouslyoscillating about the x-axis. The period of the function is the distance (on the x-axis) between successive cycles. So, the period of is

FIGURE 14.21

1

1

x

Amplitude = 1

Period = 2

f(x) = sin x

y

26

4

3

2

3 23

4 3

2 5

3 7

4

6115

6 7

6 5

4 4

3

2.fx sin x

2,

fx sin x

1.sin x

fx sin x


Section 14.3

Graphs ofTrigonometricFunctions

x 0

6

4

3

223

34

56

sin x 0.00 0.50 0.71 0.87 1.00 0.87 0.71 0.50 0.00

DISCOVERY

When the real number line iswrapped around the unit circle,each real number correspondswith a point on the circle. You can visualizethis graphically by setting yourgraphing utility to simultaneousmode. For instance, using radianand parametric modes as well,let

X1T = cos(T)Y1T = sin(T)X2T = TY2T = sin(T).

Use the viewing window settings shown below.

Tmin = 0Tmax = 6.3Tstep = .1Xmin = 2Xmax = 7Xscl = 1Ymin = 3Ymax = 3Yscl = 1

Now graph the functions. Noticehow the graphing utility tracesout the unit circle and the sinefunction simultaneously.

Try changing Y2T to cos T or tan T.

x, y cos t, sin tt

1053715_1403.qxp 11/5/08 11:51 AM Page 1042

Figure 14.22 shows the graphs of at least one cycle of all six trigonometricfunctions.

Familiarity with the graphs of the six basic trigonometric functions allowsyou to sketch graphs of more general functions such as

and

Note that the function oscillates between and a and so has anamplitude of

Amplitude of

Furthermore, because when and when itfollows that the function has a period of

Period of y a sin bx2b

.

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

y a sin bxa.

ay a sin bx

y a cos bx.

y a sin bx

SECTION 14.3 Graphs of Trigonometric Functions 1043

54

123

3

6

4

5

3

2

1

4

21

34

23

1

32

6

4

5

3

2

1

1

4

21

3

x

x

xx

xx

yyy

y = sin x y = cos x

y = tan x

1sin x

y y y

2

2

2

2

y = csc x = 1cos x

y = sec x = 1tan x

y = cot x =

Domain: all realsRange: [1, 1]Period: 2

Domain: all x nRange: (, 1] [1, ) Period: 2

Domain: all realsRange: [1, 1]Period: 2

Domain: all x + nRange: (, )Period:

2

Domain: all x nRange: (, ) Period:

Domain: all x + nRange: (, 1] [1, ) Period: 2

2

2

FIGURE 14.22 Graphs of the Six Trigonometric Functions

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4321

4321

y

x

Amplitude = 4

Period = 2

f(x) = 4 sin x

23

25

27

29

211

(0, 0)

FIGURE 14.23

32

1

1

2

3

x

Amplitude = 3

Period =

(0, 3)y

2

23

252

f(x) = 3 cos 2x

FIGURE 14.24

y

x

3Period =

f(x) = 2 tan 3x

6

6

2

65

67

32

4

FIGURE 14.25

Example 1 Graphing a Trigonometric Function

Sketch the graph of

SOLUTION The graph of has the characteristics below.

Amplitude: 4Period:

Three cycles of the graph are shown in Figure 14.23, starting with the point

CHECKPOINT 1Sketch 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 14.24, starting with the maximum point

CHECKPOINT 2Sketch the graph of


Sketch the graph of

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

Several cycles of the graph are shown in Figure 14.25, starting with the verticalasymptote

CHECKPOINT 3Sketch the graph of gx tan 4x.

x 6.

x . . . ,

6,

6,

2, 56 , . . . .

3.

fx 2 tan 3x.

gx 2 sin 4x.

0, 3.

22

fx 3 cos 2xfx 3 cos 2x.

gx 2 cos x.

0, 0.

2

fx 4 sin xf x 4 sin x.

Period 3

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Limits of Trigonometric Functions

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

When direct substitution with a trigonometric limit yields an indeterminate form,such as you can rely on technology to help evaluate the limit. The next exam-ple examines the limit 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 x-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 14.26 shows the graph of From this graph, it appears thatgets closer and closer to 1 as x approaches zero (from either side).

CHECKPOINT 4Use a calculator to evaluate the function

at several -values near Then use the result to estimate

limx0

1 cos xx

.

x 0.x

fx 1 cos xx

fxfx sin xx.

limx0

sin xx

1.

x 0.x

limx0

sin xx

.

x 0.

fx sin xx

00,

limx0

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

f(x) = sin xx

2

FIGURE 14.26

DISCOVERY

Try using the technique illustratedin Example 4 to evaluate

Can you hypothesize the limitof the general form

where is a positive integer?n

limx0

sin nxnx

limx0

sin 5x5x

.

1053715_1403.qxp 11/5/08 11:51 AM Page 1045


Applications

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

Example 5 Modeling Predator-Prey Cycles

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

and the population p 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 sizeof each population.

SOLUTION Each function has a period of 24 months. The predator populationhas an amplitude of 3000 and oscillates about the line The prey population has an amplitude of 5000 and oscillates about the line The graphs of the two models are shown in Figure 14.27. The cycles of this predator-prey population are explained by the diagram below.

FIGURE 14.27

CHECKPOINT 5Write the keystrokes required to graph correctly the predator-prey cycle fromExample 5 using a graphing utility.

20,000

15,000

10,000

5,000

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

Amplitude = 5000

Amplitude = 3000

P = 10,000 + 3000 sinPredatorPrey

Period = 24 months

Popu

latio

n

Time (in months)

Predator-Prey Cycles

2 t24

p = 15,000 + 5000 cos 2 t24

y 15,000.y 10,000.

t 0.p 15,000 5000 cos 2t24 ,

t 0P 10,000 3000 sin 2t24 ,

Predatorpopulation

increase

Preypopulationdecrease

Predatorpopulationdecrease

Preypopulation

increase

Graphing TrigonometricFunctions

A graphing utility allowsyou to explore the effects

of the constants and on the graph of a function ofthe form

After trying several values forthese constants, you can seethat determines the amplitude,

determines the period,determines the horizontal shift,and determines the verticalshift. Two examples are shownbelow. In each case, the graphof is compared with the graphof For instance, inthe first graph, notice that rela-tive to the graph of the graph of is shifted units to the right, stretched vertically by a factor of 2, andshifted up one unit. Similarly,in the second graph, notice that relative to the graph of

the graph of isshifted units to the right,stretched horizontally by a factor of stretched verticallyby a factor of 3, and shifteddown two units.

6

3y = sin x

y = 3 sin[ [ 2x 2

2 2

)) 2

2

3

3

y = sin x y = 2 sin )) 2 + 1x

2

12,

2fy sin x,

2fy sin x

y sin x.f

d

cba

f x a sinbx c d.

dc,b,a,

T E CHNO LOGY

1053715_1403.qxp 11/5/08 11:51 AM Page 1046

Example 6 Modeling Biorhythms

A popular theory that attempts to explain the ups and downs of everyday lifestates that each of us has three cycles, which begin at birth. These three cycles canbe modeled by sine waves

Physical (23 days):

Emotional (28 days):

Intellectual (33 days):

where is the number of days since birth. Describe the biorhythms during themonth of September 2007, for a person who was born on July 20, 1987.

SOLUTION Figure 14.28 shows the persons biorhythms during the month of September 2007. Note that September 1, 2007 was the 7348th day of the personslife.

CHECKPOINT 6Use a graphing utility to describe the biorhythms of the person in Example 6during the month of January 2007. Assume that January 1, 2007 is the 7105thday of the persons 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 2007

33-day cycle

Goodday

Badday

t = 7348July 20, 1987

7355 7369

Physical cycleEmotional cycleIntellectual cycle

FIGURE 14.28

1. What is the amplitude of

2. What is the period of

3. What does the amplitude of a sine function or a cosine function represent?

4. What does the period of a sine function or a cosine function represent?

fx cos x?

fx sin x?


1053715_1403.qxp 11/5/08 11:51 AM Page 1047

In Exercises 114, find the period and amplitude.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13.

14.

In Exercises 1520, find the period of the function.

15.16.17.18.

19.

20. y 5 tan 2x3

y cot x

6

y csc 4xy 3 sec 5xy 7 tan 2xy 3 tan x

y 23 cos

x

10

y 3 sin 4x

y 5 cos x4y 12 sin

2x3

y 13 sin 8xy 2 sin 10x

321

123

x

y

2

321

123

x

y

y cos 2x3y 2 sin x

8642

321

123

x

y

2

1

2

1

8642x

y

y 52 cos

x

2y 12 cos x

321

123

x

y

4

321

123

x

y

2 4

y 2 sin x3y 32 cos

x

2

321

123

x

y

321

321

x

y

2

y 3 cos 3xy 2 sin 2x



In Exercises 1 and 2, find the limit.

1. 2.

In Exercises 310, evaluate the trigonometric function without using a calculator.

3. 4. 5. 6.

7. 8. 9. 10.

In Exercises 1118, use a calculator to evaluate the trigonometric function tofour decimal places.

11. 12. 13. 14.15. 16. 17. 18. tan 140tan 327cos 72sin 103

cos 310sin 275sin 220cos 15

sin 43cos 53cos

56sin

116

cot 23tan

54sin cos

2

limx3

x3 2x2 1limx2

x2 4x 2

Skills Review 14.3


1053715_1403.qxp 11/5/08 11:51 AM Page 1048


In Exercises 2126, match the trigonometric functionwith the correct graph and give the period of the function. [The graphs are labeled (a)(f).]

(a) (b)

(c) (d)

(e) (f)

21. 22.

23. 24.

25. 26.

In Exercises 2736, sketch the graph of the function byhand. Use a graphing utility to verify your sketch.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

In Exercises 3746, sketch the graph of the function.

37. 38.

39. 40.

41. 42.

43. 44.45. 46.

In Exercises 4756, complete the table (using aspreadsheet or a graphing utility set in radian mode)to estimate

47. 48.

49. 50.

51. 52.

53. 54.

55. 56.

In Exercises 5760, use a graphing utility to graph thefunction and find

57. 58.

59. 60.

Graphical Reasoning In Exercises 6164, find and for such that the graph of matchesthe figure.

61. 62.

63. 64.

12

5

1

f

y

x

2

468

10

f

y

x

34

2

f

y

x

1

1

4

2

y

x

f

2

ffx a cos x 1 dda

f x tan 2x3xf x sin 5xsin 2x

f x sin 5x2xf x sin x2x

limx0

f x.f

f x 1 cos2 x

2x f x sin2 x

x

f x tan 4x3x f x tan 2x

x

f x 2 sinx4x

f x 31 cos xx

f x 1 cos 2xx

f x sin x5x

f x sin 2xsin 3xf x

sin 4x2x

limx0

f x.

y tan xy csc 2xy sec xy 2 sec 2x

y csc x

4y csc 2x3

y 3 tan xy cot 2x

y 10 cos x6y sin 2x

3

y 2 cot xy 2 tan x

y 32 sin

x

4y cos 2x

y 3 cos 4xy 2 sin 6x

y 32 cos

2x3y 2 cos

x

3

y 4 sin x3y sin x

2

y tan x

2y 2 csc x

2

y sec xy cot x

2

y 12 csc 2xy sec 2x

x

y

1 1 2 3 4

23

1233

21

123

x

y

4

45

1

x

y

2

2

1

x

y

2

4

x

y

3

1

2

x

y

x 0.1 0.01 0.001 0.001 0.01 0.1

f x

1053715_1403.qxp 11/5/08 11:51 AM Page 1049

Phase Shift In Exercises 6568, match the functionwith the correct graph. [The graphs are labeled (a)(d).]

(a) (b)

(c) (d)

65. 66.

67. 68.

69. Health For a person at rest, the velocity v (in liters persecond) of air flow into and out of the lungs during arespiratory 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.

70. Health After exercising for a few minutes, a person hasa respiratory cycle for which the velocity of air flow isapproximated by

Use this model to repeat Exercise 69.71. Music When tuning a piano, a technician strikes a tuning

fork for the A above middle C and sets up wave motion thatcan be approximated by where isthe 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.

72. Health The function approx-imates the blood pressure (in millimeters of mercury) attime 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.

73. Biology: Predator-Prey Cycle The population of apredator at time (in months) is modeled by

and the population p of its prey is modeled by

(a) Use a graphing utility to graph both models in the sameviewing window.

(b) Explain the oscillations in the size of each population.74. Biology: Predator-Prey Cycle The population of a

predator at time (in months) is modeled by

and the population p of its prey is modeled by

(a) Use a graphing utility to graph both models in the sameviewing window.

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

Sales In Exercises 75 and 76, sketch the graph of thesales function over 1 year where is sales in thousandsof units and t is the time in months, with corresponding to January.

75. 76.

77. Biorhythms For the person born on July 20, 1987, usethe biorhythm cycles given in Example 6 to calculate thispersons three energy levels on December 31, 2011. Assumethis is the 8930th day of the persons life.

78. Biorhythms Use your birthday and the concept ofbiorhythms as given in Example 6 to calculate your threeenergy levels on December 31, 2011. Use a spreadsheet tocalculate the number of days between your birthday andDecember 31, 2011.

79. MAKE A DECISION: CONSTRUCTION WORKERS Thenumber (in thousands) of construction workers employedin the United States during 2006 can be modeled by

where is the time in months, with corresponding toJanuary 1. (Source: U.S. Bureau of Labor Statistics)(a) Use a graphing utility to graph (b) Did the number of construction workers exceed

8 million in 2006? If so, during which month(s)?

W.

t 1tW 7594 455.2 sin0.41t 1.713

W

S 74.50 43.75 sin t6S 22.3 3.4 cos t6

t 1S

p 9800 2750 cos 2t24.

P 5700 1200 sin 2t24

tP

p 12,000 4000 cos 2t24.

P 8000 2500 sin 2t24

tP

tP

P 100 20 cos5t3

f 1p?fp

ty 0.001 sin 880t,

y 1.75 sin t2.

v < 0.v > 0,t

v 0.9 sin t3

y sinx 32 y sinx y sinx 2y sin x

y

x

2

1

1

y

x

2

1

1

y

x

23

2

1

1

y

x

23

2

1

1


1053715_1403.qxp 11/5/08 11:51 AM Page 1050


80. MAKE A DECISION: SALES The snowmobile sales (inunits) at a dealership are modeled by

where is the time in months, with corresponding toJanuary 1.(a) Use a graphing utility to graph (b) Will the sales exceed 75 units during any month? If so,

during which month(s)?81. Meteorology The normal monthly high temperatures

for Erie, Pennsylvania are approximated by

and the normal monthly low temperatures are approximated by

where t is the time in months, with corresponding toJanuary. (Source: NOAA) Use the figure to answer thequestions below.

(a) During what part of the year is the difference betweenthe normal high and low temperatures greatest? Whenis it smallest?

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

82. Finance: Cyclical Stocks The term cyclical stockdescribes the stock of a company whose profits are greatlyinfluenced by changes in the economic business cycle. Themarket prices of cyclical stocks mirror the general state ofthe economy and reflect the various phases of the businesscycle. Give a description and sketch the graph of a givencorporations stock prices during recurrent periods ofprosperity and recession. (Source: Adapted fromGarman/Forgue, Personal Finance, Eighth Edition)

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

(a) Which graph (A or B) has a longer wavelength, or period?(b) Which graph (A or B) has a greater amplitude?(c) The frequency of a graph is the number of oscillations or

cycles that occur during a given period of time. Whichgraph (A or B) has a greater frequency?

(d) Based on the definition of frequency, how are frequencyand period related?

(Source: Adapted from Shipman/Wilson/Todd, An Intro-duction to Physical Science, Eleventh Edition)

84. Biology: Predator-Prey Cycles The graph belowdemonstrates snowshoe hare and lynx population fluctua-tions. The cycles of each population follow a periodicpattern. Describe several factors that could be contributingto the cyclical patterns. (Source: Adapted from Levine/Miller, Biology: Discovering Life, Second Edition)

True or False? In Exercises 8588, determine whetherthe statement is true or false. If it is false, explain whyor give an example that shows it is false.

85. The amplitude of is

86. The period of is

87. 88. One solution of is 54.tan x 1lim

x0

sin 5x3x

53

32

.f x 5 cot4x3 3.f x 3 cos 2x

160 12

120 9

80 6

40 3

1860 1880 1900 1920 1940

Predator-Prey Cycles

Snowshoe hare

Lynx

Har

e po

pula

tion

(in th

ousa

nds

)

Lynx

pop

ula

tion

(in th

ousa

nds

)

Year

A

B

Wave amplitude

Wave amplitude

One wavelength

One wavelength

Particledisplacement

Velocity

Velocity

Particledisplacement

One wavelength

Meteorology

t

908070605040302010

2 431 65 87 9 10 11 12

H(t)

L(t)

Month (1 January)

Tem

pera

ture

(in de

gree

s Fa

hren

heit)

t 1

Lt 41.80 17.13 cos t6 13.39 sin t6

Ht 56.94 20.86 cos t6 11.58 sin t6

S.

t 1t

S 58.3 32.5 cos t6

S

1053715_1403.qxp 11/5/08 11:51 AM Page 1051


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

AC

B

500 ft d

= 35

Figure for 21

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

In Exercises 14, express the angle in radian measure as a multiple of Use acalculator to verify your result.

1. 2. 3. 4.

In Exercises 58, express the angle in degree measure. Use a calculator to verify your result.

5. 6. 7. 8.

In Exercises 914, evaluate the trigonometric function without using a calculator.

9. 10. 11.

12. 13. 14.

In Exercises 1517, solve the equation for

15.16.17.

In Exercises 1820, find the indicated side and/or angle.

18. 19. 20.

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

In Exercises 2224, (a) sketch the graph and (b) determine the period of thefunction.

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, with corresponding to January 1.(a) Use a graphing utility to graph (b) Use the graph to determine the months of maximum and minimum sales.

S.

t 1tS

S 53.5 40.5 cost6

y tan x

3y 2 cos 4xy 3 sin x

d?35d

404

a

50

16

a

6010

a

5

sin2 3 cos2 cos2 2 cos 1 0tan 1 0

0 } } 2.

csc32sec60cot 45

tan56cos 210sin

4

1112

43

415

23

358010515

.

1053715_1403.qxp 11/5/08 11:51 AM Page 1052

SECTION 14.4 Derivatives of Trigonometric Functions 1053

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 twoimportant trigonometric 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 14.29. Note that theslope of the sine curve determines the value of the cosine curve. If u is a functionof x, the Chain Rule version of this differentiation rule is

The Chain Rule versions of the differentiation rules for all six trigonometricfunctions are listed below.

ddx sin u cos u

dudx

.

cos x.

cos x1 sin x0

cos x limx0 sin xx sin x limx0

1 cos xx

limx0

cos x sin xx sin x 1 cos x

x lim

x0

sin x cos x cos x sin x sin xx

ddx sin x limx0

sinx x sin xx

limx0

1 cos xx

0.limx0

sin xx

1

Section 14.4

Derivatives ofTrigonometricFunctions

Derivatives of Trigonometric Functions

ddx csc u csc u cot u

dudx

ddx sec u sec u tan u

dudx

ddx cot u csc

2 u

dudx

ddx tan u sec

2 u

dudx

ddx cos u sin u

dudx

ddx sin u cos u

dudx

S T U D Y T I PTo help you remember these differentiation rules, note thateach trigonometric function thatbegins with a c has a negativesign in its derivative.

1

1

1

1

y

x

x

y

ddx

2

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

2

FIGURE 14.29

1053715_1404.qxp 11/5/08 11:52 AM Page 1053


Example 1 Differentiating Trigonometric Functions

Differentiate each function.

a. b. c.

SOLUTION

a. Letting you obtain

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

c. Letting you have which implies that

Example 2 Differentiating a Trigonometric Function

Differentiate

SOLUTION Letting you obtain

Apply Cosine Differentiation Rule.

Substitute for u.

Simplify.


Differentiate

SOLUTION By the Power Rule, you can write

CHECKPOINT 3Differentiate each function.

a.

b. y cos4 2x

y sin3 x

12 tan3 3x sec2 3x. 4tan3 3x3sec2 3x

ddx tan 3x

4 4tan 3x3 ddx tan 3x

f x tan4 3x.

6x sin 3x2. sin 3x26x

3x2 sin 3x2 ddx 3x2

fx sin u dudx

u 3x2,

f x cos 3x2.

dydx 3 sec

2 3x.

u 3,u 3x,

dydx sinx 1.

u 1.u x 1,

dydx cos u

dudx cos 2x

ddx 2x cos 2x2 2 cos 2x.

u 2x,

y tan 3xy cosx 1y sin 2x


a.

b.

c.

y tanx

2

y sinx 2 1

y cos 4x


a.

b. gx 2 cos x 3gx sinx

When you use a symbolicdifferentiation utility to

differentiate trigonometric functions, you can easily obtain results that appear to bedifferent from those you wouldobtain by hand. Try using asymbolic differentiation utilityto differentiate the function inExample 3. How does yourresult compare with the givensolution?

T E C H N O L O G Y

1053715_1404.qxp 11/5/08 11:52 AM Page 1054


Differentiate

SOLUTION

Write original function.

Apply Cosecant Differentiation Rule.

Simplify.


a. b.


Differentiate

SOLUTION Begin by rewriting the function in rational exponent form. Thenapply the General Power Rule to find the derivative.

Rewrite with rational exponent.

Apply General Power Rule.

Simplify.


Differentiate SOLUTION Using the Product Rule, you can write


Apply Product Rule.

Simplify.


a. b. y t sin 2ty x 2 cos x

x cos x sin x.

dydx x

ddx sin x sin x

ddx x

y x sin x

y x sin x.

2 cos 4tsin 4t

12sin 4t12 4 cos 4t ft 12sin 4t12

ddt sin 4t

ft sin 4t12

f t sin 4t.

y cot x 2y sec 4x

12 csc

x

2 cot x

2

dydx csc

x

2 cot x

2 ddx

x

2

y csc x

2

y csc x

2.



a.

b. fx 3tan 3x fx cos 2x

S T U D Y T I PNotice that all of the differentiation rules that youlearned in earlier chapters in the text can be applied totrigonometric functions. Forinstance, Example 5 uses theGeneral Power Rule and Example 6 uses the Product Rule.

1053715_1404.qxp 11/5/08 11:52 AM Page 1055


Relative Extrema of Trigonometric Functions

Example 7 Finding Relative Extrema

Find the relative extrema of

on the interval

SOLUTION To find the relative extrema of the function, begin by finding itscritical numbers. The derivative of y is

By setting the derivative equal to zero, you obtain So, in the intervalthe critical numbers are and Using the First-

Derivative Test, you can conclude that yields a relative minimum and yields a relative maximum, as shown in Figure 14.30.

Example 8 Finding Relative Extrema

Find the relative extrema of on 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 are relative maxima, and and are relative minima, asshown in Figure 14.31.

CHECKPOINT 8Find the relative extrema of on the interval 0, 2.y 12 sin 2x cos x

116, 3276, 3232, 12, 3

x

2, 32 ,

76 ,

116

.

0, 2,sin x 12.cos x 0

0 2cos x1 2 sin xsin 2x 2 cos x sin x 0 2 cos x 4 cos x sin x

0 2 cos x 2 sin 2x fx 2 cos x 2 sin 2x fx 2 sin x cos 2x

0, 2.fx 2 sin x cos 2x

533x 53.x 30, 2,cos x

12.

dydx

12 cos x.

0, 2.

y x

2 sin x

CHECKPOINT 7Find the relative extrema of

on the interval 0, 2.

y x

2 cos x

2

1

1

2

3

4

x

Relativemaximum

minimumRelative

x

2

y

23

43

5

y = sin x

FIGURE 14.30

x

2

4

3

2

1

1

2

3

y

f(x) = 2 sin x cos 2xRelativemaxima

Relativeminima

(0, 1)

, 3

, 1

( )

( )( )

, 67

23

2

32 ( ), 611 32

2

(2 , 1)

FIGURE 14.31

S T U D Y T I PRecall that the critical numbers of a function are the -values forwhich or is undefined.fx fx 0

xy fx

1053715_1404.qxp 11/5/08 11:52 AM Page 1056

Applications

Example 9 Modeling Seasonal Sales

A fertilizer manufacturer finds that the sales of one of its fertilizer brands followsa 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 November30. From the graph in Figure 14.32, you can see that, according to the model, themaximum sales occur on May 31.

FIGURE 14.32

CHECKPOINT 9Using the model from Example 9, find the rate at which sales are changingwhen t 59.

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec31 59 90 120 151 181 212 243 273 304 334 365Fe

rtiliz

er so

ld (i

n po u

nds

)

t

200,000

150,000100,000

50,000

F

Maximum sales

Minimum sales

Seasonal Pattern for Fertilizer Sales

F = 100,000 1 + sin[ [2 (t 60)365

t 3365

4 60 334 t 3654 60 151

t 60 33654 t 60 3654

2t 60365

32

2t 60365

2

32,2

cos 2t 60

365 0.

dFdt 100,000

2365 cos

2t 60365

.

t 1tF

t 0F 100,0001 sin 2t 60365 ,


Graphing TrigonometricFunctions

Because of the difficulty ofsolving some trigonometric

equations, it can be difficult to find the critical numbers of atrigonometric function. For example, consider the function

Setting the derivative of this function equalto zero produces

This equation is difficult to solveanalytically. So, it is difficult tofind the relative extrema of analytically. With a graphing utility,however, you can easily graph thefunction and use the zoom featureto estimate the relative extrema. Inthe graph shown below, notice thatthe function has three relative minima and three relative maximain the interval

Once you have obtained roughapproximations of the relativeextrema, you can further refine the approximations by applyingother approximation techniques,such as Newtons Method, which is discussed in Section 15.8, to theequation

Try using technology to locate therelative extrema of the function

How many relative extrema doesthis function have in the interval0, 2?

fx 2 sin x cos 4x.

fx 2 cos x 3 sin 3x 0.

3

0 2

3

f(x) = 2 sin x cos 3x

x = 4.28

x = 6.05

x = 5.38x = 2.91

x = 2.24x = 1.14

0, 2.

f

fx 2 cos x 3 sin 3x 0.

f x 2 sin x cos 3x.

T E C H N O L O G Y

1053715_1404.qxp 11/5/08 11:52 AM Page 1057


Example 10MAKE A DECISION Modeling Temperature Change

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

where t is the time (in hours), with corresponding to midnight, as shown inFigure 14.33. Find the rate at which the temperature is changing at 6 A.M.

FIGURE 14.33

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

3.4 per hour.

54 32

1512 cos

212

54 cos

6t 6,

dTdt

1512 cos

t 812

.

A.M. P.M.

T

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

1009080706050

T = 70 + 15 sin (t 8)12

Temperature Cycle over a 24-Hour Period

Rate of changeTem

pera

ture

(in de

gree

s Fa

hren

heit)

Time (in hours)

t 0

t 0T 70 15 sin t 812,

T

CHECKPOINT 10In Example 10, find the rate atwhich the temperature is changingat 8 P.M.

1. Given you know that the slope of the sine curve

determines the value of what curve?

2. In the differentiation rules for all six trigonometric functions, identifyeach trigonometric function that has a negative sign in its derivative.What do these functions have in common?

3. Can the General Power Rule and the Product Rule be applied to the differentiation of trigonometric functions?

4. Identify the trigonometric function whose derivative is sin u dudx

.

ddx[sin u] cos u du

dx,


1053715_1404.qxp 11/5/08 11:52 AM Page 1058


In Exercises 126, find the derivative of the function.

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.

In Exercises 2738, find the derivative of the functionand simplify your answer by using the trigonometricidentities listed in Section 14.2.

27. 28.

29. 30.

31. 32.33. 34.

35. 36.

37. 38.

In Exercises 3946, find an equation of the tangentline to the graph of the function at the given point.

Function Point

39.

40.

41.

42.

43.

44.

45.

46. 6, 22 y sin x32 , 0y lnsin x 232 , 0y sin x cos x34 , 1y cot x2, 1y csc2 x, 0y sin 4x

3, 2y sec x4, 1y tan x

y 12 lncos2 xy lnsin2 x

y sec7 x

7 sec5 x

5y sin3 x

3 sin5 x

5

y cot x xy tan x x

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

y x

2 sin 2x

4y cos2 x sin2 x

y 14 sin2 2xy cos2 x

y ex cos x

2y e2x

sin 2x

y sin4 2xy 2 tan2 4xy tan exy 3 tan 4x

y x2 cos 1x

y x sin 1x

y 12 csc 2xy sec xy csc2 x cos 2xy cos 3x sin2 xy ex sin xy ex2 sec xy x3 cot xy tan x x2

f x sin xx

gt cos tt

f x x 1 cos xf t t 2 cos tf x sin x cos xf x 4x 3 cos x

gt cos t 1t 2

y x2 cos x

y 5 sin xy 12 3 sin x

The following warm-up exercises involve skills that were covered in earlier sections. You willuse these 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 14, 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 710, solve the trigonometric equation for where

7. 8. 9. 10. sin x2 22cos

x

2 0cos x 12sin x

32

0 } x } 2.x

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

gx 2xx2 5f x x 1x

2 2x 3

gx x3 44f x 3x3 2x2 4x 7

Skills Review 14.4


1053715_1404.qxp 11/5/08 11:52 AM Page 1059

In Exercises 47 and 48, use implicit differentiation tofind and evaluate the derivative at the givenpoint.

Function Point

47.

48.

In Exercises 4952, show that the function satisfies thedifferential equation.

49.

50.

51. 52.

In Exercises 5358, find the slope of the tangent line to the given sine function at the origin. Compare this value with the number of complete cycles in theinterval

53. 54.

55. 56.

57. 58.

In Exercises 5964, determine the relative extrema ofthe function on the interval Use a graphingutility to confirm your result.

59. 60.61. 62.

63. 64.

65. Biology Plants do not grow at constant rates during anormal 24-hour period because their growth is affected bysunlight. Suppose that the growth of a certain plant speciesin a controlled environment is given by the model

where h is the height of the plant in inches and t is the timein days, with corresponding to midnight of day 1 (seefigure). During what time of day is the rate of growth ofthis plant(a) a maximum? (b) a minimum?

66. Meteorology The normal average daily temperature indegrees Fahrenheit for a city is given by

where t is the time in days, with corresponding toJanuary 1. Find the expected date of(a) the warmest day. (b) the coldest day.

67. Construction Workers The numbers (in thousands)of construction workers employed in the United Statesduring 2006 can be modeled by

where is the time in months, with corresponding toJanuary 1. Approximate the month in which the numberof construction workers employed was a maximum. Whatwas the maximum number of construction workersemployed? (Source: U.S. Bureau of Labor Statistics)

68. Amusement Park Workers The numbers (in thousands) of amusement park workers employed in theUnited States during 2006 can be modeled by

where is the time in months, with corresponding toJanuary 1. Approximate the month in which the numberof amusement park workers employed was a maximum.What was the maximum number of amusement park work-ers employed? (Source: U.S. Bureau of Labor Statistics)

tt 1t

W 139.8 37.33 sin0.612t 2.66

W

tt 1t

W 7594 455.2 sin0.4t 1.713

W

t 1

T 55 21 cos 2 t 32365

1.00.90.80.70.60.50.40.30.20.1

t

h

Time (in days)

Plant Growth

Hei

ght (

in inc

hes) h = 0.2t + 0.03 sin 2 t

1 2 3

t 0

h 0.2t 0.03 sin 2t

y sec x

2y ex cos x

y ex sin xy x 2 sin xy 2 cos x cos 2xy 2 sin x sin 2x

0, 2.

1

1

x

y

2 3

2 2

1

1x

y

2

y sin x2y sin x

1

1

x

y

32

2 2

1

1x

y

2

y sin 3x2y sin 2x

1

12

x

y

2

1

1

x

y

y sin 5x2y sin 5x4

[0, 2].

y 2y 2y 0y 4y 0y e x sin xy cos 2x sin 2x

xy y sin x

y 10 cos x

x

y y 0y 2 sin x 3 cos x

0, 0tanx y x

2,

4sin x cos 2y 1

dy/dx


1053715_1404.qxp 11/5/08 11:52 AM Page 1060


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

where represents the month, with corresponding toJanuary 1. Find the month in which New Orleans has themaximum number of daylight hours. What is this maximum number of daylight hours? (Source: U.S.Naval Observatory)

70. Tides Throughout the day, the depth of water in metersat the end of a dock varies with the tides. The depth for oneparticular day can be modeled by

where represents midnight.(a) Determine (b) Evaluate for and and interpret your

results.(c) Find the time(s) when the water depth is the greatest

and 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.

In Exercises 7176, use a graphing utility (a) to graph and on the same coordinate axes over the specifiedinterval, (b) to find the critical numbers of and (c) tofind the interval(s) on which is positive and the interval(s) on which it is negative. Note the behaviorof in relation to the sign of

Function Interval71.

72.

73.74.75.76.

In Exercises 7782, use a graphing utility to find therelative extrema of the trigonometric function. Let

77.

78.

79.

80.81.82.

True or False? In Exercises 8386, determine whetherthe statement is true or false. If it is false, explain whyor give an example that shows it is false.

83. If then 84. If then 85. If then 86. The minimum value of is

87. Extended Application To work an extended applicationanalyzing the mean monthly temperature and precipitation in Honolulu, Hawaii, visit this texts website at college.hmco.com. (Source: National Oceanic andAtmospheric Administration)

1.y 3 sin x 2y 2x sin x.y x sin2 x,

fx 2sin 2xcos 2x. f x sin22x,y 121 cos x12.y 1 cos x12,

f x sin xf x sin 0.1x2f x ln x sin x

f x ln x cos x

f x x2 2sin x 5x

f x xsin x

0 < x < 2.

0, 4f x 4e0.5x sin x0, 2f x 2x sin x0, f x x sin x0, f x sin x 13 sin 3x 15 sin 5x

0, 4f x x2 cos x

2

0, 2f t t 2 sin t

f.f

ff,

ff

t 20t 4dDdtdDdt.

t 0

0 t 24D 3.5 1.5 cos t6 ,

D

tt 0t

0 t 12D 12.13 1.87 cos t 0.076 ,

D

B u s i n e s s C a p s u l e

After a successful career as a critical carenurse, Grandee Ann Ray started Grand Ideas,a corporate gift, specialty, and promotionalproducts firm in Charleston, South Carolina. The company offers a wide variety of items,including office accessories, apparel, and glass-ware, that bear the logo of the client company.Ray started Grand Ideas from her home in 2001with little more than a cell phone, a fax machine,and minimal inventory. Today, the company hassales approaching $1.5 million per year, and shehas a team of 12 women working as full-timeand part-time employees and independent contractors.

88. Research Project Use your schools library, theInternet, or some other reference source to gatherinformation on a company that offers unique products or services to its customers. Collect dataabout the revenue that the company has generated,and find a mathematical model of the data. Write ashort paper that summarizes your findings.

Photo courtesy of Grandee Ann Ray/www.grandideas.net

1053715_1404.qxp 11/5/08 11:52 AM Page 1061


Section 14.5

Integrals ofTrigonometricFunctions

Find the six basic trigonometric integrals.

Solve trigonometric integrals.

Use trigonometric integrals to solve real-life problems.

The Six Basic Trigonometric Integrals

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

is the integration rule

The list below contains the integration formulas that correspond to the six basictrigonometric differentiation rules.

sin u du cos u C.

ddx cos u sin u

dudx

Integrals Involving Trigonometric Functions

Differentiation Rule Integration Rule

csc u cot u du csc u Cddx csc u csc u cot u dudx csc2 u du cot u Cddx cot u csc2 u dudx sec u tan u du sec u Cddx sec u sec u tan u dudx sec2 u du tan u Cddx tan u sec2 u dudx sin u du cos u Cddx cos u sin u dudx cos u du sin u Cddx sin u cos u dudx

S T U D Y T I PNote that this list gives you formulas for integrating only two of the sixtrigonometric functions: the sine function and the cosine function. The listdoes not show you how to integrate the other four trigonometric functions.Rules for integrating those functions are discussed later in this section.

1053715_1405.qxp 11/5/08 11:52 AM Page 1062

If you have access to asymbolic integration

utility, try using it to integratethe functions in Examples 1, 2,and 3. Does your utility givethe same results that are givenin the examples?

T E C H N O L O G Y

CHECKPOINT 1

Find

5 sin x dx.

CHECKPOINT 2

Find

4x3 cos x4 dx.

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.


Find

SOLUTION Let Then

Multiply and divide by 3.

Substitute for 3x and 3 dx.

Integrate.

Substitute for u.

CHECKPOINT 3

Find

sec2 5x dx.

13 sec 3x C

13 sec u C

13 sec u tan u du

sec 3x tan 3x dx 13 sec 3x tan 3x3 dxdu 3 dx.u 3x.

sec 3x tan 3x dx.

cos x3 C cos u C

3x2 dx.x 3 sin u du 3x2 sin x3 dx sin x33x2 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.

SECTION 14.5 Integrals of Trigonometric Functions 1063

1053715_1405.qxp 11/5/08 11:52 AM Page 1063

S T U D Y T I PIt is a good idea to check youranswers to integration problemsby differentiating. In Example 5,for instance, try differentiatingthe answer

You should obtain the originalintegrand, as shown.

sin2 4x cos 4x

y 112 3sin 4x

2cos 4x4

y 1

12 sin3 4x C.

CHECKPOINT 4

Find

2 csc 2x cot 2x dx.


Find

SOLUTION Let Then

Rewrite integrand.

Substitute for and

Integrate.

Substitute for u.

The next two examples use the General Power Rule for integration and theGeneral Log 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 u. Forinstance, in the next example, the proper choice for u is

Example 5 Using the General Power Rule

Find

SOLUTION Let Then

Rewrite integrand.

Integrate.

Substitute for u.

Simplify.

CHECKPOINT 5

Find

cos3 2x sin 2x dx.

112 sin

3 4x C

14

sin 4x33 C

14

u3

3 C

14 u2 du

sin2 4x cos 4x dx 14 sin 4x2 4 cos 4x dxdudx 4 cos 4x.u sin 4x.

sin2 4x cos 4x dx.

sin 4x.

dudxu

dx lnu C.

n 1 un dudx dx un1n 1 C,

tan ex C tan u C

ex dx.ex sec2 u du ex sec2 ex dx sec2 exex dx

du ex dx.u ex.

ex sec2 ex dx.


u2 dudx

Substitute for and4 cos 4x dx.

sin 4x

1053715_1405.qxp 11/5/08 11:52 AM Page 1064

1x

y = sin x

y

2

FIGURE 14.34

CHECKPOINT 6

Find

cos xsin x dx.

CHECKPOINT 7

Find

20

sin 2x dx.

Example 6 Using the Log Rule

Find

SOLUTION Let Then

Rewrite integrand.

Substitute for and

Apply Log Rule.

Substitute for u.

Example 7 Evaluating a Definite Integral

Evaluate

SOLUTION

Example 8 Finding Area by Integration

Find the area of the region bounded by the x-axis and one arc of the graph of

SOLUTION As indicated in Figure 14.34, this area is given by

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

CHECKPOINT 8Find the area of the region bounded by the graphs of and for

0 x 2.

y 0y cos x

2. 1 1

cos x

0

Area 0

sin x dx

y sin x.

12 0

12

40

cos 2x dx 12 sin 2x

4

0

40

cos 2x dx.

lncos x C lnu C

sin x.cos x dudxu

dx

sin xcos x

dx sin xcos x

dx

dudx sin x.u cos x.

sin xcos x

dx.


1053715_1405.qxp 11/5/08 11:52 AM Page 1065

Integrals of Trigonometric Functions

csc u du lncsc u cot u C cot u du lnsin u C sec u du lnsec u tan u C tan u du lncos u C

Other Trigonometric Integrals

At the beginning of this section, the integration rules for the sine and cosinefunctions were 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 developedin a similar way. For instance, to obtain the integration formula for the secantfunction, you can integrate as shown.

These formulas, and integration formulas for the other two trigonometricfunctions, are summarized below.


Find

SOLUTION Let Then

Rewrite integrand.

Substitute for 4x and 4 dx.

Integrate.

Substitute for u.

CHECKPOINT 9

Find

sec 2x dx.

14 lncos 4x C

14 lncos u C

14 tan u du

tan 4x dx 14 tan 4x4 dxdu 4 dx.u 4x.

tan 4x dx.

lnsec x tan x C sec2 x sec x tan x

sec x tan x dx

sec x dx sec xsec x tan xsec x tan x

dx

tan x dx sin xcos x

dx lncos x C.


Use substitution withu sec x tan x.

1053715_1405.qxp 11/5/08 11:52 AM Page 1066

Application

In the next example, recall from Section 11.4 that the average value of a functionf over an interval is given by

Example 10MAKE A DECISION Finding an Average Temperature

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

where t is the time (in hours), with corresponding to midnight. Will theaverage temperature during the four-hour period from noon to 4 P.M. be greaterthan

SOLUTION To find the average temperature A, use the formula for the averagevalue of a function over an interval.

So, the average temperature is as indicated in Figure 14.35. No, the aver-age temperature from noon to 4 P.M. will not be greater than 90.

89.2,

72 54

89.2

14 288

216

147216 18

12

12 7212 18

12

12

1472t 18

12 cos

t 812

16

12

A 1416

12 72 18 sin t 812 dt

90?

t 0

T 72 18 sin t 812

1b a

b

a

fx dx.

a, b


100908070605040302010

t

T

Average 89.2

T = 72 + 18 sin (t 8)12

Time (in hours)

Tem

pera

ture

(in d

egre

es F

ahre

nhei

t)

4 8 12 16 20 24

Average Temperature

FIGURE 14.35

CHECKPOINT 10Use the function in Example 10 tofind the average temperature from 9 A.M. to noon.

1. For each trigonometric differentiation rule, is there a correspondingintegration rule?

2. For the differentiation rule what is the correspondingintegration rule?

3. For the differentiation rule what is the correspondingintegration rule?

4. For the integration rule what is the corresponding

differentiation rule?

sec2 u du tan u 1 C,

ddx

[cos u] sin ududx

,

ddx

[sin u] cos ududx

,


1053715_1405.qxp 11/5/08 11:52 AM Page 1067

In Exercises 134, find the indefinite integral.

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.

33. 34. 1 tan 2 d sin x cos x2 dx esec x sec x tan x dx esin x cos x dx ex tan ex dx ex sin ex dx 1 cos sin d csc2 xcot3 x dx sin xx dx sin x1 cos x dx cos t1 sin t dt sec x tan xsec x 1 dx sin xcos2 x dx sec2 xtan x dx sec x2 dx csc 2x dx tan 5x dx cot x dx

cot x csc2 x dx tan3 x sec2 x dx csc x3 cot x3 dx tan 3x dx csc2 4x dx sec2 x2 dx 2x sin x2 dx 2x cos x2 dx cos 6x dx sin 2x dx sec y tan y sec2 y dy csc2 cos d 2 sec2 d 1 csc t cot t dt t 2 sin t dt 2 sin x 3 cos x dx



In Exercises 18, evaluate the trigonometric function.

1. 2. 3. 4.

5. 6. 7. 8.

In Exercises 916, simplify the expression using the trigonometric identities.

9. 10.11. 12.

13. 14.

15. 16.

In Exercises 1720, evaluate the definite integral.

17. 18.

19. 20. 10

x9 x 2 dx20

x4 x2 dx

11

1 x 2 dx40

x2 3x 4 dx

cot xsin2 xcot x sec x

cot x cos 2 xsec x sin

2 xsin2 xcsc2 x 1cos2 xsec2 x 1csc x cos xsin x sec x

cos

2sec cot 53tan

56

cos6sin

3sin 76cos

54

Skills Review 14.5


1053715_1405.qxp 11/5/08 11:52 AM Page 1068


In Exercises 3538, use integration by parts to find theindefinite integral.

35. 36.

37. 38.

In Exercises 3946, evaluate the definite integral. Usea symbolic integration utility to verify your results.

39. 40.

41. 42.

43.

44.

45.

46.

In Exercises 4752, determine the area of the region.

47. 48.

49. 50.

51. 52.

53. Consumer Trends Energy consumption in the UnitedStates is seasonal. For instance, primary residential energyconsumption can be approximated by the model

where is the monthly consumption (in trillion Btu) and isthe time in months, with corresponding to January.Find the average consumption rate of domestic energy dur-ing a year. (Source: Energy Information Administration)

54. Seasonal Sales The monthly sales (in millions of units)of snow blowers can be modeled by

where is the time in months, with corresponding toJanuary. Find the average monthly sales(a) during a year.(b) from July through December.

55. Inventory The stockpile level of liquefied petroleumgases in the United States in 2006 can be approximated bythe model

where Q is measured in millions of barrels and t is the timein months, with corresponding to January. Find theaverage levels given by this model during(a) the first quarter (b) the second quarter (c) the entire year (Source: Energy Information Administration)

56. Construction Workers The number (in thousands)of construction workers employed in the United States during 2006 can be modeled by

where is the time in months, with corresponding toJanuary. Use a graphing utility to estimate the averagenumber of construction workers during(a) the first quarter (b) the second quarter (c) the entire year (Source: U.S. Bureau of Labor Statistics)

57. Meteorology The average monthly precipitation P ininches, including rain, snow, and ice, for Sacramento,California, can be modeled by

where t is the time in months, with corresponding toJanuary. Find the total annual precipitation for Sacramento.(Source: National Oceanic and AtmosphericAdministration)

t 10 t 12P 2.47 sin0.40t 1.80 2.08,

0 t 12.3 t 6.

0 t 3.

t 1tW 7594 455.2 sin0.41t 1.713)

W

0 t 12.3 t 6.

0 t 3.

t 1

Q 109 32 cos t 36

t 1t

0 t 12S 15 6 sint 86 ,

t 1tQ

0 t 12Q 588 390 cos0.46t 0.25,

x

y

3

2

1

2

1

x

y

2

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

x

y

2

0.51

1.54321

x

y

2

y x

2 cos xy x sin x

x

y

1

1

8

4

1

x

y

2

y tan xy cos x

4

40

sec x tan x dx

10

tan1 x dx

80

sin 2x cos 2x dx

412

csc 2x cot 2x dx

20

x cos x dx232

sec2 x

2 dx

60

sin 6x dx40

cos 4x3 dx

sec tan d x sec2 x dx x sin x dx x cos x dx

1053715_1405.qxp 11/5/08 11:52 AM Page 1069

58. Meteorology The average monthly precipitation P ininches, including rain, snow, and ice, for Bismarck, NorthDakota, can be modeled by

where t is the time in months, with corresponding toJanuary. (Source: National Oceanic and AtmosphericAdministration)(a) Find the maximum and minimum precipitation and the

month in which each occurs.(b) Determine the average monthly precipitation for the

year.(c) Find the total annual precipitation for Bismarck.

59. Cost Suppose that the temperature in degrees Fahrenheitis given by

where t is the time in hours, with corresponding tomidnight. Furthermore, suppose that it costs $0.30 to coola particular house for 1 hour.(a) Use the integration capabilities of a graphing utility to

find the cost C of cooling this house between 8 A.M.and 8 P.M., if the thermostat is set at (see figure) andthe cost is given by

(b) Use the integration capabilities of a graphing utility tofind the savings realized by resetting the thermostat to

(see figure) by evaluating the integral

60. Health For a person at rest, the velocity v (in liters per second) of air flow into and out of the lungs during arespiratory cycle is approximated by

where t is the time in seconds. Find the volume in liters ofair inhaled during one cycle by integrating this functionover the interval

61. Health After exercising for a few minutes, a person hasa respiratory cycle for which the velocity of air flow isapproximated by

How much does the lung capacity of a person increase as aresult of exercising? Use the results of Exercise 60 todetermine how much more air is inhaled during a cycleafter exercising than is inhaled during a cycle at rest. (Notethat the cycle is shorter and you must integrate over theinterval )

62. Sales In Example 9 in Section 14.4, the sales of a season-al product were approximated by the model

where F was measured in pounds and t was the time in days,with corresponding to January 1. The manufacturerof this product wants to set up a manufacturing schedule toproduce a uniform amount each day. What should thisamount be? (Assume that there are 200 production daysduring the year.)

In Exercises 6366, use a graphing utility andSimpsons Rule to approximate the integral.

Integral n

63. 8

64. 8

65. 20

66. 20

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

67.

68. 4 sin x cos x dx 0b

a

sin x dx b2a

sin x dx

20

4 x sin x dx

0

1 cos2 x dx

20

cos x dx

20

x sin x dx

t 1

t 0F 100,0001 sin 2 t 60365 ,

0, 2.

v 1.75 sin t2 .

0, 3.

v 0.9 sin t3

Tem

pera

ture

(in de

gree

s Fa

hren

heit)

Time (in hours)

8478726660

2416 201284

T

t

C 0.31810

72 12 sin t 812 78 dt.78

Tem

pera

ture

(in de

gree

s Fa

hren

heit)

Time (in hours)

8478726660

2416 201284

T

t

T = 72 + 12 sin (t 8)12

C 0.3208

72 12 sin t 812 72 dt.

72

1

t 0

T 72 12 sin t 812

t 10 t 12P 1.07 sin0.59t 3.94 1.52,


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Algebra Review 1071

Algebra Review

Solving Trigonometric EquationsSolving a trigonometric equation requires the use of trigonometry, but it also requires theuse of algebra. Some examples of solving trigonometric equations were presented onpages 1037 and 1038. Here are several others.

Example 1 Solving a Trigonometric Equation

Solve each trigonometric equation.

a.

b.

c.

SOLUTION

a. Write original equation.

Add sin x to, and subtract from, each side.

Combine like terms.

Divide each side by 2.

b. Write original equation.


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 interval as shown.

and

No solution is obtained from because are outside the range of thecosine function.

2cos x 2

cos x 2

cos2 x 2 x 2, 32

cos2 x 2 0 cot x 0

0 x 2 cot xcos2 x 2 0

cot x cos2 x 2 cot x 0 cot x cos2 x 2 cot x

0 x 2 x 6, 56 ,

76 ,

116 ,

tan x 33

tan2 x 13

3 tan2 x 1

0 x 2 x 54 , 74 ,

sin x 22

2 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

1053715_140R.qxp 11/5/08 11:53 AM Page 1071


Example 2 Solving a Trigonometric Equation

Solve each trigonometric equation.

a.

b.

c.

SOLUTION

a. Write original equation.

Factor.


and

b. Write original equation.Pythagorean Identity

Combine like terms.

Multiply each side by

Factor.


and

c. Write original equation.

Add 1 to each side.


In the interval there are four other solutions.0 t 2,

0 t 23 t

9, 59 ,

0 3t 2 3t 3, 53 ,

cos 3t 12

2 cos 3t 1 2 cos 3t 1 0

x

3, 53

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 1cos x 1 0

1. 2 cos2 x 3 cos x 1 0 2 cos2 x 3 cos x 1 0

21 cos2 x 3 cos x 3 0 2 sin2 x 3 cos x 3 0

x

2 x 76 ,

116

sin x 1 sin x 12

sin x 1 0 2 sin x 1 0

0, 2 2 sin x 1sin x 1 0

2 sin2 x sin x 1 0

2 cos 3t 1 0

2 sin2 x 3 cos x 3 0

2 sin2 x sin x 1 0

1053715_140R.qxp 11/5/08 11:53 AM Page 1072

Chapter Summary and Study Strategies 1073

Chapter Summary and Study Strategies

After studying this chapter, you should have acquired the following skills. The exercise numbers are keyed to the Review Exercises that begin on page 1075.Answers to odd-numbered Review Exercises are given in the back of the

Trigonometric Functions

Documents

angle coterminal

angle measures

acute angle

obtuse angle

radian measure of angles

isa right angle

angles initial

positive angles