8.5: Polar Coordinates - Graphs

Learning Objectives

In this section you will:

• Test polar equations for symmetry.
• Graph polar equations by plotting points.

The planets move through space in elliptical, periodic orbits about the sun, as shown in Figure 1. They are in constant motion, so fixing an exact position of any planet is valid only for a moment. In other words, we can fix only a planet’s instantaneous position. This is one application of polar coordinates, represented as (r,θ).(r,θ). We interpret rr as the distance from the sun and θθ as the planet’s angular bearing, or its direction from a fixed point on the sun. In this section, we will focus on the polar system and the graphs that are generated directly from polar coordinates.

Figure 1 Planets follow elliptical paths as they orbit around the Sun. (credit: modification of work by NASA/JPL-Caltech)

Testing Polar Equations for Symmetry

Just as a rectangular equation such as y=x2y=x2 describes the relationship between xx and yy on a Cartesian grid, a polar equation describes a relationship between rr and θθ on a polar grid. Recall that the coordinate pair (r,θ)(r,θ) indicates that we move counterclockwise from the polar axis (positive x-axis) by an angle of θ,θ, and extend a ray from the pole (origin) rr units in the direction of θ.θ. All points that satisfy the polar equation are on the graph.

Symmetry is a property that helps us recognize and plot the graph of any equation. If an equation has a graph that is symmetric with respect to an axis, it means that if we folded the graph in half over that axis, the portion of the graph on one side would coincide with the portion on the other side. By performing three tests, we will see how to apply the properties of symmetry to polar equations. Further, we will use symmetry (in addition to plotting key points, zeros, and maximums of r)r) to determine the graph of a polar equation.

In the first test, we consider symmetry with respect to the line θ=π2θ=π2 (y-axis). We replace (r,θ)(r,θ) with (−r,−θ)(−r,−θ) to determine if the new equation is equivalent to the original equation. For example, suppose we are given the equation r=2sinθ;r=2sinθ;

r=2sinθ−r=2sin(−θ)−r=−2sinθr=2sinθReplace(r,θ)with (−r,−θ).Identity: sin(−θ)=−sinθ.Multiply both sides by−1.r=2sinθ−r=2sin(−θ)Replace(r,θ)with (−r,−θ).−r=−2sinθIdentity: sin(−θ)=−sinθ.r=2sinθMultiply both sides by−1.

This equation exhibits symmetry with respect to the line θ=π2.θ=π2.

In the second test, we consider symmetry with respect to the polar axis ( xx-axis). We replace (r,θ)(r,θ) with (r,−θ)(r,−θ) or (−r,π−θ)(−r,π−θ) to determine equivalency between the tested equation and the original. For example, suppose we are given the equation r=1−2cosθ.r=1−2cosθ.

r=1−2cosθr=1−2cos(−θ)r=1−2cosθReplace (r,θ)with(r,−θ).Even/Odd identityr=1−2cosθr=1−2cos(−θ)Replace (r,θ)with(r,−θ).r=1−2cosθEven/Odd identity

The graph of this equation exhibits symmetry with respect to the polar axis.

In the third test, we consider symmetry with respect to the pole (origin). We replace (r,θ)(r,θ) with (−r,θ)(−r,θ) to determine if the tested equation is equivalent to the original equation. For example, suppose we are given the equation r=2sin(3θ).r=2sin(3θ).

r=2sin(3θ)−r=2sin(3θ)r=2sin(3θ)−r=2sin(3θ)

The equation has failed the symmetry test, but that does not mean that it is not symmetric with respect to the pole. Passing one or more of the symmetry tests verifies that symmetry will be exhibited in a graph. However, failing the symmetry tests does not necessarily indicate that a graph will not be symmetric about the line θ=π2,θ=π2, the polar axis, or the pole. In these instances, we can confirm that symmetry exists by plotting reflecting points across the apparent axis of symmetry or the pole. Testing for symmetry is a technique that simplifies the graphing of polar equations, but its application is not perfect.

SYMMETRY TESTS

A polar equation describes a curve on the polar grid. The graph of a polar equation can be evaluated for three types of symmetry, as shown in Figure 2.

Figure 2 (a) A graph is symmetric with respect to the line θ=π2θ=π2 (y-axis) if replacing (r,θ)(r,θ) with (−r,−θ)(−r,−θ) yields an equivalent equation. (b) A graph is symmetric with respect to the polar axis (x-axis) if replacing (r,θ)(r,θ) with (r,−θ)(r,−θ) or (−r,π−θ)(−r,π−θ) yields an equivalent equation. (c) A graph is symmetric with respect to the pole (origin) if replacing (r,θ)(r,θ) with (−r,θ)(−r,θ) yields an equivalent equation.

HOW TO

Given a polar equation, test for symmetry.

1. Substitute the appropriate combination of components for (r,θ):(r,θ): (−r,−θ)(−r,−θ) for θ=π2θ=π2 symmetry; (r,−θ)(r,−θ) for polar axis symmetry; and (−r,θ)(−r,θ) for symmetry with respect to the pole.
2. If the resulting equations are equivalent in one or more of the tests, the graph produces the expected symmetry.

EXAMPLE 1

Testing a Polar Equation for Symmetry

Test the equation r=2sinθr=2sinθ for symmetry.

Answer

Analysis

Using a graphing calculator, we can see that the equation r=2sinθr=2sinθ is a circle centered at (0,1)(0,1) with radius r=1r=1 and is indeed symmetric to the line θ=π2.θ=π2. We can also see that the graph is not symmetric with the polar axis or the pole. See Figure 3.

Figure 3

TRY IT #1

Test the equation for symmetry: r=−2cosθ.r=−2cosθ.

Graphing Polar Equations by Plotting Points

To graph in the rectangular coordinate system we construct a table of xx and yy values. To graph in the polar coordinate system we construct a table of θθ and rr values. We enter values of θθ into a polar equation and calculate r.r. However, using the properties of symmetry and finding key values of θθ and rr means fewer calculations will be needed.

Finding Zeros and Maxima

To find the zeros of a polar equation, we solve for the values of θθ that result in r=0.r=0. Recall that, to find the zeros of polynomial functions, we set the equation equal to zero and then solve for x.x. We use the same process for polar equations. Set r=0,r=0, and solve for θ.θ.

For many of the forms we will encounter, the maximum value of a polar equation is found by substituting those values of θθ into the equation that result in the maximum value of the trigonometric functions. Consider r=5cosθ;r=5cosθ; the maximum distance between the curve and the pole is 5 units. The maximum value of the cosine function is 1 when θ=0,θ=0, so our polar equation is 5cosθ,5cosθ, and the value θ=0θ=0 will yield the maximum |r|.| r |.

Similarly, the maximum value of the sine function is 1 when θ=π2,θ=π2, and if our polar equation is r=5sinθ,r=5sinθ, the value θ=π2θ=π2 will yield the maximum |r|.| r |. We may find additional information by calculating values of rr when θ=0.θ=0. These points would be polar axis intercepts, which may be helpful in drawing the graph and identifying the curve of a polar equation.

EXAMPLE 2

Finding Zeros and Maximum Values for a Polar Equation

Using the equation in Example 1, find the zeros and maximum |r|| r | and, if necessary, the polar axis intercepts of r=2sinθ.r=2sinθ.

Answer

Analysis

The point (2,π2)(2,π2) will be the maximum value on the graph. Let’s plot a few more points to verify the graph of a circle. See Table 2 and Figure 4.

θθ	r=2sinθr=2sinθ	rr
0	r=2sin(0)=0r=2sin(0)=0	00
π6π6	r=2sin(π6)=1r=2sin(π6)=1	11
π3π3	r=2sin(π3)≈1.73r=2sin(π3)≈1.73	1.731.73
π2π2	r=2sin(π2)=2r=2sin(π2)=2	22
2π32π3	r=2sin(2π3)≈1.73r=2sin(2π3)≈1.73	1.731.73
5π65π6	r=2sin(5π6)=1r=2sin(5π6)=1	11
ππ	r=2sin(π)=0r=2sin(π)=0	00

Table 2

Figure 4

TRY IT #2

Without converting to Cartesian coordinates, test the given equation for symmetry and find the zeros and maximum values of |r|:| r |: r=3cosθ.r=3cosθ.

Investigating Circles

Now we have seen the equation of a circle in the polar coordinate system. In the last two examples, the same equation was used to illustrate the properties of symmetry and demonstrate how to find the zeros, maximum values, and plotted points that produced the graphs. However, the circle is only one of many shapes in the set of polar curves.

There are five classic polar curves: cardioids, limaҫons, lemniscates, rose curves, and Archimedes’ spirals. We will briefly touch on the polar formulas for the circle before moving on to the classic curves and their variations.

FORMULAS FOR THE EQUATION OF A CIRCLE

Some of the formulas that produce the graph of a circle in polar coordinates are given by r=acosθr=acosθ and r=asinθ,r=asinθ, where aa is the diameter of the circle or the distance from the pole to the farthest point on the circumference. The radius is |a|2,| a |2, or one-half the diameter. For r=acosθ, r=acosθ,  the center is (a2,0).(a2,0). For r=asinθ,r=asinθ, the center is (a2,π2).(a2,π2). Figure 5 shows the graphs of these four circles.

Figure 5

EXAMPLE 3

Sketching the Graph of a Polar Equation for a Circle

Sketch the graph of r=4cosθ.r=4cosθ.

Answer

Investigating Cardioids

While translating from polar coordinates to Cartesian coordinates may seem simpler in some instances, graphing the classic curves is actually less complicated in the polar system. The next curve is called a cardioid, as it resembles a heart. This shape is often included with the family of curves called limaçons, but here we will discuss the cardioid on its own.

FORMULAS FOR A CARDIOID

The formulas that produce the graphs of a cardioid are given by r=a±bcosθr=a±bcosθ and r=a±bsinθr=a±bsinθ where a>0,a>0, b>0,b>0, and ab=1.ab=1. The cardioid graph passes through the pole, as we can see in Figure 7.

Figure 7

HOW TO

Given the polar equation of a cardioid, sketch its graph.

1. Check equation for the three types of symmetry.
2. Find the zeros. Set r=0.r=0.
3. Find the maximum value of the equation according to the maximum value of the trigonometric expression.
4. Make a table of values for rr and θ.θ.
5. Plot the points and sketch the graph.

EXAMPLE 4

Sketching the Graph of a Cardioid

Sketch the graph of r=2+2cosθ.r=2+2cosθ.

Answer

Investigating Limaçons

The word limaçon is Old French for “snail,” a name that describes the shape of the graph. As mentioned earlier, the cardioid is a member of the limaçon family, and we can see the similarities in the graphs. The other images in this category include the one-loop limaçon and the two-loop (or inner-loop) limaçon. One-loop limaçons are sometimes referred to as dimpled limaçons when 1<ab<21<ab<2 and convex limaçons when ab≥2.ab≥2.

FORMULAS FOR ONE-LOOP LIMAÇONS

The formulas that produce the graph of a dimpled one-loop limaçon are given by r=a±bcosθr=a±bcosθ and r=a±bsinθr=a±bsinθ where a>0,b>0,a>0,b>0, and 1<ab<2.and 1<ab<2. All four graphs are shown in Figure 9.

Figure 9 Dimpled limaçons

HOW TO

Given a polar equation for a one-loop limaçon, sketch the graph.

1. Test the equation for symmetry. Remember that failing a symmetry test does not mean that the shape will not exhibit symmetry. Often the symmetry may reveal itself when the points are plotted.
2. Find the zeros.
3. Find the maximum values according to the trigonometric expression.
4. Make a table.
5. Plot the points and sketch the graph.

EXAMPLE 5

Sketching the Graph of a One-Loop Limaçon

Graph the equation r=4−3sinθ.r=4−3sinθ.

Answer

Analysis

This is an example of a curve for which making a table of values is critical to producing an accurate graph. The symmetry tests fail; the zero is undefined. While it may be apparent that an equation involving sinθsinθ is likely symmetric with respect to the line θ=π2,θ=π2, evaluating more points helps to verify that the graph is correct.

TRY IT #3

Sketch the graph of r=3−2cosθ.r=3−2cosθ.

Another type of limaçon, the inner-loop limaçon, is named for the loop formed inside the general limaçon shape. It was discovered by the German artist Albrecht Dürer(1471-1528), who revealed a method for drawing the inner-loop limaçon in his 1525 book Underweysung der Messing. A century later, the father of mathematician Blaise Pascal, Étienne Pascal(1588-1651), rediscovered it.

FORMULAS FOR INNER-LOOP LIMAÇONS

The formulas that generate the inner-loop limaçons are given by r=a±bcosθr=a±bcosθ and r=a±bsinθr=a±bsinθ where a>0,a>0, b>0,b>0, and a<b.a<b. The graph of the inner-loop limaçon passes through the pole twice: once for the outer loop, and once for the inner loop. See Figure 11 for the graphs.

Figure 11

EXAMPLE 6

Sketching the Graph of an Inner-Loop Limaçon

Sketch the graph of r=2+5cosθ.r=2+5cosθ.

Answer

Investigating Lemniscates

The lemniscate is a polar curve resembling the infinity symbol ∞∞ or a figure 8. Centered at the pole, a lemniscate is symmetrical by definition.

FORMULAS FOR LEMNISCATES

The formulas that generate the graph of a lemniscate are given by r2=a2cos2θr2=a2cos2θ and r2=a2sin2θr2=a2sin2θ where a≠0.a≠0. The formula r2=a2sin2θr2=a2sin2θ is symmetric with respect to the pole. The formula r2=a2cos2θr2=a2cos2θ is symmetric with respect to the pole, the line θ=π2,θ=π2, and the polar axis. See Figure 13 for the graphs.

Figure 13

EXAMPLE 7

Sketching the Graph of a Lemniscate

Sketch the graph of r2=4cos2θ.r2=4cos2θ.

Answer

Analysis

Making a substitution such as u=2θu=2θ is a common practice in mathematics because it can make calculations simpler. However, we must not forget to replace the substitution term with the original term at the end, and then solve for the unknown.

Some of the points on this graph may not show up using the Trace function on the TI-84 graphing calculator, and the calculator table may show an error for these same points of r.r. This is because there are no real square roots for these values of θ.θ. In other words, the corresponding r-values of 4cos(2θ)−−−−−−−√4cos(2θ) are complex numbers because there is a negative number under the radical.

Investigating Rose Curves

The next type of polar equation produces a petal-like shape called a rose curve. Although the graphs look complex, a simple polar equation generates the pattern.

ROSE CURVES

The formulas that generate the graph of a rose curve are given by r=acosnθr=acosnθ and r=asinnθr=asinnθ where a≠0.a≠0. If nn is even, the curve has 2n2n petals. If nn is odd, the curve has nn petals. See Figure 15.

Figure 15

EXAMPLE 8

Sketching the Graph of a Rose Curve (n Even)

Sketch the graph of r=2cos4θ.r=2cos4θ.

Answer

Analysis

When these curves are drawn, it is best to plot the points in order, as in the Table 8. This allows us to see how the graph hits a maximum (the tip of a petal), loops back crossing the pole, hits the opposite maximum, and loops back to the pole. The action is continuous until all the petals are drawn.

TRY IT #4

Sketch the graph of r=4sin(2θ).r=4sin(2θ).

EXAMPLE 9

Sketching the Graph of a Rose Curve (n Odd)

Sketch the graph of r=2sin(5θ).r=2sin(5θ).

Answer

TRY IT #5

Sketch the graph of r=3cos(3θ).r=3cos(3θ).

Investigating the Archimedes’ Spiral

The final polar equation we will discuss is the Archimedes’ spiral, named for its discoverer, the Greek mathematician Archimedes (c. 287 BCE-c. 212 BCE), who is credited with numerous discoveries in the fields of geometry and mechanics.

ARCHIMEDES’ SPIRAL

The formula that generates the graph of the Archimedes’ spiral is given by r=θr=θ for θ≥0.θ≥0. As θθ increases, rr increases at a constant rate in an ever-widening, never-ending, spiraling path. See Figure 18.

Figure 18

HOW TO

Given an Archimedes’ spiral over [0,2π],[ 0,2π ], sketch the graph.

1. Make a table of values for rr and θθ over the given domain.
2. Plot the points and sketch the graph.

EXAMPLE 10

Sketching the Graph of an Archimedes’ Spiral

Sketch the graph of r=θr=θ over [0,2π].[0,2π].

Answer

Analysis

The domain of this polar curve is [0,2π].[ 0,2π ]. In general, however, the domain of this function is (−∞,∞).(−∞,∞). Graphing the equation of the Archimedes’ spiral is rather simple, although the image makes it seem like it would be complex.

TRY IT #6

Sketch the graph of r=−θr=−θ over the interval [0,4π].[ 0,4π ].

Summary of Curves

We have explored a number of seemingly complex polar curves in this section. Figure 20 and Figure 21 summarize the graphs and equations for each of these curves.

Figure 20

Figure 21

MEDIA

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

• Graphing Polar Equations Part 1
• Graphing Polar Equations Part 2
• Animation: The Graphs of Polar Equations
• Graphing Polar Equations on the TI-84

8.4 Section Exercises

Verbal

1.

Describe the three types of symmetry in polar graphs, and compare them to the symmetry of the Cartesian plane.

2.

Which of the three types of symmetries for polar graphs correspond to the symmetries with respect to the x-axis, y-axis, and origin?

3.

What are the steps to follow when graphing polar equations?

4.

Describe the shapes of the graphs of cardioids, limaçons, and lemniscates.

5.

What part of the equation determines the shape of the graph of a polar equation?

Graphical

For the following exercises, test the equation for symmetry.

6.

r=5cos3θr=5cos3θ

7.

r=3−3cosθr=3−3cosθ

8.

r=3+2sinθr=3+2sinθ

9.

r=3sin2θr=3sin2θ

10.

r=4r=4

11.

r=2θr=2θ

12.

r=4cosθ2r=4cosθ2

13.

r=2θr=2θ

14.

r=31−cos2θ−−−−−−−√r=31−cos2θ

15.

r=5sin2θ−−−−−−√r=5sin2θ

For the following exercises, graph the polar equation. Identify the name of the shape.

16.

r=3cosθr=3cosθ

17.

r=4sinθr=4sinθ

18.

r=2+2cosθr=2+2cosθ

19.

r=2−2cosθr=2−2cosθ

20.

r=5−5sinθr=5−5sinθ

21.

r=3+3sinθr=3+3sinθ

22.

r=3+2sinθr=3+2sinθ

23.

r=7+4sinθr=7+4sinθ

24.

r=4+3cosθr=4+3cosθ

25.

r=5+4cosθr=5+4cosθ

26.

r=10+9cosθr=10+9cosθ

27.

r=1+3sinθr=1+3sinθ

28.

r=2+5sinθr=2+5sinθ

29.

r=5+7sinθr=5+7sinθ

30.

r=2+4cosθr=2+4cosθ

31.

r=5+6cosθr=5+6cosθ

32.

r2=36cos(2θ)r2=36cos(2θ)

33.

r2=10cos(2θ)r2=10cos(2θ)

34.

r2=4sin(2θ)r2=4sin(2θ)

35.

r2=10sin(2θ)r2=10sin(2θ)

36.

r=3sin(2θ)r=3sin(2θ)

37.

r=3cos(2θ)r=3cos(2θ)

38.

r=5sin(3θ)r=5sin(3θ)

39.

r=4sin(4θ)r=4sin(4θ)

40.

r=4sin(5θ)r=4sin(5θ)

41.

r=−θr=−θ

42.

r=2θr=2θ

43.

r=−3θr=−3θ

Technology

For the following exercises, use a graphing calculator to sketch the graph of the polar equation.

44.

r=1θr=1θ

45.

r=1θ√r=1θ

46.

r=2sinθtanθ,r=2sinθtanθ, a cissoid

47.

r=21−sin2θ−−−−−−−√r=21−sin2θ, a hippopede

48.

r=5+cos(4θ)r=5+cos(4θ)

49.

r=2−sin(2θ)r=2−sin(2θ)

50.

r=θ2r=θ2

51.

r=θ+1r=θ+1

52.

r=θsinθr=θsinθ

53.

r=θcosθr=θcosθ

For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of [0,4π][ 0,4π ] and then explain the differences shown in the graphs.

54.

r=θ,r=−θr=θ,r=−θ

55.

r=θ,r=θ+sinθr=θ,r=θ+sinθ

56.

r=sinθ+θ,r=sinθ−θr=sinθ+θ,r=sinθ−θ

57.

r=2sin(θ2),r=θsin(θ2)r=2sin(θ2),r=θsin(θ2)

58.

r=sin(cos(3θ))r=sin(3θ)r=sin(cos(3θ))r=sin(3θ)

59.

On a graphing utility, graph r=sin(165θ)r=sin(165θ) on [0[0, 4π]4π], [0[0, 8π]8π], [0[0, 12π]12π], and [0[0, 16π].16π]. Describe the effect of increasing the width of the domain.

60.

On a graphing utility, graph and sketch r=sinθ+(sin(52θ))3r=sinθ+(sin(52θ))3 on [0,4π].[ 0,4π ].

61.

On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.

r1=3sin(3θ)r2=2sin(3θ)r3=sin(3θ)r1=3sin(3θ)r2=2sin(3θ)r3=sin(3θ)

62.

On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.

r1=3+3cosθr2=2+2cosθr3=1+cosθr1=3+3cosθr2=2+2cosθr3=1+cosθ

63.

On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.

r1=3θr2=2θr3=θr1=3θr2=2θr3=θ

Extensions

For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.

64.

r1=3+2sinθ,r2=2r1=3+2sinθ,r2=2

65.

r1=6−4cosθ,r2=4r1=6−4cosθ,r2=4

66.

r1=1+sinθ,r2=3sinθr1=1+sinθ,r2=3sinθ

67.

r1=1+cosθ,r2=3cosθr1=1+cosθ,r2=3cosθ

68.

r1=cos(2θ),r2=sin(2θ)r1=cos(2θ),r2=sin(2θ)

69.

r1=sin2(2θ)r1=sin2(2θ), r2=1−cos(4θ)r2=1−cos(4θ)

70.

r1=3–√,r2=2sin(θ)r1=3,r2=2sin(θ)

71.

r12=sinθ,r22=cosθr12=sinθ,r22=cosθ

72.

r1=1+cosθr1=1+cosθ, r2=1−sinθ