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SECTION 10.4 Polar Coordinates and Polar Graphs 729 Section 10.4 Polar Coordinates and Polar Graphs Understand the polar coordinate system. Rewrite rectangular coordinates and equations in polar form and vice versa. Sketch the graph of an equation given in polar form. Find the slope of a tangent line to a polar graph. Identify several types of special polar graphs. Polar Coordinates So far, you have been representing graphs as collections of points on the rec- tangular coordinate system. The corresponding equations for these graphs have been in either rectangular or parametric form. In this section you will study a coordinate system called the polar coordinate system. To form the polar coordinate system in the plane, fix a point called the pole (or origin), and construct from an initial ray called the polar axis, as shown in Figure 10.36. Then each point in the plane can be assigned polar coordinates as follows. Figure 10.37 shows three points on the polar coordinate system. Notice that in this system, it is convenient to locate points with respect to a grid of concentric circles intersected by radial lines through the pole. With rectangular coordinates, each point has a unique representation. This is not true with polar coordinates. For instance, the coordinates and represent the same point [see parts (b) and (c) in Figure 10.37]. Also, because is a directed distance, the coordinates and represent the same point. In general, the point can be written as or where is any integer. Moreover, the pole is represented by where is any angle. 0, , n r, r, 2n 1 r, r, 2n r, r, r, r r, 2 r, x, y directed angle, counterclockwise from polar axis to segment OP r directed distance from O to P r, , P O O, x, y POLAR COORDINATES The mathematician credited with first using polar coordinates was James Bernoulli, who introduced them in 1691. However, there is some evidence that it may have been Isaac Newton who first used them. O θ = directed angle Polar axis θ P = (r, ) r = directed distance 2 3 3, ) ( 6 π 11 = θ 6 π 11 0 π 2 π 3 π 2 Polar coordinates Figure 10.36 0 π 2 π 3 = θ π 3 2, ) ( π 3 1 2 3 π 2 = θ 2 3 3, ) ( π π 6 6 0 π 2 π 3 π 2 (a) Figure 10.37 (b) (c) 332460_1004.qxd 11/2/04 3:35 PM Page 729

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SECTION 10.4 Polar Coordinates and Polar Graphs 729

Section 10.4 Polar Coordinates and Polar Graphs

• Understand the polar coordinate system.• Rewrite rectangular coordinates and equations in polar form and vice versa.• Sketch the graph of an equation given in polar form.• Find the slope of a tangent line to a polar graph.• Identify several types of special polar graphs.

Polar Coordinates

So far, you have been representing graphs as collections of points on the rec-tangular coordinate system. The corresponding equations for these graphs have beenin either rectangular or parametric form. In this section you will study a coordinatesystem called the polar coordinate system.

To form the polar coordinate system in the plane, fix a point called the pole(or origin), and construct from an initial ray called the polar axis, as shown inFigure 10.36. Then each point in the plane can be assigned polar coordinatesas follows.

Figure 10.37 shows three points on the polar coordinate system. Notice that in thissystem, it is convenient to locate points with respect to a grid of concentric circlesintersected by radial lines through the pole.

With rectangular coordinates, each point has a unique representation. Thisis not true with polar coordinates. For instance, the coordinates and represent the same point [see parts (b) and (c) in Figure 10.37]. Also, because is adirected distance, the coordinates and represent the same point. Ingeneral, the point can be written as

or

where is any integer. Moreover, the pole is represented by where is anyangle.

��0, ��,n

�r, �� � ��r, � � �2n � 1���

�r, �� � �r, � � 2n��

�r, ����r, � � ���r, ��

r�r, 2� � ���r, ��

�x, y�

� � directed angle, counterclockwise from polar axis to segment OP

r � directed distance from O to P

�r, ��,PO

O,

�x, y�

POLAR COORDINATES

The mathematician credited with first usingpolar coordinates was James Bernoulli, whointroduced them in 1691. However, there issome evidence that it may have been IsaacNewton who first used them.

O

θ = directed angle

Polaraxis

θP = (r, )

r = dire

cted dista

nce

2 3

3, )( 6π11

=θ6π11

2π3

π2

Polar coordinatesFigure 10.36

2π3

=θ π3

2, )( π3

1 2 3

π2

2 3

3, )( π

π

6

6−

2π3

π2

(a)Figure 10.37

(b) (c)

332460_1004.qxd 11/2/04 3:35 PM Page 729

730 CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates

Coordinate Conversion

To establish the relationship between polar and rectangular coordinates, let the polaraxis coincide with the positive axis and the pole with the origin, as shown in Figure10.38. Because lies on a circle of radius it follows that Moreover, for the definition of the trigonometric functions implies that

and

If you can show that the same relationships hold.

EXAMPLE 1 Polar-to-Rectangular Conversion

a. For the point

and

So, the rectangular coordinates are

b. For the point

and

So, the rectangular coordinates are See Figure 10.39.

EXAMPLE 2 Rectangular-to-Polar Conversion

a. For the second quadrant point

Because was chosen to be in the same quadrant as you should use a posi-tive value of

This implies that one set of polar coordinates is

b. Because the point lies on the positive axis, choose andand one set of polar coordinates is

See Figure 10.40.

�r, �� � �2, ��2�.r � 2,� � ��2y-�x, y� � �0, 2�

�r, �� � ��2, 3��4�. � �2

� ���1�2 � �1�2

r � �x2 � y2

r.�x, y�,�

� �3�

4.tan � �

yx

� �1

�x, y� � ��1, 1�,

�x, y� � �3�2, �3�2�.

y � �3 sin �

6�

�32

.x � �3 cos �

6�

32

�r, �� � ��3, ��6�,�x, y� � ��2, 0�.

y � r sin � � 2 sin � � 0.x � r cos � � 2 cos � � �2

�r, �� � �2, ��,

r < 0,

sin � �yr .cos � �

xr ,tan � �

yx ,

r > 0,r2 � x2 � y2.r,�x, y�

x-

yr

y

x

x

θPole

Polar axis(x-axis)(Origin)

(x, y)(r, )θ

Relating polar and rectangular coordinatesFigure 10.38

x1

1

2

2

−1

−1

−2

−2

(x, y) = (−2, 0)

(r, ) = (2, )πθ

(r, ) =θ ( , )π6

(x, y) = ( , )3223

3

y

To convert from polar to rectangular coordi-nates, let cos and sin Figure 10.39

�.y � r�x � r

1

2

(x, y) = (−1, 1)

(x, y) = (0, 2)

(r, ) =θ ( , )2π2

(r, ) =θ ( , )3π4

x1 2−1−2

2

y

To convert from rectangular to polar coordi-nates, let tan and Figure 10.40

r � �x 2 � y 2 .� � y�x

THEOREM 10.10 Coordinate Conversion

The polar coordinates of a point are related to the rectangular coordinatesof the point as follows.

1. 2.

r2 � x2 � y2y � r sin �

tan � �yx

x � r cos �

�x, y��r, ��

332460_1004.qxd 11/2/04 3:35 PM Page 730

SECTION 10.4 Polar Coordinates and Polar Graphs 731

Polar Graphs

One way to sketch the graph of a polar equation is to convert to rectangular coordi-nates and then sketch the graph of the rectangular equation.

EXAMPLE 3 Graphing Polar Equations

Describe the graph of each polar equation. Confirm each description by converting toa rectangular equation.

a. b. c.

Solution

a. The graph of the polar equation consists of all points that are two units fromthe pole. In other words, this graph is a circle centered at the origin with a radiusof 2. [See Figure 10.41(a).] You can confirm this by using the relationship

to obtain the rectangular equation

Rectangular equation

b. The graph of the polar equation consists of all points on the line thatmakes an angle of with the positive axis. [See Figure 10.41(b).] You canconfirm this by using the relationship to obtain the rectangularequation

Rectangular equation

c. The graph of the polar equation is not evident by simple inspection, soyou can begin by converting to rectangular form using the relationship

Polar equation

Rectangular equation

From the rectangular equation, you can see that the graph is a vertical line. [See Figure10.41(c.)]

x � 1

r cos � � 1

r � sec �

r cos � � x.r � sec �

y � �3 x.

tan � � y�xx-��3

� � ��3

x2 � y2 � 22.

r2 � x2 � y2

r � 2

r � sec �� ��

3r � 2

−9 9

−6

6

Spiral of ArchimedesFigure 10.42

1 2 30π

2π3

π2

1 2 30π

2π3

π2

1 2 30π

2π3

π2

(a) Circle: r � 2

(b) Radial line: � ��

3

(c) Vertical line:Figure 10.41

r � sec �

TECHNOLOGY Sketching the graphs of complicated polar equations by handcan be tedious. With technology, however, the task is not difficult. If your graphingutility has a polar mode, use it to graph the equations in the exercise set. If yourgraphing utility doesn’t have a polar mode, but does have a parametric mode, youcan graph by writing the equation as

For instance, the graph of shown in Figure 10.42 was produced with agraphing calculator in parametric mode. This equation was graphed using theparametric equations

with the values of varying from to This curve is of the form andis called a spiral of Archimedes.

r � a�4�.�4��

y �12

� sin �

x �12

� cos �

r �12�

y � f ��� sin �.

x � f ��� cos �

r � f ���

332460_1004.qxd 11/2/04 3:35 PM Page 731

732 CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates

EXAMPLE 4 Sketching a Polar Graph

Sketch the graph of

Solution Begin by writing the polar equation in parametric form.

and

After some experimentation, you will find that the entire curve, which is called a rosecurve, can be sketched by letting vary from 0 to as shown in Figure 10.43. If youtry duplicating this graph with a graphing utility, you will find that by letting varyfrom 0 to you will actually trace the entire curve twice.

Use a graphing utility to experiment with other rose curves they are of the formor For instance, Figure 10.44 shows the graphs of two

other rose curves.r � a sin n��.r � a cos n�

2�,�

�,�

y � 2 cos 3� sin �x � 2 cos 3� cos �

r � 2 cos 3�.NOTE One way to sketch the graph ofby hand is to make a table

of values.

By extending the table and plotting thepoints, you will obtain the curve shownin Example 4.

r � 2 cos 3�

1 20π

2π3

π2

1 20π

2π3

π2

1 20π

2π3

π2

1 20π

2π3

π2

2

−2

−3 3

r = 2 sin 5θ

Generated by Derive

r = 0.5 cos 2θ

0.2 0.3 0.4

0

π2

1 20π

2π3

π2

1 20π

2π3

π2

0 ≤ � ≤�

6

Figure 10.43

0 ≤ � ≤2�

30 ≤ � ≤

5�

60 ≤ � ≤ �

0 ≤ � ≤�

30 ≤ � ≤

2

Rose curvesFigure 10.44

0

r 2 0 0 2�2

2�

3�

2�

3�

6�

332460_1004.qxd 11/2/04 3:36 PM Page 732

SECTION 10.4 Polar Coordinates and Polar Graphs 733

Slope and Tangent Lines

To find the slope of a tangent line to a polar graph, consider a differentiable functiongiven by To find the slope in polar form, use the parametric equations

and

Using the parametric form of given in Theorem 10.7, you have

which establishes the following theorem.

From Theorem 10.11, you can make the following observations.

1. Solutions to yield horizontal tangents, provided that

2. Solutions to yield vertical tangents, provided that

If and are simultaneously 0, no conclusion can be drawn about tangentlines.

EXAMPLE 5 Finding Horizontal and Vertical Tangent Lines

Find the horizontal and vertical tangent lines of

Solution Begin by writing the equation in parametric form.

and

Next, differentiate and with respect to and set each derivative equal to 0.

So, the graph has vertical tangent lines at and and it hashorizontal tangent lines at and as shown in Figure 10.46.�1, ��2�,�0, 0�

��2�2, 3��4�,��2�2, ��4�

� � 0, �

2dyd�

� 2 sin � cos � � sin 2� � 0

� ��

4,

3�

4dxd�

� cos2� � sin2� � cos 2� � 0

�yx

y � r sin � � sin � sin � � sin2 �

x � r cos � � sin � cos �

0 ≤ � ≤ �.r � sin �,

dx�d�dy�d�

dyd�

� 0.dxd�

� 0

dxd�

� 0.dyd�

� 0

�f ��� cos � � f���� sin �

�f ��� sin � � f���� cos �

dydx

�dy�d�

dx�d�

dy�dx

y � r sin � � f ��� sin �.x � r cos � � f ��� cos �

r � f ���.

θ

0

(r, )

Tangent lineθr = f( )

π2

π

2π3

Tangent line to polar curveFigure 10.45

2π3

(0, 0) 12

1,( )π2

( ), π42( ),

2 4π32 2

π2

Horizontal and vertical tangent lines ofsin

Figure 10.46�r �

THEOREM 10.11 Slope in Polar Form

If is a differentiable function of then the slope of the tangent line to thegraph of at the point is

provided that at See Figure 10.45.���r, ��.dx�d� � 0

dydx

�dy�d�

dx�d��

f ��� cos � � f���� sin ��f ��� sin � � f���� cos �

�r, ��r � f ����,f

332460_1004.qxd 11/2/04 3:36 PM Page 733

734 CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates

EXAMPLE 6 Finding Horizontal and Vertical Tangent Lines

Find the horizontal and vertical tangents to the graph of

Solution Using differentiate and set equal to 0.

So, and and you can conclude that whenand 0. Similarly, using you have

So, or and you can conclude that when and From these results, and from the graph shown in Figure 10.47, you

can conclude that the graph has horizontal tangents at and andhas vertical tangents at and This graph is called a cardioid.Note that both derivatives and are 0 when Using thisinformation alone, you don’t know whether the graph has a horizontal or verticaltangent line at the pole. From Figure 10.47, however, you can see that the graph has acusp at the pole.

Theorem 10.11 has an important consequence. Suppose the graph of passes through the pole when and Then the formula for simplifies as follows.

So, the line is tangent to the graph at the pole,

Theorem 10.12 is useful because it states that the zeros of can be usedto find the tangent lines at the pole. Note that because a polar curve can cross the polemore than once, it can have more than one tangent line at the pole. For example, therose curve

has three tangent lines at the pole, as shown in Figure 10.48. For this curve,cos is 0 when is and Moreover, the derivative

is not 0 for these values of �.f���� � �6 sin 3�5��6.��2,��6,�3�f ��� � 2

f ��� � 2 cos 3�

r � f ���

�0, ��.� � �

dydx

�f���� sin � � f ��� cos �

f���� cos � � f ��� sin ��

f���� sin � � 0f���� cos � � 0

�sin �cos �

� tan �

dy�dxf���� � 0.� � �r � f���

� � 0.dx�d���dy�d��4, ��.�1, 5��3�,�1, ��3�,

�3, 4��3�,�3, 2��3�5��3.�, ��3,

� � 0,dx�d� � 0cos � �12,sin � � 0

dxd�

� �2 sin � � 4 cos � sin � � 2 sin ��2 cos � � 1� � 0.

x � r cos � � 2 cos � � 2 cos2 �

x � r cos �,4��3,� � 2��3,dy�d� � 0cos � � 1,cos � � �

12

� �2�2 cos � � 1��cos � � 1� � 0

dyd�

� 2 ��1 � cos ���cos �� � sin ��sin ���

y � r sin � � 2�1 � cos �� sin �

dy�d�y � r sin �,

r � 2�1 � cos ��.

1,( )π3

1,( )π3

5

3,( )π3

2

3,( )π3

4

π(4, )0π

2π3

π2

Horizontal and vertical tangent lines ofcos

Figure 10.47��r � 2�1 �

2

f( ) = 2 cos 3θθ

2π3

π2

This rose curve has three tangent linesand

at the pole.Figure 10.48

� � 5��6��� � ��6, � � �� 2,

THEOREM 10.12 Tangent Lines at the Pole

If and then the line is tangent at the pole to thegraph of r � f ���.

� � �f���� � 0,f ��� � 0

332460_1004.qxd 11/2/04 3:36 PM Page 734

SECTION 10.4 Polar Coordinates and Polar Graphs 735

Special Polar Graphs

Several important types of graphs have equations that are simpler in polar form thanin rectangular form. For example, the polar equation of a circle having a radius of and centered at the origin is simply Later in the text you will come to appreci-ate this benefit. For now, several other types of graphs that have simpler equations inpolar form are shown below. (Conics are considered in Section 10.6.)

r � a.a

Generated by Maple

2π3

π2

2π3

π2

2π3

π2

2π3

n = 3

a

π2

2π3

n = 4

a

π2

n = 5

a

2π3

π2

a

2π3

π2

a

2π3

π2

a0π

2π3

π2

FOR FURTHER INFORMATION For more information on rose curves and related curves, seethe article “A Rose is a Rose . . .” by Peter M. Maurer in The American Mathematical Monthly.The computer-generated graph at the left is the result of an algorithm that Maurer calls “TheRose.” To view this article, go to the website www.matharticles.com.

Limaçon with inner loop

ab

< 1

Cardioid(heart-shaped)

ab

� 1

Dimpled limaçon

1 <ab

< 2

cos Rose curve

n�r � a

cos Circle

�r � a sin Circle

�r � a sin Lemniscate

2�r2 � a2

cos Rose curve

n�r � a sin Rose curve

n�r � a

2π3

π2

n = 2

a0π

2π3

π2

a

2π3

π2

Convex limaçon

ab

≥ 2

cos Lemniscate

2�r2 � a2

sin Rose curve

n�r � a

Limaçons

�a > 0, b > 0�r � a ± b sin �

r � a ± b cos �

petals if is odd

petals if is even�n ≥ 2�

n2n

nn

Rose Curves

Circles and Lemniscates

TECHNOLOGY The rose curves described above are of the form or where is a positive integer that is greater than or equal to 2. Usea graphing utility to graph or for some noninteger valuesof Are these graphs also rose curves? For example, try sketching the graph of

0 ≤ � ≤ 6�.r � cos 23�,n.

r � a sin n�r � a cos n�nr � a sin n�,

r � a cos n�

332460_1004.qxd 11/2/04 3:36 PM Page 735

736 CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates

In Exercises 1–6, plot the point in polar coordinates and findthe corresponding rectangular coordinates for the point.

1. 2.

3. 4.

5. 6.

In Exercises 7–10, use the feature of a graphing utility tofind the rectangular coordinates for the point given in polarcoordinates. Plot the point.

7. 8.

9. 10.

In Exercises 11–16, the rectangular coordinates of a point aregiven. Plot the point and find sets of polar coordinates forthe point for

11. 12.

13. 14.

15. 16.

In Exercises 17–20, use the feature of a graphing utility tofind one set of polar coordinates for the point given in rectan-gular coordinates.

17. 18.

19. 20.

21. Plot the point if the point is given in (a) rectangularcoordinates and (b) polar coordinates.

22. Graphical Reasoning

(a) Set the window format of a graphing utility to rectangularcoordinates and locate the cursor at any position off theaxes. Move the cursor horizontally and vertically. Describeany changes in the displayed coordinates of the points.

(b) Set the window format of a graphing utility to polarcoordinates and locate the cursor at any position off theaxes. Move the cursor horizontally and vertically. Describeany changes in the displayed coordinates of the points.

(c) Why are the results in parts (a) and (b) different?

In Exercises 23–26, match the graph with its polar equation.[The graphs are labeled (a), (b), (c), and (d).]

(a) (b)

(c) (d)

23. 24.

25. 26.

In Exercises 27–34, convert the rectangular equation to polarform and sketch its graph.

27. 28.

29. 30.

31.

32.

33.

34.

In Exercises 35– 42, convert the polar equation to rectangularform and sketch its graph.

35. 36.

37. 38.

39. 40.

41. 42.

In Exercises 43–52, use a graphing utility to graph the polarequation. Find an interval for over which the graph is tracedonly once.

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

53. Convert the equation

to rectangular form and verify that it is the equation of a circle.Find the radius and the rectangular coordinates of the center ofthe circle.

r � 2�h cos � � k sin ��

r 2 �1�

r 2 � 4 sin 2�

r � 3 sin�5�

2 �r � 2 cos�3�

2 �

r �2

4 � 3 sin �r �

21 � cos �

r � 4 � 3 cos �r � 2 � sin �

r � 5�1 � 2 sin ��r � 3 � 4 cos �

r � 2 csc �r � 3 sec �

� �5�

6r � �

r � 5 cos �r � sin �

r � �2r � 3

�x2 � y 2�2 � 9�x2 � y 2� � 0

y 2 � 9x

xy � 4

3x � y � 2 � 0

x � 10y � 4

x2 � y 2 � 2ax � 0x2 � y 2 � a2

r � 2 sec �r � 3�1 � cos ��r � 4 cos 2�r � 2 sin �

01 3

π2

021

π2

04

π2

02 4

π2

�4, 3.5�

�0, �5��52, 43�

�3�2, 3�2 ��3, �2�

angle

�3, ��3���3, �1��4, �2���3, 4��0, �5��1, 1�

0 ≤ � < 2�.two

�8.25, 1.3���3.5, 2.5���2, 11��6��5, 3��4�

angle

��3, �1.57���2, 2.36��0, �7��6���4, ���3���2, 7��4��4, 3��6�

E x e r c i s e s f o r S e c t i o n 1 0 . 4 See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

332460_1004.qxd 11/2/04 3:36 PM Page 736

SECTION 10.4 Polar Coordinates and Polar Graphs 737

54. Distance Formula

(a) Verify that the Distance Formula for the distance betweenthe two points and in polar coordinates is

(b) Describe the positions of the points relative to each other ifSimplify the Distance Formula for this case. Is the

simplification what you expected? Explain.

(c) Simplify the Distance Formula if Is thesimplification what you expected? Explain.

(d) Choose two points on the polar coordinate system and findthe distance between them. Then choose different polarrepresentations of the same two points and apply theDistance Formula again. Discuss the result.

In Exercises 55–58, use the result of Exercise 54 to approximatethe distance between the two points in polar coordinates.

55. 56.

57. 58.

In Exercises 59 and 60, find and the slopes of the tangentlines shown on the graph of the polar equation.

59. 60.

In Exercises 61–64, use a graphing utility to (a) graph the polarequation, (b) draw the tangent line at the given value of and(c) find at the given value of Hint: Let the incrementbetween the values of equal

61. 62.

63. 64.

In Exercises 65 and 66, find the points of horizontal and verti-cal tangency (if any) to the polar curve.

65. 66.

In Exercises 67 and 68, find the points of horizontal tangency (if any) to the polar curve.

67. 68.

In Exercises 69–72, use a graphing utility to graph the polarequation and find all points of horizontal tangency.

69. 70.

71. 72.

In Exercises 73–80, sketch a graph of the polar equation andfind the tangents at the pole.

73. 74.

75. 76.

77. 78.

79. 80.

In Exercises 81–92, sketch a graph of the polar equation.

81. 82.

83. 84.

85. 86.

87. 88.

89. 90.

91. 92.

In Exercises 93 –96, use a graphing utility to graph the equationand show that the given line is an asymptote of the graph.

93. Conchoid

94. Conchoid

95. Hyperbolic spiral

96. Strophoid

101. Sketch the graph of over each interval.

(a) (b) (c)

102. Think About It Use a graphing utility to graph the polarequation for (a) (b) and (c) Use the graphs to describe the effect of theangle Write the equation as a function of sin for part (c).�.

� ��2. � ��4, � 0,r � 6 �1 � cos�� � ��

��

2≤ � ≤

2�

2≤ � ≤ �0 ≤ � ≤

2

r � 4 sin �

x � �2r � 2 cos 2� sec �

y � 2r � 2��

y � 1r � 2 � csc �

x � �1r � 2 � sec �

AsymptotePolar Equation Name of Graph

r 2 � 4 sin �r 2 � 4 cos 2�

r �1�

r � 2�

r �6

2 sin � � 3 cos �r � 3 csc �

r � 5 � 4 sin �r � 3 � 2 cos �

r � 1 � sin �r � 4�1 � cos ��r � 2r � 5

r � 3 cos 2�r � 3 sin 2�

r � �sin 5�r � 2 cos 3�

r � 3�1 � cos ��r � 2�1 � sin ��r � 3 cos �r � 3 sin �

r � 2 cos�3� � 2�r � 2 csc � � 5

r � 3 cos 2� sec �r � 4 sin � cos2 �

r � a sin � cos2 �r � 2 csc � � 3

r � a sin �r � 1 � sin �

r � 4, � ��

4r � 3 sin �, � �

3

r � 3 � 2 cos �, � � 0r � 3�1 � cos ��, � ��

2

� /24.��.dy /dx

�,

24, π3( )

63, π7( )

(2, 0)

1 2 30

π2

2−1, π3( )

π25,( )

(2, )π0

2 3

π2

r � 2�1 � sin ��r � 2 � 3 sin �

dy/dx

�12, 1��4, 2.5�,�7, 1.2��2, 0.5�,

�3, ���10, 7�

6 �,�2, �

6��4, 2�

3 �,

�1 � �2 � 90.

�1 � �2.

d � �r12 � r2

2 � 2r1r2 cos��1 � �2� .

�r2, �2��r1, �1�

Writing About Concepts97. Describe the differences between the rectangular coordi-

nate system and the polar coordinate system.

98. Give the equations for the coordinate conversion fromrectangular to polar coordinates and vice versa.

99. For constants and describe the graphs of the equationsand in polar coordinates.

100. How are the slopes of tangent lines determined in polarcoordinates? What are tangent lines at the pole and howare they determined?

� � br � ab,a

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738 CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates

103. Verify that if the curve whose polar equation is isrotated about the pole through an angle then an equation forthe rotated curve is

104. The polar form of an equation for a curve is Show that the form becomes

(a) if the curve is rotated counterclockwiseradians about the pole.

(b) if the curve is rotated counterclockwise radians about the pole.

(c) if the curve is rotated counterclockwise radians about the pole.

In Exercises 105–108, use the results of Exercises 103 and 104.

105. Write an equation for the limaçon after it hasbeen rotated by the given amount. Use a graphing utility tograph the rotated limaçon.

(a) (b) (c) (d)

106. Write an equation for the rose curve after it hasbeen rotated by the given amount. Verify the results by usinga graphing utility to graph the rotated rose curve.

(a) (b) (c) (d)

107. Sketch the graph of each equation.

(a) (b)

108. Prove that the tangent of the angle betweenthe radial line and the tangent line at the point on thegraph of (see figure) is given by tan

In Exercises 109–114, use the result of Exercise 108 to find theangle between the radial and tangent lines to the graph forthe indicated value of Use a graphing utility to graph thepolar equation, the radial line, and the tangent line for theindicated value of Identify the angle

109.

110.

111.

112.

113.

114.

True or False? In Exercises 115–118, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.

115. If and represent the same point on the polarcoordinate system, then

116. If and represent the same point on the polarcoordinate system, then for some integer

117. If then the point on the rectangular coordinatesystem can be represented by on the polar coordinatesystem, where and

118. The polar equations and have thesame graph.

r � �sin 2�r � sin 2�

� � arctan�y�x�.r � �x2 � y 2�r, ��

�x, y�x > 0,

n.�1 � �2 � 2�n�r, �2��r, �1�

�r1� � �r2�.�r2, �2��r1, �1�

� � ��6r � 5

� � 2��3r �6

1 � cos �

� � ��6r � 4 sin 2�

� � ��4r � 2 cos 3�

� � 3��4r � 3�1 � cos ��� � �r � 2�1 � cos ��Value of �Polar Equation

�.�.

�.�

0A

θ

P = (r, )θ

Tangent line

Radial line

θ= ( )r f

Polar axis

O

ψ

Polar curve:

π2

� � �r��dr�d���.r � f ����r, ��

� �0 ≤ � ≤ ��2�

r � 1 � sin�� ��

4�r � 1 � sin �

�2�

3�

2�

6

r � 2 sin 2�

3�

2�

2�

4

r � 2 � sin �

3��2r � f �cos ��

�r � f ��sin ����2r � f ��cos ��

r � f �sin ��.r � f �� � �.

,r � f ���

Section Project: Anamorphic Art

Use the anamorphic transformations

and

to sketch the transformed polar image of the rectangular graph.When the reflection (in a cylindrical mirror centered at the pole) ofeach polar image is viewed from the polar axis, the viewer will seethe original rectangular image.

(a) (b)

(c) (d)

This example of anamorphic art is from the Museum of Science andIndustry in Manchester, England. When the reflection of the trans-formed “polar painting”is viewed in the mirror, the viewer sees faces.

FOR FURTHER INFORMATION For more information on anamor-phic art, see the article “Anamorphisms” by Philip Hickin in theMathematical Gazette.

x2 � �y � 5�2 � 52y � x � 5

x � 2y � 3

�3�

4≤ � ≤

3�

4� � �

8x,r � y � 16

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