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Calculus Volume 2

7.3 Polar Coordinates

Calculus Volume 27.3 Polar Coordinates
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  1. Preface
  2. 1 Integration
    1. Introduction
    2. 1.1 Approximating Areas
    3. 1.2 The Definite Integral
    4. 1.3 The Fundamental Theorem of Calculus
    5. 1.4 Integration Formulas and the Net Change Theorem
    6. 1.5 Substitution
    7. 1.6 Integrals Involving Exponential and Logarithmic Functions
    8. 1.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  3. 2 Applications of Integration
    1. Introduction
    2. 2.1 Areas between Curves
    3. 2.2 Determining Volumes by Slicing
    4. 2.3 Volumes of Revolution: Cylindrical Shells
    5. 2.4 Arc Length of a Curve and Surface Area
    6. 2.5 Physical Applications
    7. 2.6 Moments and Centers of Mass
    8. 2.7 Integrals, Exponential Functions, and Logarithms
    9. 2.8 Exponential Growth and Decay
    10. 2.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  4. 3 Techniques of Integration
    1. Introduction
    2. 3.1 Integration by Parts
    3. 3.2 Trigonometric Integrals
    4. 3.3 Trigonometric Substitution
    5. 3.4 Partial Fractions
    6. 3.5 Other Strategies for Integration
    7. 3.6 Numerical Integration
    8. 3.7 Improper Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  5. 4 Introduction to Differential Equations
    1. Introduction
    2. 4.1 Basics of Differential Equations
    3. 4.2 Direction Fields and Numerical Methods
    4. 4.3 Separable Equations
    5. 4.4 The Logistic Equation
    6. 4.5 First-order Linear Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  6. 5 Sequences and Series
    1. Introduction
    2. 5.1 Sequences
    3. 5.2 Infinite Series
    4. 5.3 The Divergence and Integral Tests
    5. 5.4 Comparison Tests
    6. 5.5 Alternating Series
    7. 5.6 Ratio and Root Tests
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Chapter Review Exercises
  7. 6 Power Series
    1. Introduction
    2. 6.1 Power Series and Functions
    3. 6.2 Properties of Power Series
    4. 6.3 Taylor and Maclaurin Series
    5. 6.4 Working with Taylor Series
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  8. 7 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 7.1 Parametric Equations
    3. 7.2 Calculus of Parametric Curves
    4. 7.3 Polar Coordinates
    5. 7.4 Area and Arc Length in Polar Coordinates
    6. 7.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. 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

  • 7.3.1. Locate points in a plane by using polar coordinates.
  • 7.3.2. Convert points between rectangular and polar coordinates.
  • 7.3.3. Sketch polar curves from given equations.
  • 7.3.4. Convert equations between rectangular and polar coordinates.
  • 7.3.5. 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 7.27. 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 7.27 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 7.4

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,
7.7
r2=x2+y2andtanθ=yx.r2=x2+y2andtanθ=yx.
7.8

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

Example 7.10

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)

Solution

  1. Use x=1x=1 and y=1y=1 in Equation 7.8:
    r2=x2+y2=12+12r=2andtanθ=yx=11=1θ=π4.r2=x2+y2=12+12r=2andtanθ=yx=11=1θ=π4.

    Therefore this point can be represented as (2,π4)(2,π4) in polar coordinates.
  2. Use x=−3x=−3 and y=4y=4 in Equation 7.8:
    r2=x2+y2=(−3)2+(4)2r=5andtanθ=yx=43θ=arctan(43)2.21.r2=x2+y2=(−3)2+(4)2r=5andtanθ=yx=43θ=arctan(43)2.21.

    Therefore this point can be represented as (5,2.21)(5,2.21) in polar coordinates.
  3. Use x=0x=0 and y=3y=3 in Equation 7.8:
    r2=x2+y2=(3)2+(0)2=9+0r=3andtanθ=yx=30.r2=x2+y2=(3)2+(0)2=9+0r=3andtanθ=yx=30.

    Direct application of the second equation leads to division by zero. Graphing the point (0,3)(0,3) on the rectangular coordinate system reveals that the point is located on the positive y-axis. The angle between the positive x-axis and the positive y-axis is π2.π2. Therefore this point can be represented as (3,π2)(3,π2) in polar coordinates.
  4. Use x=53x=53 and y=−5y=−5 in Equation 7.8:
    r2=x2+y2=(53)2+(−5)2=75+25r=10andtanθ=yx=−553=33θ=π6.r2=x2+y2=(53)2+(−5)2=75+25r=10andtanθ=yx=−553=33θ=π6.

    Therefore this point can be represented as (10,π6)(10,π6) in polar coordinates.
  5. Use r=3r=3 and θ=π3θ=π3 in Equation 7.7:
    x=rcosθ=3cos(π3)=3(12)=32andy=rsinθ=3sin(π3)=3(32)=332.x=rcosθ=3cos(π3)=3(12)=32andy=rsinθ=3sin(π3)=3(32)=332.

    Therefore this point can be represented as (32,332)(32,332) in rectangular coordinates.
  6. Use r=2r=2 and θ=3π2θ=3π2 in Equation 7.7:
    x=rcosθ=2cos(3π2)=2(0)=0andy=rsinθ=2sin(3π2)=2(−1)=−2.x=rcosθ=2cos(3π2)=2(0)=0andy=rsinθ=2sin(3π2)=2(−1)=−2.

    Therefore this point can be represented as (0,−2)(0,−2) in rectangular coordinates.
  7. Use r=6r=6 and θ=5π6θ=5π6 in Equation 7.7:
    x=rcosθ=6cos(5π6)=6(32)=−33andy=rsinθ=6sin(5π6)=6(12)=−3.x=rcosθ=6cos(5π6)=6(32)=−33andy=rsinθ=6sin(5π6)=6(12)=−3.

    Therefore this point can be represented as (−33,−3)(−33,−3) in rectangular coordinates.
Checkpoint 7.10

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 7.8:

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 7.28 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 7.28 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 7.11

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)

Solution

The three points are plotted in the following figure.

Three points are marked on a polar coordinate plane, specifically (2, π/4) in the first quadrant, (4, 5π/4) in the third quadrant, and (−3, 2π/3) in the fourth quadrant.
Figure 7.29 Three points plotted in the polar coordinate system.
Checkpoint 7.11

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 7.12

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.

Solution

Because the function is a multiple of a sine function, it is periodic with period 2π,2π, so use values for θθ between 0 and 2π.2π. The result of steps 1–3 appear in the following table. Figure 7.30 shows the graph based on this table.

θθ r=4sinθr=4sinθ θθ r=4sinθr=4sinθ
0 0 ππ 0
π6π6 22 7π67π6 −2−2
π4π4 222.8222.8 5π45π4 −22−2.8−22−2.8
π3π3 233.4233.4 4π34π3 −23−3.4−23−3.4
π2π2 44 3π23π2 44
2π32π3 233.4233.4 5π35π3 −23−3.4−23−3.4
3π43π4 222.8222.8 7π47π4 −22−2.8−22−2.8
5π65π6 22 11π611π6 −2−2
2π2π 0
On the polar coordinate plane, a circle is drawn with radius 2. It touches the origin, (2 times the square root of 2, π/4), (4, π/2), and (2 times the square root of 2, 3π/4).
Figure 7.30 The graph of the function r=4sinθr=4sinθ is a circle.

This is the graph of a circle. The equation r=4sinθr=4sinθ can be converted into rectangular coordinates by first multiplying both sides by r.r. This gives the equation r2=4rsinθ.r2=4rsinθ. Next use the facts that r2=x2+y2r2=x2+y2 and y=rsinθ.y=rsinθ. This gives x2+y2=4y.x2+y2=4y. To put this equation into standard form, subtract 4y4y from both sides of the equation and complete the square:

x2+y24y=0x2+(y24y)=0x2+(y24y+4)=0+4x2+(y2)2=4.x2+y24y=0x2+(y24y)=0x2+(y24y+4)=0+4x2+(y2)2=4.

This is the equation of a circle with radius 2 and center (0,2)(0,2) in the rectangular coordinate system.

Checkpoint 7.12

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

The graph in Example 7.12 was that of a circle. The equation of the circle can be transformed into rectangular coordinates using the coordinate transformation formulas in Equation 7.8. Example 7.14 gives some more examples of functions for transforming from polar to rectangular coordinates.

Example 7.13

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

Solution

  1. Take the tangent of both sides. This gives tanθ=tan(π/3)=3.tanθ=tan(π/3)=3. Since tanθ=y/xtanθ=y/x we can replace the left-hand side of this equation by y/x.y/x. This gives y/x=3,y/x=3, which can be rewritten as y=x3.y=x3. This is the equation of a straight line passing through the origin with slope 3.3. In general, any polar equation of the form θ=Kθ=K represents a straight line through the pole with slope equal to tanK.tanK.
  2. First, square both sides of the equation. This gives r2=9.r2=9. Next replace r2r2 with x2+y2.x2+y2. This gives the equation x2+y2=9,x2+y2=9, which is the equation of a circle centered at the origin with radius 3. In general, any polar equation of the form r=kr=k where k is a positive constant represents a circle of radius k centered at the origin. (Note: when squaring both sides of an equation it is possible to introduce new points unintentionally. This should always be taken into consideration. However, in this case we do not introduce new points. For example, (−3,π3)(−3,π3) is the same point as (3,4π3).)(3,4π3).)
  3. Multiply both sides of the equation by r.r. This leads to r2=6rcosθ8rsinθ.r2=6rcosθ8rsinθ. Next use the formulas
    r2=x2+y2,x=rcosθ,y=rsinθ.r2=x2+y2,x=rcosθ,y=rsinθ.

    This gives
    r2=6(rcosθ)8(rsinθ)x2+y2=6x8y.r2=6(rcosθ)8(rsinθ)x2+y2=6x8y.

    To put this equation into standard form, first move the variables from the right-hand side of the equation to the left-hand side, then complete the square.
    x2+y2=6x8yx26x+y2+8y=0(x26x)+(y2+8y)=0(x26x+9)+(y2+8y+16)=9+16(x3)2+(y+4)2=25.x2+y2=6x8yx26x+y2+8y=0(x26x)+(y2+8y)=0(x26x+9)+(y2+8y+16)=9+16(x3)2+(y+4)2=25.

    This is the equation of a circle with center at (3,−4)(3,−4) and radius 5. Notice that the circle passes through the origin since the center is 5 units away.
Checkpoint 7.13

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 7.31
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 7.32

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 7.33 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 7.34(a)). However, if the coefficient is irrational, then the curve never closes (Figure 7.34(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 7.34 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 7.34(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 7.14

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 7.35 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 7.35 How can we describe a spiral graph mathematically?

Solution

As the point P travels around the spiral in a counterclockwise direction, its distance d from the origin increases. Assume that the distance d is a constant multiple k of the angle θθ that the line segment OP makes with the positive x-axis. Therefore d(P,O)=kθ,d(P,O)=kθ, where OO is the origin. Now use the distance formula and some trigonometry:

d(P,O)=kθ(x0)2+(y0)2=karctan(yx)x2+y2=karctan(yx)arctan(yx)=x2+y2ky=xtan(x2+y2k).d(P,O)=kθ(x0)2+(y0)2=karctan(yx)x2+y2=karctan(yx)arctan(yx)=x2+y2ky=xtan(x2+y2k).

Although this equation describes the spiral, it is not possible to solve it directly for either x or y. However, if we use polar coordinates, the equation becomes much simpler. In particular, d(P,O)=r,d(P,O)=r, and θθ is the second coordinate. Therefore the equation for the spiral becomes r=kθ.r=kθ. Note that when θ=0θ=0 we also have r=0,r=0, so the spiral emanates from the origin. We can remove this restriction by adding a constant to the equation. Then the equation for the spiral becomes r=a+kθr=a+kθ for arbitrary constants aa and k.k. This is referred to as an Archimedean spiral, after the Greek mathematician Archimedes.

Another type of spiral is the logarithmic spiral, described by the function r=a·bθ.r=a·bθ. A graph of the function r=1.2(1.25θ)r=1.2(1.25θ) is given in Figure 7.36. This spiral describes the shell shape of the chambered nautilus.

This figure has two figures. The first is a shell with many chambers that increase in size from the center out. The second is a spiral with equation r = 1.2(1.25θ).
Figure 7.36 A logarithmic spiral is similar to the shape of the chambered nautilus shell. (credit: modification of work by Jitze Couperus, Flickr)

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 7.31 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 7.5

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 7.15

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.

Solution

Suppose the point (r,θ)(r,θ) is on the graph of r=3sin(2θ).r=3sin(2θ).

  1. To test for symmetry about the polar axis, first try replacing θθ with θ.θ. This gives r=3sin(2(θ))=−3sin(2θ).r=3sin(2(θ))=−3sin(2θ). Since this changes the original equation, this test is not satisfied. However, returning to the original equation and replacing rr with rr and θθ with πθπθ yields
    r=3sin(2(πθ))r=3sin(2π2θ)r=3sin(−2θ)r=−3sin2θ.r=3sin(2(πθ))r=3sin(2π2θ)r=3sin(−2θ)r=−3sin2θ.

    Multiplying both sides of this equation by −1−1 gives r=3sin2θ,r=3sin2θ, which is the original equation. This demonstrates that the graph is symmetric with respect to the polar axis.
  2. To test for symmetry with respect to the pole, first replace rr with r,r, which yields r=3sin(2θ).r=3sin(2θ). Multiplying both sides by −1 gives r=−3sin(2θ),r=−3sin(2θ), which does not agree with the original equation. Therefore the equation does not pass the test for this symmetry. However, returning to the original equation and replacing θθ with θ+πθ+π gives
    r=3sin(2(θ+π))=3sin(2θ+2π)=3(sin2θcos2π+cos2θsin2π)=3sin2θ.r=3sin(2(θ+π))=3sin(2θ+2π)=3(sin2θcos2π+cos2θsin2π)=3sin2θ.

    Since this agrees with the original equation, the graph is symmetric about the pole.
  3. To test for symmetry with respect to the vertical line θ=π2,θ=π2, first replace both rr with rr and θθ with θ.θ.
    r=3sin(2(θ))r=3sin(−2θ)r=−3sin2θ.r=3sin(2(θ))r=3sin(−2θ)r=−3sin2θ.

    Multiplying both sides of this equation by −1−1 gives r=3sin2θ,r=3sin2θ, which is the original equation. Therefore the graph is symmetric about the vertical line θ=π2.θ=π2.

This graph has symmetry with respect to the polar axis, the origin, and the vertical line going through the pole. To graph the function, tabulate values of θθ between 0 and π/2π/2 and then reflect the resulting graph.

θθ rr
00 00
π6π6 3322.63322.6
π4π4 33
π3π3 3322.63322.6
π2π2 00

This gives one petal of the rose, as shown in the following graph.

A single petal is graphed with equation r = 3 sin(2θ) for 0 ≤ θ ≤ π/2. It starts at the origin and reaches a maximum distance from the origin of 3.
Figure 7.37 The graph of the equation between θ=0θ=0 and θ=π/2.θ=π/2.

Reflecting this image into the other three quadrants gives the entire graph as shown.

A four-petaled rose is graphed with equation r = 3 sin(2θ). Each petal starts at the origin and reaches a maximum distance from the origin of 3.
Figure 7.38 The entire graph of the equation is called a four-petaled rose.
Checkpoint 7.14

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

Section 7.3 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) (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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