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Precalculus

8.3 Polar Coordinates

Precalculus8.3 Polar Coordinates
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Learning Objectives

In this section, you will:
  • Plot points using polar coordinates.
  • Convert from polar coordinates to rectangular coordinates.
  • Convert from rectangular coordinates to polar coordinates.
  • Transform equations between polar and rectangular forms.
  • Identify and graph polar equations by converting to rectangular equations.

Over 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a 16-knot wind (see Figure 1). How can the sailor indicate his location to the Coast Guard? In this section, we will investigate a method of representing location that is different from a standard coordinate grid.

An illustration of a boat on the polar grid.
Figure 1

Plotting Points Using Polar Coordinates

When we think about plotting points in the plane, we usually think of rectangular coordinates ( x,y ) ( x,y )in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates, which are points labeled ( r,θ ) ( r,θ )and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.

The polar grid is scaled as the unit circle with the positive x-axis now viewed as the polar axis and the origin as the pole. The first coordinate r ris the radius or length of the directed line segment from the pole. The angle θ, θ, measured in radians, indicates the direction of r. r.We move counterclockwise from the polar axis by an angle of θ, θ,and measure a directed line segment the length of r rin the direction of θ. θ.Even though we measure θ θfirst and then r, r, the polar point is written with the r-coordinate first. For example, to plot the point ( 2, π 4 ), ( 2, π 4 ),we would move π 4 π 4 units in the counterclockwise direction and then a length of 2 from the pole. This point is plotted on the grid in Figure 2.

Polar grid with point (2, pi/4) plotted.
Figure 2

Example 1

Plotting a Point on the Polar Grid

Plot the point ( 3, π 2 ) ( 3, π 2 )on the polar grid.

Try It #1

Plot the point ( 2, π 3 ) ( 2, π 3 )in the polar grid.

Example 2

Plotting a Point in the Polar Coordinate System with a Negative Component

Plot the point ( 2, π 6 ) ( 2, π 6 )on the polar grid.

Try It #2

Plot the points ( 3, π 6 ) ( 3, π 6 )and ( 2, 9π 4 ) ( 2, 9π 4 )on the same polar grid.

Converting from Polar Coordinates to Rectangular Coordinates

When given a set of polar coordinates, we may need to convert them to rectangular coordinates. To do so, we can recall the relationships that exist among the variables x,y,r, x,y,r,and θ. θ.

cosθ= x r x=rcosθ sinθ= y r y=rsinθ cosθ= x r x=rcosθ sinθ= y r y=rsinθ

Dropping a perpendicular from the point in the plane to the x-axis forms a right triangle, as illustrated in Figure 5. An easy way to remember the equations above is to think of cosθ cosθ as the adjacent side over the hypotenuse and sinθ sinθas the opposite side over the hypotenuse.

Comparison between polar coordinates and rectangular coordinates. There is a right triangle plotted on the x,y axis. The sides are a horizontal line on the x-axis of length x, a vertical line extending from thex-axis to some point in quadrant 1, and a hypotenuse r extending from the origin to that same point in quadrant 1. The vertices are at the origin (0,0), some point along the x-axis at (x,0), and that point in quadrant 1. This last point is (x,y) or (r, theta), depending which system of coordinates you use.
Figure 5

Converting from Polar Coordinates to Rectangular Coordinates

To convert polar coordinates ( r,θ ) ( r,θ )to rectangular coordinates ( x,y ), ( x,y ), let

cosθ= x r x=rcosθ cosθ= x r x=rcosθ
sinθ= y r y=rsinθ sinθ= y r y=rsinθ

How To

Given polar coordinates, convert to rectangular coordinates.

  1. Given the polar coordinate ( r,θ ), ( r,θ ), write x=rcosθ x=rcosθand y=rsinθ. y=rsinθ.
  2. Evaluate cosθ cosθand sinθ. sinθ.
  3. Multiply cosθ cosθby r rto find the x-coordinate of the rectangular form.
  4. Multiply sinθ sinθby r rto find the y-coordinate of the rectangular form.

Example 3

Writing Polar Coordinates as Rectangular Coordinates

Write the polar coordinates ( 3, π 2 ) ( 3, π 2 )as rectangular coordinates.

Example 4

Writing Polar Coordinates as Rectangular Coordinates

Write the polar coordinates ( 2,0 ) ( 2,0 )as rectangular coordinates.

Try It #3

Write the polar coordinates ( 1, 2π 3 ) ( 1, 2π 3 )as rectangular coordinates.

Converting from Rectangular Coordinates to Polar Coordinates

To convert rectangular coordinates to polar coordinates, we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.

Converting from Rectangular Coordinates to Polar Coordinates

Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated in Figure 8.

cosθ= x r  orx=rcosθ sinθ= y r  ory=rsinθ r 2 = x 2 + y 2 tanθ= y x cosθ= x r  orx=rcosθ sinθ= y r  ory=rsinθ r 2 = x 2 + y 2 tanθ= y x
Figure 8

Example 5

Writing Rectangular Coordinates as Polar Coordinates

Convert the rectangular coordinates ( 3,3 ) ( 3,3 )to polar coordinates.

Analysis

There are other sets of polar coordinates that will be the same as our first solution. For example, the points ( 3 2 , 5π 4 ) ( 3 2 , 5π 4 )and ( 3 2 , 7π 4 ) ( 3 2 , 7π 4 )will coincide with the original solution of ( 3 2 , π 4 ). ( 3 2 , π 4 ).The point ( 3 2 , 5π 4 ) ( 3 2 , 5π 4 )indicates a move further counterclockwise by π, π,which is directly opposite π 4 . π 4 .The radius is expressed as 3 2 . 3 2 .However, the angle 5π 4 5π 4 is located in the third quadrant and, as r ris negative, we extend the directed line segment in the opposite direction, into the first quadrant. This is the same point as ( 3 2 , π 4 ). ( 3 2 , π 4 ).The point ( 3 2 , 7π 4 ) ( 3 2 , 7π 4 )is a move further clockwise by 7π 4 , 7π 4 ,from π 4 . π 4 .The radius, 3 2 , 3 2 ,is the same.

Transforming Equations between Polar and Rectangular Forms

We can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms. Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems. We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation.

How To

Given an equation in polar form, graph it using a graphing calculator.

  1. Change the MODE to POL, representing polar form.
  2. Press the Y= button to bring up a screen allowing the input of six equations: r 1 , r 2 ,..., r 6 . r 1 , r 2 ,..., r 6 .
  3. Enter the polar equation, set equal to r. r.
  4. Press GRAPH.

Example 6

Writing a Cartesian Equation in Polar Form

Write the Cartesian equation x 2 + y 2 =9 x 2 + y 2 =9in polar form.

Example 7

Rewriting a Cartesian Equation as a Polar Equation

Rewrite the Cartesian equation x 2 + y 2 =6y x 2 + y 2 =6yas a polar equation.

Example 8

Rewriting a Cartesian Equation in Polar Form

Rewrite the Cartesian equation y=3x+2 y=3x+2as a polar equation.

Try It #4

Rewrite the Cartesian equation y 2 =3 x 2 y 2 =3 x 2 in polar form.

Identify and Graph Polar Equations by Converting to Rectangular Equations

We have learned how to convert rectangular coordinates to polar coordinates, and we have seen that the points are indeed the same. We have also transformed polar equations to rectangular equations and vice versa. Now we will demonstrate that their graphs, while drawn on different grids, are identical.

Example 9

Graphing a Polar Equation by Converting to a Rectangular Equation

Covert the polar equation r=2secθ r=2secθ to a rectangular equation, and draw its corresponding graph.

Example 10

Rewriting a Polar Equation in Cartesian Form

Rewrite the polar equation r= 3 12cosθ r= 3 12cosθ as a Cartesian equation.

Analysis

In this example, the right side of the equation can be expanded and the equation simplified further, as shown above. However, the equation cannot be written as a single function in Cartesian form. We may wish to write the rectangular equation in the hyperbola’s standard form. To do this, we can start with the initial equation.

                              x 2 + y 2 = (3+2x) 2           x 2 + y 2 (3+2x) 2 =0 x 2 + y 2 (9+12x+4 x 2 )=0    x 2 + y 2 912x4 x 2 =0             3 x 2 12x+ y 2 =9 Multiply through by −1.                3 x 2 +12x y 2 =9     3( x 2 +4x+) y 2 =9 Organize terms to complete the square forx.         3( x 2 +4x+4) y 2 =9+12                   3 (x+2) 2 y 2 =3                     (x+2) 2 y 2 3 =1                               x 2 + y 2 = (3+2x) 2           x 2 + y 2 (3+2x) 2 =0 x 2 + y 2 (9+12x+4 x 2 )=0    x 2 + y 2 912x4 x 2 =0             3 x 2 12x+ y 2 =9 Multiply through by −1.                3 x 2 +12x y 2 =9     3( x 2 +4x+) y 2 =9 Organize terms to complete the square forx.         3( x 2 +4x+4) y 2 =9+12                   3 (x+2) 2 y 2 =3                     (x+2) 2 y 2 3 =1
Try It #5

Rewrite the polar equation r=2sinθ r=2sinθ in Cartesian form.

Example 11

Rewriting a Polar Equation in Cartesian Form

Rewrite the polar equation r=sin( 2θ ) r=sin( 2θ )in Cartesian form.

Media

Access these online resources for additional instruction and practice with polar coordinates.

8.3 Section Exercises

Verbal

1.

How are polar coordinates different from rectangular coordinates?

2.

How are the polar axes different from the x- and y-axes of the Cartesian plane?

3.

Explain how polar coordinates are graphed.

4.

How are the points ( 3, π 2 ) ( 3, π 2 )and ( 3, π 2 ) ( 3, π 2 )related?

5.

Explain why the points ( 3, π 2 ) ( 3, π 2 )and ( 3, π 2 ) ( 3, π 2 )are the same.

Algebraic

For the following exercises, convert the given polar coordinates to Cartesian coordinates with r>0 r>0and 0θ2π. 0θ2π.Remember to consider the quadrant in which the given point is located when determining θ θfor the point.

6.

( 7, 7π 6 ) ( 7, 7π 6 )

7.

( 5,π ) ( 5,π )

8.

( 6, π 4 ) ( 6, π 4 )

9.

( 3, π 6 ) ( 3, π 6 )

10.

( 4, 7π 4 ) ( 4, 7π 4 )

For the following exercises, convert the given Cartesian coordinates to polar coordinates with r>0,0θ<2π. r>0,0θ<2π.Remember to consider the quadrant in which the given point is located.

11.

( 4,2 ) ( 4,2 )

12.

( 4,6 ) ( 4,6 )

13.

( 3,−5 ) ( 3,−5 )

14.

( −10,−13 ) ( −10,−13 )

15.

( 8,8 ) ( 8,8 )

For the following exercises, convert the given Cartesian equation to a polar equation.

16.

x=3 x=3

17.

y=4 y=4

18.

y=4 x 2 y=4 x 2

19.

y=2 x 4 y=2 x 4

20.

x 2 + y 2 =4y x 2 + y 2 =4y

21.

x 2 + y 2 =3x x 2 + y 2 =3x

22.

x 2 y 2 =x x 2 y 2 =x

23.

x 2 y 2 =3y x 2 y 2 =3y

24.

x 2 + y 2 =9 x 2 + y 2 =9

25.

x 2 =9y x 2 =9y

26.

y 2 =9x y 2 =9x

27.

9xy=1 9xy=1

For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

28.

r=3sinθ r=3sinθ

29.

r=4cosθ r=4cosθ

30.

r= 4 sinθ+7cosθ r= 4 sinθ+7cosθ

31.

r= 6 cosθ+3sinθ r= 6 cosθ+3sinθ

32.

r=2secθ r=2secθ

33.

r=3cscθ r=3cscθ

34.

r= rcosθ+2 r= rcosθ+2

35.

r 2 =4secθcscθ r 2 =4secθcscθ

36.

r=4 r=4

37.

r 2 =4 r 2 =4

38.

r= 1 4cosθ3sinθ r= 1 4cosθ3sinθ

39.

r= 3 cosθ5sinθ r= 3 cosθ5sinθ

Graphical

For the following exercises, find the polar coordinates of the point.

40.
Polar coordinate system with a point located on the third concentric circle and pi/2.
41.
Polar coordinate system with a point located on the third concentric circle and midway between pi/2 and pi in the second quadrant.
42.
Polar coordinate system with a point located midway between the first and second concentric circles and a third of the way between pi and 3pi/2 (closer to pi).
43.
Polar coordinate system with a point located on the fifth concentric circle and pi.
44.
Polar coordinate system with a point located on the fourth concentric circle and a third of the way between 3pi/2 and 2pi (closer to 3pi/2).

For the following exercises, plot the points.

45.

( 2, π 3 ) ( 2, π 3 )

46.

( 1, π 2 ) ( 1, π 2 )

47.

( 3.5, 7π 4 ) ( 3.5, 7π 4 )

48.

( 4, π 3 ) ( 4, π 3 )

49.

( 5, π 2 ) ( 5, π 2 )

50.

( 4, 5π 4 ) ( 4, 5π 4 )

51.

( 3, 5π 6 ) ( 3, 5π 6 )

52.

( 1.5, 7π 6 ) ( 1.5, 7π 6 )

53.

( 2, π 4 ) ( 2, π 4 )

54.

( 1, 3π 2 ) ( 1, 3π 2 )

For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.

55.

5xy=6 5xy=6

56.

2x+7y=3 2x+7y=3

57.

x 2 + ( y1 ) 2 =1 x 2 + ( y1 ) 2 =1

58.

( x+2 ) 2 + ( y+3 ) 2 =13 ( x+2 ) 2 + ( y+3 ) 2 =13

59.

x=2 x=2

60.

x 2 + y 2 =5y x 2 + y 2 =5y

61.

x 2 + y 2 =3x x 2 + y 2 =3x

For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.

62.

r=6 r=6

63.

r=4 r=4

64.

θ= 2π 3 θ= 2π 3

65.

θ= π 4 θ= π 4

66.

r=secθ r=secθ

67.

r=−10sinθ r=−10sinθ

68.

r=3cosθ r=3cosθ

Technology

69.

Use a graphing calculator to find the rectangular coordinates of ( 2, π 5 ). ( 2, π 5 ).Round to the nearest thousandth.

70.

Use a graphing calculator to find the rectangular coordinates of ( 3, 3π 7 ). ( 3, 3π 7 ).Round to the nearest thousandth.

71.

Use a graphing calculator to find the polar coordinates of ( 7,8 ) ( 7,8 )in degrees. Round to the nearest thousandth.

72.

Use a graphing calculator to find the polar coordinates of ( 3,4 ) ( 3,4 )in degrees. Round to the nearest hundredth.

73.

Use a graphing calculator to find the polar coordinates of ( 2,0 ) ( 2,0 )in radians. Round to the nearest hundredth.

Extensions

74.

Describe the graph of r=asecθ;a>0. r=asecθ;a>0.

75.

Describe the graph of r=asecθ;a<0. r=asecθ;a<0.

76.

Describe the graph of r=acscθ;a>0. r=acscθ;a>0.

77.

Describe the graph of r=acscθ;a<0. r=acscθ;a<0.

78.

What polar equations will give an oblique line?

For the following exercise, graph the polar inequality.

79.

r<4 r<4

80.

0θ π 4 0θ π 4

81.

θ= π 4 ,r2 θ= π 4 ,r2

82.

θ= π 4 ,r−3 θ= π 4 ,r−3

83.

0θ π 3 ,r<2 0θ π 3 ,r<2

84.

π 6 <θ π 3 ,3<r<2 π 6 <θ π 3 ,3<r<2

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