5.3 solving trigonometric equationsthspapprecal.weebly.com/uploads/7/0/8/1/7081416/... · section...

Section 5.3 Solving Trigonometric Equations 389

IntroductionTo solve a trigonometric equation, use standard algebraic techniques such ascollecting like terms and factoring. Your preliminary goal in solving a trigono-metric equation is to isolate the trigonometric function involved in the equation.For example, to solve the equation divide each side by 2 to obtain

To solve for note in Figure 5.3 that the equation has solutions and in the interval Moreover, because has a period of there are infinitely many other solutions, which can be written as

and General solution

where is an integer, as shown in Figure 5.3.

Another way to show that the equation has infinitely manysolutions is indicated in Figure 5.4. Any angles that are coterminal with or

will also be solutions of the equation.

FIGURE 5.4

When solving trigonometric equations, you should write your answer(s)using exact values rather than decimal approximations.

sin + 2 =n5566

6

12

ππ

π

π( ) sin + 2 =n6

12

π π( )

5��6��6

sin x �12

n

x �5�

6� 2n�x �

�

6� 2n�

2�,sin x�0, 2��.x � 5��6x � ��6sin x �

12x,

sin x �1

2.

2 sin x � 1,

What you should learn• Use standard algebraic

techniques to solve trigonometric equations.

• Solve trigonometric equationsof quadratic type.

• Solve trigonometric equationsinvolving multiple angles.

• Use inverse trigonometricfunctions to solve trigonometric equations.

Why you should learn itYou can use trigonometric equations to solve a variety ofreal-life problems. For instance,in Exercise 72 on page 398,you can solve a trigonometric equation to help answer questions about monthly salesof skiing equipment.

Solving Trigonometric Equations

Tom Stillo/Index Stock Imagery

5.3

x

x = − 2

x = − 2

x =

x =

ππ

π π

π

π

66

6 6

x = + 2π π61

−1

y = 12

y = sin x

ππ−

5 x = + 2π π655

y

FIGURE 5.3

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Collecting Like Terms

Solve

SolutionBegin by rewriting the equation so that is isolated on one side of theequation.

Write original equation.

Add to each side.

Subtract from each side.

Combine like terms.

Divide each side by 2.

Because has a period of first find all solutions in the interval These solutions are and Finally, add multiples of toeach of these solutions to get the general form


where is an integer.

Now try Exercise 7.

Extracting Square Roots

Solve

SolutionBegin by rewriting the equation so that is isolated on one side of theequation.


Add 1 to each side.


Extract square roots.

Because has a period of first find all solutions in the interval These solutions are and Finally, add multiples of to eachof these solutions to get the general form



Now try Exercise 11.

n

x �5�

6� n�x �

�

6� n�

�x � 5��6.x � ��6�0, ��.�,tan x

tan x � ±1�3

� ±�3

3

tan2 x �1

3

3 tan2 x � 1

3 tan2 x � 1 � 0

tan x

3 tan2 x � 1 � 0.

n

x �7�

4� 2n�x �

5�

4� 2n�

2�x � 7��4.x � 5��4�0, 2��.2�,sin x

sin x � ��2

2

2 sin x � ��2

�2 sin x � sin x � ��2

sin xsin x � sin x � �2 � 0

sin x � �2 � �sin x

sin x

sin x � �2 � �sin x.

390 Chapter 5 Analytic Trigonometry

Example 1

Example 2

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The equations in Examples 1 and 2 involved only one trigonometric function.When two or more functions occur in the same equation, collect all terms on oneside and try to separate the functions by factoring or by using appropriate identities. This may produce factors that yield no solutions, as illustrated inExample 3.

Factoring

Solve

SolutionBegin by rewriting the equation so that all terms are collected on one side of theequation.


Subtract from each side.

Factor.

By setting each of these factors equal to zero, you obtain

and

The equation has the solution [in the interval Nosolution is obtained for because are outside the range of thecosine function. Because has a period of the general form of the solutionis obtained by adding multiples of to to get

General solution

where is an integer. You can confirm this graphically by sketching the graph ofas shown in Figure 5.5. From the graph you can see

that the intercepts occur at and so on. These intercepts correspond to the solutions of


Equations of Quadratic TypeMany trigonometric equations are of quadratic type Here area couple of examples.

Quadratic in sin x Quadratic in sec x

To solve equations of this type, factor the quadratic or, if this is not possible, usethe Quadratic Formula.

�sec x�2 � 3�sec x� � 2 � 02�sin x�2 � sin x � 1 � 0

sec2 x � 3 sec x � 2 � 02 sin2 x � sin x � 1 � 0

ax2 � bx � c � 0.

cot x cos2 x � 2 cot x � 0.x-3��2,��2,��2,�3��2,x-

y � cot x cos2 x � 2 cot x,n

x ��

2� n�

x � ��2,��,cot x±�2cos x � ±�2

�0, ��].x � ��2cot x � 0

cos x � ±�2.

cos2 x � 2 x ��

2

cos2 x � 2 � 0 cot x � 0

cot x�cos2 x � 2� � 0

2 cot x cot x cos2 x � 2 cot x � 0

cot x cos2 x � 2 cot x

cot x cos2 x � 2 cot x.


Example 3

x

1

−1

−2

−3

− ππ

y

y = cot x cos2 x − 2 cot x

FIGURE 5.5

Using the equation fromExample 3, explain what wouldhappen if you divided each sideof the equation by Is thisa correct method to use whensolving equations?

cot x.

Exploration

333202_0503.qxd 12/5/05 9:03 AM 91

Rewriting with a Single Trigonometric Function

Solve

SolutionThis equation contains both sine and cosine functions. You can rewrite the equa-tion so that it has only cosine functions by using the identity


Pythagorean identity

Multiply each side by

Factor.

Set each factor equal to zero to find the solutions in the interval

Because has a period of the general form of the solution is obtained byadding multiples of to get

General solution



n

x �5�

3� 2n�x �

�

3� 2n�,x � 2n�,

2�2�,cos x

x � 0cos x � 1cos x � 1 � 0

x ��

3,

5�

3cos x �

12

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�1 � cos2 x� � 3 cos x � 3 � 0

2 sin2 x � 3 cos x � 3 � 0

1 � cos2 x.sin2 x �

2 sin2 x � 3 cos x � 3 � 0.


Example 5

Factoring an Equation of Quadratic Type

Find all solutions of in the interval �0, 2��.2 sin2 x � sin x � 1 � 0

Example 4

Algebraic SolutionBegin by treating the equation as a quadratic in andfactoring.


Factor.

Setting each factor equal to zero, you obtain the following solutions in the interval

and


x ��

2x �

7�

6,

11�

6

sin x � 1sin x � �1

2

sin x � 1 � 02 sin x � 1 � 0

�0, 2��.

�2 sin x � 1��sin x � 1� � 0

2 sin2 x � sin x � 1 � 0

sin xGraphical SolutionUse a graphing utility set in radian mode to graph

for as shown inFigure 5.6. Use the zero or root feature or the zoom and tracefeatures to approximate the intercepts to be

and

These values are the approximate solutions ofin the interval �0, 2��.2 sin2 x � sin x � 1 � 0

x � 5.760 �11�

6.x � 3.665 �

7�

6,x � 1.571 �

�

2,

x-

0 ≤ x < 2�,y � 2 sin2 x � sin x � 1

The algebraic model can be difficult forstudents to visualize. Encourage them towrite or visualize the algebraic model,rewrite the corresponding trigonometricequation in terms of a single trigono-metric function, and then solve for x.

0 2

−2

�

3y = 2 sin2x − sin x − 1

FIGURE 5.6

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Sometimes you must square each side of an equation to obtain a quadratic,as demonstrated in the next example. Because this procedure can introduceextraneous solutions, you should check any solutions in the original equation tosee whether they are valid or extraneous.

Squaring and Converting to Quadratic Type

Find all solutions of in the interval

SolutionIt is not clear how to rewrite this equation in terms of a single trigonometricfunction. Notice what happens when you square each side of the equation.


Square each side.


Rewrite equation.

Combine like terms.

Factor.

Setting each factor equal to zero produces

and

Because you squared the original equation, check for extraneous solutions.

Check

Substitute for

Solution checks. ✓

Check

Substitute for

Solution does not check.

Check

Substitute for

Solution checks. ✓

Of the three possible solutions, is extraneous. So, in the intervalthe only two solutions are and


x � �.x � ��2�0, 2��,x � 3��2

�1 � 1 � 0

x.�cos � � 1 �?

sin �

x � �

0 � 1 � �1

x.3��2 cos 3�

2� 1 �

?sin

3�

2

x � 3�/2 0 � 1 � 1

x.��2cos �

2� 1 �

?sin

�

2

x � �/2

x � �.x ��

2,

3�

2

cos x � �1cos x � 0

cos x � 1 � 02 cos x � 0

2 cos x�cos x � 1� � 0

2 cos2 x � 2 cos x � 0

cos2 x � cos2 x � 2 cos x � 1 � 1 � 0

cos2 x � 2 cos x � 1 � 1 � cos2 x

cos2 x � 2 cos x � 1 � sin2 x

cos x � 1 � sin x

�0, 2��.cos x � 1 � sin x


Example 6

Use a graphing utility toconfirm the solutions found inExample 6 in two differentways. Do both methods producethe same -values? Whichmethod do you prefer? Why?

1. Graph both sides of theequation and find the -coordinates of the points

at which the graphs intersect.

Left side:

Right side:

2. Graph the equation

and find the -intercepts ofthe graph.

x

y � cos x � 1 � sin x

y � sin x

y � cos x � 1

x

x

Exploration

You square each side of theequation in Example 6 becausethe squares of the sine andcosine functions are related bya Pythagorean identity. Thesame is true for the squares ofthe secant and tangent functionsand the cosecant and cotangentfunctions.

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Functions Involving Multiple AnglesThe next two examples involve trigonometric functions of multiple angles of theforms sin and cos To solve equations of these forms, first solve theequation for then divide your result by

Functions of Multiple Angles

Solve

SolutionWrite original equation.

Add 1 to each side.


In the interval you know that and are the onlysolutions, so, in general, you have

and

Dividing these results by 3, you obtain the general solution




Functions of Multiple Angles

Solve

Solution


Subtract 3 from each side.


In the interval you know that is the only solution, so, ingeneral, you have

Multiplying this result by 2, you obtain the general solution

General solution



n

x �3�

2� 2n�

x

2�

3�

4� n�.

x�2 � 3��4�0, ��,

tan x

2� �1

3 tan x

2� �3

3 tan x

2� 3 � 0

3 tan x

2� 3 � 0.

n

t �5�

9�

2n�

3t �

�

9�

2n�

3

3t �5�

3� 2n�.3t �

�

3� 2n�

3t � 5��33t � ��3�0, 2��,

cos 3t �1

2

2 cos 3t � 1

2 cos 3t � 1 � 0

2 cos 3t � 1 � 0.

k.ku,ku.ku


Example 7

Example 8

Additional Examples

Find all solutions for .

a.

Solution:

b.

Solution:

x ��

16, 5�

16, 9�

16, 13�

16, 17�

16, 21�

16, 25�

16, 29�

16

tan 4x � 1

x ��

8,

3�

8,

5�

8,

7�

8,

9�

8,

11�

8,

13�

8, 15�

8

2 sin2 2x � 1

0 ≤ x < 2�

Consider having your students checktheir solutions with a graphing utility.This will help to reinforce the fact thatthere are many solutions, and willdemonstrate why.

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Using Inverse FunctionsIn the next example, you will see how inverse trigonometric functions can beused to solve an equation.

Using Inverse Functions

Solve

SolutionWrite original equation.


Combine like terms.

Factor.

Setting each factor equal to zero, you obtain two solutions in the interval[Recall that the range of the inverse tangent function is

and

Finally, because has a period of you obtain the general solution byadding multiples of


where is an integer. You can use a calculator to approximate the value of


arctan 3.n

x � ��

4� n�x � arctan 3 � n�

��,tan x

x � ��

4 x � arctan 3

tan x � �1 tan x � 3

tan x � 1 � 0tan x � 3 � 0

��2, ��2�.]��2, ��2�.

�tan x � 3��tan x � 1� � 0

tan2 x � 2 tan x � 3 � 0

1 � tan2 x � 2 tan x � 4 � 0

sec2 x � 2 tan x � 4

sec2 x � 2 tan x � 4.


Example 9

Activities1. Find all solutions of the equation

without using a calculator.

Answer:

2. Solve for withoutusing a calculator.

Answer:

3. Solve for

Answer: 5.4424�

3,

2�

3,x � 0.841,

6 cos2 x � cos x � 2 � 0

0 ≤ x < 2�.x,

3�

25�

6,x �

�

6,

sin2 3x � 2 sin 3x � 1 � 0

0 ≤ x < 2�,x,

3�

4� n�x �

�

4� n�,

tan2 x � 1

W RITING ABOUT MATHEMATICS

Equations with No Solutions One of the following equations has solutions and theother two do not. Which two equations do not have solutions?

a.

b.

c.

Find conditions involving the constants b and c that will guarantee that the equation

has at least one solution on some interval of length 2� .

sin2 x � b sin x � c � 0

sin2 x � 5 sin x � 6 � 0

sin2 x � 4 sin x � 6 � 0

sin2 x � 5 sin x � 6 � 0

333202_0503.qxd 12/5/05 9:03 AM Page 395

In Exercises 1– 6, verify that the -values are solutions ofthe equation.

1.

(a) (b)

2.

(a) (b)

3.

(a) (b)

4.

(a) (b)

5.

(a) (b)

6.

(a) (b)

In Exercises 7–20, solve the equation.

7. 8.

9. 10.

11. 12.

13.

14.

15. 16.

17. 18.

19. 20.

In Exercises 21–34, find all solutions of the equation in theinterval

21. 22.

23. 24.

25. 26.

27. 28.

29.

30.

31.

32.

33.

34.

In Exercises 35– 40, solve the multiple-angle equation.

35. 36.

37. 38.

39. 40.

In Exercises 41– 44, find the -intercepts of the graph.

41. 42.

43. 44.

x1

−2

2

−1−3 1 3

y

x1

−2

2

−1−3 1 3

y

y � sec4�� x

8 � � 4y � tan2��x

6 � � 3

x

1

1 21 52 2

y

x1

−2

23

−1−2 1 2 3 4

y

y � sin � x � cos � xy � sin �x

2� 1

x

sin x2

� ��32

cos x

2�

�2

2

sec 4x � 2tan 3x � 1

sin 2x � ��3

2cos 2x �

12

sin x � 2 � cos x � 2

csc x � cot x � 1

cos x � sin x tan x � 2

2 sec2 x � tan2 x � 3 � 0

2 sin2 x � 3 sin x � 1 � 0

2 cos2 x � cos x � 1 � 0

sec x � tan x � 12 sin x � csc x � 0

sec x csc x � 2 csc xsec2 x � sec x � 2

2 sin2 x � 2 � cos x3 tan3 x � tan x

sec2 x � 1 � 0cos3 x � cos x

[0, 2�.

cos 2x�2 cos x � 1� � 0tan 3x�tan x � 1� � 0

tan2 3x � 32 sin2 2x � 1

sin2 x � 3 cos2 x4 cos2 x � 1 � 0

�3 tan2 x � 1��tan2 x � 3� � 0

sin x�sin x � 1� � 0

3 cot2 x � 1 � 03 sec2 x � 4 � 0

tan x � �3 � 0�3 csc x � 2 � 0

2 sin x � 1 � 02 cos x � 1 � 0

x �5�

6x �

�

6

csc4 x � 4 csc2 x � 0

x �7�

6x �

�

2

2 sin2 x � sin x � 1 � 0

x �3�

16x �

�

16

2 cos2 4x � 1 � 0

x �5�

12x �

�

12

3 tan2 2x � 1 � 0

x �5�

3x �

�

3

sec x � 2 � 0

x �5�

3x �

�

3

2 cos x � 1 � 0

x


Exercises 5.3

VOCABULARY CHECK: Fill in the blanks.

1. The equation has the solutions and which are called ________ solutions.

2. The equation is a trigonometric equation that is of ________ type.

3. A solution to an equation that does not satisfy the original equation is called an ________ solution.

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.

2 tan2 x � 3 tan x � 1 � 0

� �11�

6� 2n�,� �

7�

6� 2n�2 sin � � 1 � 0

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In Exercises 45– 54, use a graphing utility to approximatethe solutions (to three decimal places) of the equation inthe interval

45.

46.

47.

48.

49. 50.

51.

52.

53.

54.

In Exercises 55–58, use the Quadratic Formula to solve theequation in the interval Then use a graphing utilityto approximate the angle

55.

56.

57.

58.

In Exercises 59–62, use inverse functions where needed tofind all solutions of the equation in the interval

59.

60.

61.

62.

In Exercises 63 and 64, (a) use a graphing utility to graphthe function and approximate the maximum and minimumpoints on the graph in the interval and (b) solve the trigonometric equation and demonstrate that itssolutions are the -coordinates of the maximum andminimum points of (Calculus is required to find thetrigonometric equation.)

Function Trigonometric Equation

63.

64.

Fixed Point In Exercises 65 and 66, find the smallestpositive fixed point of the function A fixed point of afunction is a real number such that

65. 66.

67. Graphical Reasoning Consider the function given by

and its graph shown in the figure.

(a) What is the domain of the function?

(b) Identify any symmetry and any asymptotes of thegraph.

(c) Describe the behavior of the function as

(d) How many solutions does the equation

have in the interval Find the solutions.

(e) Does the equation have a greatestsolution? If so, approximate the solution. If not,explain why.

68. Graphical Reasoning Consider the function given by

and its graph shown in the figure.

(a) What is the domain of the function?

(b) Identify any symmetry and any asymptotes of thegraph.

(c) Describe the behavior of the function as

(d) How many solutions does the equation

have in the interval Find the solutions.��8, 8?

sin x

x� 0

x → 0.

x− ππ −1

23

−2−3

y

f �x� �sin x

x

cos�1�x� � 0

��1, 1?

cos 1

x� 0

x → 0.

x− ππ

1

2

−2

y

f �x� � cos 1

x

f �x� � cos xf �x� � tan �x

4

f �c � c.]cf[f.

2 cos x � 4 sin x cos x � 0f �x� � 2 sin x � cos 2x

cos x � sin x � 0f �x� � sin x � cos x

f.x

[0, 2�,

2 sin2 x � 7 sin x � 3 � 0

2 cos2 x � 5 cos x � 2 � 0

sec2 x � tan x � 3 � 0

tan2 x � 6 tan x � 5 � 0

[0, 2�.

4 cos2 x � 4 cos x � 1 � 0

tan2 x � 3 tan x � 1 � 0

3 tan2 x � 4 tan x � 4 � 0

12 sin2 x � 13 sin x � 3 � 0

x.[0, 2�.

6 sin2 x � 7 sin x � 2 � 0

2 tan2 x � 7 tan x � 15 � 0

csc2 x � 0.5 cot x � 5 � 0

sec2 x � 0.5 tan x � 1 � 0

x cos x � 1 � 0x tan x � 1 � 0

cos x cot x

1 � sin x� 3

1 � sin x

cos x�

cos x

1 � sin x� 4

4 sin3 x � 2 sin2 x � 2 sin x � 1 � 0

2 sin x � cos x � 0

[0, 2�.

333202_0503.qxd 12/5/05 9:03 AM Page 397

69. Harmonic Motion A weight is oscillating on the end of aspring (see figure). The position of the weight relative to thepoint of equilibrium is given by where is the displacement (in meters) and is the time (inseconds). Find the times when the weight is at the point ofequilibrium for

70. Damped Harmonic Motion The displacement fromequilibrium of a weight oscillating on the end of a spring is given by where is the displacement (in feet) and is the time (in seconds). Use agraphing utility to graph the displacement function for

Find the time beyond which the displacementdoes not exceed 1 foot from equilibrium.

71. Sales The monthly sales (in thousands of units) of aseasonal product are approximated by

where is the time (in months), with correspondingto January. Determine the months when sales exceed100,000 units.

72. Sales The monthly sales (in hundreds of units) ofskiing equipment at a sports store are approximated by

where is the time (in months), with correspondingto January. Determine the months when sales exceed 7500units.

73. Projectile Motion A batted baseball leaves the bat at anangle of with the horizontal and an initial velocity of

feet per second. The ball is caught by an outfielder 300 feet from home plate (see figure). Find ifthe range of a projectile is given by

74. Projectile Motion A sharpshooter intends to hit a targetat a distance of 1000 yards with a gun that has a muzzlevelocity of 1200 feet per second (see figure). Neglectingair resistance, determine the gun’s minimum angle ofelevation if the range is given by

75. Ferris Wheel A Ferris wheel is built such that the height(in feet) above ground of a seat on the wheel at time (in

minutes) can be modeled by

The wheel makes one revolution every 32 seconds. Theride begins when

(a) During the first 32 seconds of the ride, when will aperson on the Ferris wheel be 53 feet above ground?

(b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts 160 seconds, how many times will a person be at thetop of the ride, and at what times?

t � 0.

h�t� � 53 � 50 sin� �

16t �

�

2�.

th

r = 1000 ydθ

Not drawn to scale

r �1

32v0

2 sin 2�.

r�

r = 300 ft

θ

Not drawn to scale

r �132 v0

2 sin 2�.r�

v0 � 100�

t � 1t

S � 58.3 � 32.5 cos � t6

S

t � 1t

S � 74.50 � 43.75 sin � t

6

S

0 ≤ t ≤ 10.

tyy � 1.56e�0.22t cos 4.9t,

y

Equilibrium

0 ≤ t ≤ 1.�y � 0�

tyy �

112�cos 8t � 3 sin 8t�,


76. Data Analysis: Unemployment Rate The tableshows the unemployment rates in the United Statesfor selected years from 1990 through 2004. The time is measured in years, with corresponding to1990. (Source: U.S. Bureau of Labor Statistics)

(a) Create a scatter plot of the data.

t � 0t

r

Model It

Time, t Rate, r

8 4.5

10 4.0

12 5.8

14 5.5

Time, t Rate, r

0 5.6

2 7.5

4 6.1

6 5.4

333202_0503.qxd 12/5/05 9:03 AM Page 398


77. Geometry The area of a rectangle (see figure) inscribedin one arc of the graph of is given by

(a) Use a graphing utility to graph the area function, andapproximate the area of the largest inscribed rectangle.

(b) Determine the values of for which

78. Quadratic Approximation Consider the function givenby

(a) Approximate the zero of the function in the interval

(b) A quadratic approximation agreeing with at isUse a graphing

utility to graph and in the same viewing window.Describe the result.

(c) Use the Quadratic Formula to find the zeros of Compare the zero in the interval with the resultof part (a).

Synthesis

True or False? In Exercises 79 and 80, determine whetherthe statement is true or false. Justify your answer.

79. The equation has four times the numberof solutions in the interval as the equation

80. If you correctly solve a trigonometric equation to the state-ment then you can finish solving the equationby using an inverse function.

In Exercises 81 and 82, use the graph to approximate thenumber of points of intersection of the graphs of and

81. 82.

.

Skills Review

In Exercises 83 and 84, solve triangle by finding allmissing angle measures and side lengths.

83.

84.

In Exercises 85–88, use reference angles to find the exactvalues of the sine, cosine, and tangent of the angle with thegiven measure.

85. 86.

87. 88.

89. Angle of Depression Find the angle of depression fromthe top of a lighthouse 250 feet above water level to thewater line of a ship 2 miles offshore.

90. Height From a point 100 feet in front of a public library,the angles of elevation to the base of the flagpole and thetop of the pole are and respectively. The flag-pole is mounted on the front of the library’s roof. Find theheight of the flagpole.

91. Make a Decision To work an extended applicationanalyzing the normal daily high temperatures in Phoenixand in Seattle, visit this text’s website at college.hmco.com.(Data Source: NOAA)

45�,39�28�

�1410��1845�

600�390�

14.6A C

71°B

22.3

A C

66°B

ABC

x

2π

123

−3−4

4

y1

y2

y

x

2π

1234

y1

y2

y

y2 �12 x � 1y2 � 3x � 1

y1 � 2 sin xy1 � 2 sin x

y2.y1

sin x � 3.4,

2 sin t � 1 � 0.�0, 2��

2 sin 4t � 1 � 0

�0, 6g.

gfg�x� � �0.45x 2 � 5.52x � 13.70.

x � 5f

�0, 6.

f �x� � 3 sin�0.6x � 2�.

A ≥ 1.x

x

2−

−1

π2πx

y

0 < x <�

2.A � 2x cos x,

y � cos x

Model It (cont inued)

(b) Which of the following models best represents thedata? Explain your reasoning.

(1)

(2)

(3)

(4)

(c) What term in the model gives the averageunemployment rate? What is the rate?

(d) Economists study the lengths of business cyclessuch as unemployment rates. Based on this shortspan of time, use the model to find the length ofthis cycle.

(e) Use the model to estimate the next time theunemployment rate will be 5% or less.

r � 896 sin�0.57t � 2.05� � 6.48

r � sin�0.10t � 5.61� � 4.80

r � 1.24 sin�0.47t � 0.01� � 5.45

r � 1.24 sin�0.47t � 0.40� � 5.45

333202_0503.qxd 12/5/05 9:03 AM Page 399

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