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Calculus Volume 3

1.3 Polar Coordinates

Calculus Volume 31.3 Polar Coordinates
  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

Learning Objectives

  • 1.3.1 Locate points in a plane by using polar coordinates.
  • 1.3.2 Convert points between rectangular and polar coordinates.
  • 1.3.3 Sketch polar curves from given equations.
  • 1.3.4 Convert equations between rectangular and polar coordinates.
  • 1.3.5 Identify symmetry in polar curves and equations.

The rectangular coordinate system (or Cartesian plane) provides a means of mapping points to ordered pairs and ordered pairs to points. This is called a one-to-one mapping from points in the plane to ordered pairs. The polar coordinate system provides an alternative method of mapping points to ordered pairs. In this section we see that in some circumstances, polar coordinates can be more useful than rectangular coordinates.

Defining Polar Coordinates

To find the coordinates of a point in the polar coordinate system, consider Figure 1.27. The point PP has Cartesian coordinates (x,y).(x,y). The line segment connecting the origin to the point PP measures the distance from the origin to PP and has length r.r. The angle between the positive xx-axis and the line segment has measure θ.θ. This observation suggests a natural correspondence between the coordinate pair (x,y)(x,y) and the values rr and θ.θ. This correspondence is the basis of the polar coordinate system. Note that every point in the Cartesian plane has two values (hence the term ordered pair) associated with it. In the polar coordinate system, each point also has two values associated with it: rr and θ.θ.

A point P(x, y) is given in the first quadrant with lines drawn to indicate its x and y values. There is a line from the origin to P(x, y) marked r and this line make an angle θ with the x axis.
Figure 1.27 An arbitrary point in the Cartesian plane.

Using right-triangle trigonometry, the following equations are true for the point P:P:

cosθ=xrsox=rcosθcosθ=xrsox=rcosθ
sinθ=yrsoy=rsinθ.sinθ=yrsoy=rsinθ.

Furthermore,

r2=x2+y2andtanθ=yx.r2=x2+y2andtanθ=yx.

Each point (x,y)(x,y) in the Cartesian coordinate system can therefore be represented as an ordered pair (r,θ)(r,θ) in the polar coordinate system. The first coordinate is called the radial coordinate and the second coordinate is called the angular coordinate. Every point in the plane can be represented in this form.

Note that the equation tanθ=y/xtanθ=y/x has an infinite number of solutions for any ordered pair (x,y).(x,y). However, if we restrict the solutions to values between 00 and 2π2π then we can assign a unique solution to the quadrant in which the original point (x,y)(x,y) is located. Then the corresponding value of r is positive, so r2=x2+y2.r2=x2+y2.

Theorem 1.4

Converting Points between Coordinate Systems

Given a point PP in the plane with Cartesian coordinates (x,y)(x,y) and polar coordinates (r,θ),(r,θ), the following conversion formulas hold true:

x=rcosθandy=rsinθ,x=rcosθandy=rsinθ,
1.7
r2=x2+y2andtanθ=yx.r2=x2+y2andtanθ=yx.
1.8

These formulas can be used to convert from rectangular to polar or from polar to rectangular coordinates.

Example 1.10

Converting between Rectangular and Polar Coordinates

Convert each of the following points into polar coordinates.

  1. (1,1)(1,1)
  2. (−3,4)(−3,4)
  3. (0,3)(0,3)
  4. (53,−5)(53,−5)

Convert each of the following points into rectangular coordinates.

  1. (3,π/3)(3,π/3)
  2. (2,3π/2)(2,3π/2)
  3. (6,−5π/6)(6,−5π/6)
Checkpoint 1.10

Convert (−8,−8)(−8,−8) into polar coordinates and (4,2π3)(4,2π3) into rectangular coordinates.

The polar representation of a point is not unique. For example, the polar coordinates (2,π3)(2,π3) and (2,7π3)(2,7π3) both represent the point (1,3)(1,3) in the rectangular system. Also, the value of rr can be negative. Therefore, the point with polar coordinates (−2,4π3)(−2,4π3) also represents the point (1,3)(1,3) in the rectangular system, as we can see by using Equation 1.8:

x=rcosθ=−2cos(4π3)=−2(12)=1andy=rsinθ=−2sin(4π3)=−2(32)=3.x=rcosθ=−2cos(4π3)=−2(12)=1andy=rsinθ=−2sin(4π3)=−2(32)=3.

Every point in the plane has an infinite number of representations in polar coordinates. However, each point in the plane has only one representation in the rectangular coordinate system.

Note that the polar representation of a point in the plane also has a visual interpretation. In particular, rr is the directed distance that the point lies from the origin, and θθ measures the angle that the line segment from the origin to the point makes with the positive xx-axis. Positive angles are measured in a counterclockwise direction and negative angles are measured in a clockwise direction. The polar coordinate system appears in the following figure.

A series of concentric circles is drawn with spokes indicating different values between 0 and 2π in increments of π/12. The first quadrant starts with 0 where the x axis would be, then the next spoke is marked π/12, then π/6, π/4, π/3, 5π/12, π/2, and so on into the second, third, and fourth quadrants. The polar axis is noted near the former x axis line.
Figure 1.28 The polar coordinate system.

The line segment starting from the center of the graph going to the right (called the positive x-axis in the Cartesian system) is the polar axis. The center point is the pole, or origin, of the coordinate system, and corresponds to r=0.r=0. The innermost circle shown in Figure 1.28 contains all points a distance of 1 unit from the pole, and is represented by the equation r=1.r=1. Then r=2r=2 is the set of points 2 units from the pole, and so on. The line segments emanating from the pole correspond to fixed angles. To plot a point in the polar coordinate system, start with the angle. If the angle is positive, then measure the angle from the polar axis in a counterclockwise direction. If it is negative, then measure it clockwise. If the value of rr is positive, move that distance along the terminal ray of the angle. If it is negative, move along the ray that is opposite the terminal ray of the given angle.

Example 1.11

Plotting Points in the Polar Plane

Plot each of the following points on the polar plane.

  1. (2,π4)(2,π4)
  2. (−3,2π3)(−3,2π3)
  3. (4,5π4)(4,5π4)
Checkpoint 1.11

Plot (4,5π3)(4,5π3) and (−3,7π2)(−3,7π2) on the polar plane.

Polar Curves

Now that we know how to plot points in the polar coordinate system, we can discuss how to plot curves. In the rectangular coordinate system, we can graph a function y=f(x)y=f(x) and create a curve in the Cartesian plane. In a similar fashion, we can graph a curve that is generated by a function r=f(θ).r=f(θ).

The general idea behind graphing a function in polar coordinates is the same as graphing a function in rectangular coordinates. Start with a list of values for the independent variable (θ(θ in this case) and calculate the corresponding values of the dependent variable r.r. This process generates a list of ordered pairs, which can be plotted in the polar coordinate system. Finally, connect the points, and take advantage of any patterns that may appear. The function may be periodic, for example, which indicates that only a limited number of values for the independent variable are needed.

Problem-Solving Strategy: Plotting a Curve in Polar Coordinates
  1. Create a table with two columns. The first column is for θ,θ, and the second column is for r.r.
  2. Create a list of values for θ.θ.
  3. Calculate the corresponding rr values for each θ.θ.
  4. Plot each ordered pair (r,θ)(r,θ) on the coordinate axes.
  5. Connect the points and look for a pattern.

Media

Watch this video for more information on sketching polar curves.

Example 1.12

Graphing a Function in Polar Coordinates

Graph the curve defined by the function r=4sinθ.r=4sinθ. Identify the curve and rewrite the equation in rectangular coordinates.

Checkpoint 1.12

Create a graph of the curve defined by the function r=4+4cosθ.r=4+4cosθ.

The graph in Example 1.12 was that of a circle. The equation of the circle can be transformed into rectangular coordinates using the coordinate transformation formulas in Equation 1.8. Example 1.14 gives some more examples of functions for transforming from polar to rectangular coordinates.

Example 1.13

Transforming Polar Equations to Rectangular Coordinates

Rewrite each of the following equations in rectangular coordinates and identify the graph.

  1. θ=π3θ=π3
  2. r=3r=3
  3. r=6cosθ8sinθr=6cosθ8sinθ
Checkpoint 1.13

Rewrite the equation r=secθtanθr=secθtanθ in rectangular coordinates and identify its graph.

We have now seen several examples of drawing graphs of curves defined by polar equations. A summary of some common curves is given in the tables below. In each equation, a and b are arbitrary constants.

This table has three columns and 3 rows. The first row is a header row and is given from left to right as name, equation, and example. The second row is Line passing through the pole with slope tan K; θ = K; and a picture of a straight line on the polar coordinate plane with θ = π/3. The third row is Circle; r = a cosθ + b sinθ; and a picture of a circle on the polar coordinate plane with equation r = 2 cos(t) – 3 sin(t): the circle touches the origin but has center in the third quadrant.
Figure 1.31
This table has three columns and 3 rows. The first row is Spiral; r = a + bθ; and a picture of a spiral starting at the origin with equation r = θ/3. The second row is Cardioid; r = a(1 + cosθ), r = a(1 – cosθ), r = a(1 + sinθ), r = a(1 – sinθ); and a picture of a cardioid with equation r = 3(1 + cosθ): the cardioid looks like a heart turned on its side with a rounded bottom instead of a pointed one. The third row is Limaçon; r = a cosθ + b, r = a sinθ + b; and a picture of a limaçon with equation r = 2 + 4 sinθ: the figure looks like a deformed circle with a loop inside of it. The seventh row is Rose; r = a cos(bθ), r = a sin(bθ); and a picture of a rose with equation r = 3 sin(2θ): the rose looks like a flower with four petals, one petal in each quadrant, each with length 3 and reaching to the origin between each petal.
Figure 1.32

A cardioid is a special case of a limaçon (pronounced “lee-mah-son”), in which a=ba=b or a=b.a=b. The rose is a very interesting curve. Notice that the graph of r=3sin2θr=3sin2θ has four petals. However, the graph of r=3sin3θr=3sin3θ has three petals as shown.

A rose with three petals, one in the first quadrant, another in the second quadrant, and the third in both the third and fourth quadrants, each with length 3. Each petal starts and ends at the origin.
Figure 1.33 Graph of r=3sin3θ.r=3sin3θ.

If the coefficient of θθ is even, the graph has twice as many petals as the coefficient. If the coefficient of θθ is odd, then the number of petals equals the coefficient. You are encouraged to explore why this happens. Even more interesting graphs emerge when the coefficient of θθ is not an integer. For example, if it is rational, then the curve is closed; that is, it eventually ends where it started (Figure 1.34(a)). However, if the coefficient is irrational, then the curve never closes (Figure 1.34(b)). Although it may appear that the curve is closed, a closer examination reveals that the petals just above the positive x axis are slightly thicker. This is because the petal does not quite match up with the starting point.

This figure has two figures. The first is a rose with so many overlapping petals that there are a few patterns that develop, starting with a sharp 10 pointed star in the center and moving out to an increasingly rounded set of petals. The second figure is a rose with even more overlapping petals, so many so that it is impossible to tell what is happening in the center, but on the outer edges are a number of sharply rounded petals.
Figure 1.34 Polar rose graphs of functions with (a) rational coefficient and (b) irrational coefficient. Note that the rose in part (b) would actually fill the entire circle if plotted in full.

Since the curve defined by the graph of r=3sin(πθ)r=3sin(πθ) never closes, the curve depicted in Figure 1.34(b) is only a partial depiction. In fact, this is an example of a space-filling curve. A space-filling curve is one that in fact occupies a two-dimensional subset of the real plane. In this case the curve occupies the circle of radius 3 centered at the origin.

Example 1.14

Chapter Opener: Describing a Spiral

Recall the chambered nautilus introduced in the chapter opener. This creature displays a spiral when half the outer shell is cut away. It is possible to describe a spiral using rectangular coordinates. Figure 1.35 shows a spiral in rectangular coordinates. How can we describe this curve mathematically?

A spiral starting at the origin and continually increasing its radius to a point P(x, y).
Figure 1.35 How can we describe a spiral graph mathematically?

Suppose a curve is described in the polar coordinate system via the function r=f(θ).r=f(θ). Since we have conversion formulas from polar to rectangular coordinates given by

x=rcosθy=rsinθ,x=rcosθy=rsinθ,

it is possible to rewrite these formulas using the function

x=f(θ)cosθy=f(θ)sinθ.x=f(θ)cosθy=f(θ)sinθ.

This step gives a parameterization of the curve in rectangular coordinates using θθ as the parameter. For example, the spiral formula r=a+bθr=a+bθ from Figure 1.31 becomes

x=(a+bθ)cosθy=(a+bθ)sinθ.x=(a+bθ)cosθy=(a+bθ)sinθ.

Letting θθ range from to generates the entire spiral.

Symmetry in Polar Coordinates

When studying symmetry of functions in rectangular coordinates (i.e., in the form y=f(x)),y=f(x)), we talk about symmetry with respect to the y-axis and symmetry with respect to the origin. In particular, if f(x)=f(x)f(x)=f(x) for all xx in the domain of f,f, then ff is an even function and its graph is symmetric with respect to the y-axis. If f(x)=f(x)f(x)=f(x) for all xx in the domain of f,f, then ff is an odd function and its graph is symmetric with respect to the origin. By determining which types of symmetry a graph exhibits, we can learn more about the shape and appearance of the graph. Symmetry can also reveal other properties of the function that generates the graph. Symmetry in polar curves works in a similar fashion.

Theorem 1.5

Symmetry in Polar Curves and Equations

Consider a curve generated by the function r=f(θ)r=f(θ) in polar coordinates.

  1. The curve is symmetric about the polar axis if for every point (r,θ)(r,θ) on the graph, the point (r,θ)(r,θ) is also on the graph. Similarly, the equation r=f(θ)r=f(θ) is unchanged by replacing θθ with θ.θ.
  2. The curve is symmetric about the pole if for every point (r,θ)(r,θ) on the graph, the point (r,π+θ)(r,π+θ) is also on the graph. Similarly, the equation r=f(θ)r=f(θ) is unchanged when replacing rr with r,r, or θθ with π+θ.π+θ.
  3. The curve is symmetric about the vertical line θ=π2θ=π2 if for every point (r,θ)(r,θ) on the graph, the point (r,πθ)(r,πθ) is also on the graph. Similarly, the equation r=f(θ)r=f(θ) is unchanged when θθ is replaced by πθ.πθ.

The following table shows examples of each type of symmetry.

This table has three rows and two columns. The first row reads “Symmetry with respect to the polar axis: For every point (r, θ) on the graph, there is also a point reflected directly across the horizontal (polar) axis” and it has a picture of a cardioid with equation r = 2 – 2 cosθ: this cardioid has points marked (r, θ) and (r, −θ), which are symmetric about the x axis, and the entire cardioid is symmetric about the x axis. The second row reads “Symmetry with respect to the pole: For every point (r, θ) on the graph, there is also a point on the graph that is reflected through the pole as well” and it has a picture of a skewed infinity symbol with equation r2 = 9 cos(2θ – π/2): this figure has points marked (r, θ) and (−r, θ), which are symmetric about the pole, and the entire figure is symmetric about the pole. The third row reads “Symmetry with respect to the vertical line θ = π/2: For every point (r, θ) on the graph, there is also a point reflected directly across the vertical axis” and there is a picture of a cardioid with equation r = 2 – 2 sinθ: this figure has points marked (r, θ) and (r, π − θ), which are symmetric about the vertical line θ = π/2, and the entire cardioid is symmetric about the vertical line θ = π/2.

Example 1.15

Using Symmetry to Graph a Polar Equation

Find the symmetry of the rose defined by the equation r=3sin(2θ)r=3sin(2θ) and create a graph.

Checkpoint 1.14

Determine the symmetry of the graph determined by the equation r=2cos(3θ)r=2cos(3θ) and create a graph.

Section 1.3 Exercises

In the following exercises, plot the point whose polar coordinates are given by first constructing the angle θθ and then marking off the distance r along the ray.

125.

(3,π6)(3,π6)

126.

(−2,5π3)(−2,5π3)

127.

(0,7π6)(0,7π6)

128.

(−4,3π4)(−4,3π4)

129.

(1,π4)(1,π4)

130.

(2,5π6)(2,5π6)

131.

(1,π2)(1,π2)

For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point.

The polar coordinate plane is divided into 12 pies. Point A is drawn on the first circle on the first spoke above the θ = 0 line in the first quadrant. Point B is drawn in the fourth quadrant on the third circle and the second spoke below the θ = 0 line. Point C is drawn on the θ = π line on the third circle. Point D is drawn on the fourth circle on the first spoke below the θ = π line.
132.

Coordinates of point A.

133.

Coordinates of point B.

134.

Coordinates of point C.

135.

Coordinates of point D.

For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0,2π].(0,2π]. Round to three decimal places.

136.

(2,2)(2,2)

137.

(3,−4)(3,−4)

138.

(8,15)(8,15)

139.

(−6,8)(−6,8)

140.

(4,3)(4,3)

141.

(3,3)(3,3)

For the following exercises, find rectangular coordinates for the given point in polar coordinates.

142.

(2,5π4)(2,5π4)

143.

(−2,π6)(−2,π6)

144.

(5,π3)(5,π3)

145.

(1,7π6)(1,7π6)

146.

(−3,3π4)(−3,3π4)

147.

(0,π2)(0,π2)

148.

(−4.5,6.5)(−4.5,6.5)

For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the xx-axis, the yy-axis, or the origin.

149.

r=3sin(2θ)r=3sin(2θ)

150.

r2=9cosθr2=9cosθ

151.

r=cos(θ5)r=cos(θ5)

152.

r=2secθr=2secθ

153.

r=1+cosθr=1+cosθ

For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.

154.

r=3r=3

155.

θ=π4θ=π4

156.

r=secθr=secθ

157.

r=cscθr=cscθ

For the following exercises, convert the rectangular equation to polar form and sketch its graph.

158.

x2+y2=16x2+y2=16

159.

x2y2=16x2y2=16

160.

x=8x=8

For the following exercises, convert the rectangular equation to polar form and sketch its graph.

161.

3xy=23xy=2

162.

y2=4xy2=4x

For the following exercises, convert the polar equation to rectangular form and sketch its graph.

163.

r=4sinθr=4sinθ

164.

r=6cosθr=6cosθ

165.

r=θr=θ

166.

r=cotθcscθr=cotθcscθ

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

167.

r=1+sinθr=1+sinθ

168.

r=32cosθr=32cosθ

169.

r=22sinθr=22sinθ

170.

r=54sinθr=54sinθ

171.

r=3cos(2θ)r=3cos(2θ)

172.

r=3sin(2θ)r=3sin(2θ)

173.

r=2cos(3θ)r=2cos(3θ)

174.

r=3cos(θ2)r=3cos(θ2)

175.

r2=4cos(2θ)r2=4cos(2θ)

176.

r2=4sinθr2=4sinθ

177.

r=2θr=2θ

178.

[T] The graph of r=2cos(2θ)sec(θ).r=2cos(2θ)sec(θ). is called a strophoid. Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.

179.

[T] Use a graphing utility and sketch the graph of r=62sinθ3cosθ.r=62sinθ3cosθ.

180.

[T] Use a graphing utility to graph r=11cosθ.r=11cosθ.

181.

[T] Use technology to graph r=esin(θ)2cos(4θ).r=esin(θ)2cos(4θ).

182.

[T] Use technology to plot r=sin(3θ7)r=sin(3θ7) (use the interval 0θ14π).0θ14π).

183.

Without using technology, sketch the polar curve θ=2π3.θ=2π3.

184.

[T] Use a graphing utility to plot r=θsinθr=θsinθ for πθπ.πθπ.

185.

[T] Use technology to plot r=e−0.1θr=e−0.1θ for −10θ10.−10θ10.

186.

[T] There is a curve known as the “Black Hole.” Use technology to plot r=e−0.01θr=e−0.01θ for −100θ100.−100θ100.

187.

[T] Use the results of the preceding two problems to explore the graphs of r=e−0.001θr=e−0.001θ and r=e−0.0001θr=e−0.0001θ for |θ|>100.|θ|>100.

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