Further Applications of Trigonometry

# 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). 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
We interpretas the distance from the sun andas 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.

### Testing Polar Equations for Symmetry

Just as a rectangular equation such asdescribes the relationship betweenandon a Cartesian grid, a polar equation describes a relationship betweenandon a polar grid. Recall that the coordinate pairindicates that we move counterclockwise from the polar axis (positive x-axis) by an angle ofand extend a ray from the pole (origin)units in the direction ofAll 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 to determine the graph of a polar equation.

In the first test, we consider symmetry with respect to the line(y-axis). We replacewithto determine if the new equation is equivalent to the original equation. For example, suppose we are given the equation

This equation exhibits symmetry with respect to the line

In the second test, we consider symmetry with respect to the polar axis (-axis). We replacewithorto determine equivalency between the tested equation and the original. For example, suppose we are given the equation

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 replacewithto determine if the tested equation is equivalent to the original equation. For example, suppose we are given the equation

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 linethe 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).

### How To

Given a polar equation, test for symmetry.

1. Substitute the appropriate combination of components for$\,\left(-r,-\theta \right)\,$forsymmetry;for polar axis symmetry; andfor 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.

### Testing a Polar Equation for Symmetry

Test the equation for symmetry.

Test for each of the three types of symmetry.

 1) Replacingwithyields the same result. Thus, the graph is symmetric with respect to the line 2) Replacingwithdoes not yield the same equation. Therefore, the graph fails the test and may or may not be symmetric with respect to the polar axis. 3) Replacingwithchanges the equation and fails the test. The graph may or may not be symmetric with respect to the pole.

#### Analysis

Using a graphing calculator, we can see that the equation is a circle centered atwith radiusand is indeed symmetric to the lineWe can also see that the graph is not symmetric with the polar axis or the pole. See (Figure).

### Try It

Test the equation for symmetry:

The equation fails the symmetry test with respect to the lineand with respect to the pole. It passes the polar axis symmetry test.

### Graphing Polar Equations by Plotting Points

To graph in the rectangular coordinate system we construct a table ofandvalues. To graph in the polar coordinate system we construct a table ofandvalues. We enter values of into a polar equation and calculateHowever, using the properties of symmetry and finding key values ofandmeans fewer calculations will be needed.

#### Finding Zeros and Maxima

To find the zeros of a polar equation, we solve for the values ofthat result in Recall that, to find the zeros of polynomial functions, we set the equation equal to zero and then solve forWe use the same process for polar equations. Setand solve for

For many of the forms we will encounter, the maximum value of a polar equation is found by substituting those values ofinto the equation that result in the maximum value of the trigonometric functions. Considerthe maximum distance between the curve and the pole is 5 units. The maximum value of the cosine function is 1 whenso our polar equation isand the value will yield the maximum

Similarly, the maximum value of the sine function is 1 whenand if our polar equation isthe valuewill yield the maximumWe may find additional information by calculating values ofwhenThese points would be polar axis intercepts, which may be helpful in drawing the graph and identifying the curve of a polar equation.

### Finding Zeros and Maximum Values for a Polar Equation

Using the equation in (Figure), find the zeros and maximumand, if necessary, the polar axis intercepts of

To find the zeros, setequal to zero and solve for

Substitute any one of thevalues into the equation. We will use

The pointsandare the zeros of the equation. They all coincide, so only one point is visible on the graph. This point is also the only polar axis intercept.

To find the maximum value of the equation, look at the maximum value of the trigonometric functionwhich occurs whenresulting inSubstitutefor

#### Analysis

The pointwill be the maximum value on the graph. Let’s plot a few more points to verify the graph of a circle. See (Figure) and (Figure).

0

### Try It

Without converting to Cartesian coordinates, test the given equation for symmetry and find the zeros and maximum values of$\,r=3\mathrm{cos}\,\theta .$

Tests will reveal symmetry about the polar axis. The zero isand the maximum value is

#### 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 byand whereis the diameter of the circle or the distance from the pole to the farthest point on the circumference. The radius is or one-half the diameter. For the center isFor the center is(Figure) shows the graphs of these four circles.

### Sketching the Graph of a Polar Equation for a Circle

Sketch the graph of

First, testing the equation for symmetry, we find that the graph is symmetric about the polar axis. Next, we find the zeros and maximumforFirst, setand solve for. Thus, a zero occurs atA key point to plot is

To find the maximum value of note that the maximum value of the cosine function is 1 whenSubstituteinto the equation:

The maximum value of the equation is 4. A key point to plot is

As is symmetric with respect to the polar axis, we only need to calculate r-values forover the intervalPoints in the upper quadrant can then be reflected to the lower quadrant. Make a table of values similar to (Figure). The graph is shown in (Figure).

 0 4 3.46 2.83 2 0 −2 −2.83 −3.46 4

#### 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 byandwhereandThe cardioid graph passes through the pole, as we can see in (Figure).

### 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
3. Find the maximum value of the equation according to the maximum value of the trigonometric expression.
4. Make a table of values forand
5. Plot the points and sketch the graph.

### Sketching the Graph of a Cardioid

Sketch the graph of

First, testing the equation for symmetry, we find that the graph of this equation will be symmetric about the polar axis. Next, we find the zeros and maximums. Settingwe haveThe zero of the equation is located atThe graph passes through this point.

The maximum value ofoccurs when is a maximum, which is whenor whenSubstituteinto the equation, and solve for

The pointis the maximum value on the graph.

We found that the polar equation is symmetric with respect to the polar axis, but as it extends to all four quadrants, we need to plot values over the intervalThe upper portion of the graph is then reflected over the polar axis. Next, we make a table of values, as in (Figure), and then we plot the points and draw the graph. See (Figure).

 4 3.41 2 1 0

#### 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 whenand convex limaçons when

### Formulas for One-Loop Limaçons

The formulas that produce the graph of a dimpled one-loop limaçon are given byandwhereAll four graphs are shown in (Figure).

### 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.

### Sketching the Graph of a One-Loop Limaçon

Graph the equation

First, testing the equation for symmetry, we find that it fails all three symmetry tests, meaning that the graph may or may not exhibit symmetry, so we cannot use the symmetry to help us graph it. However, this equation has a graph that clearly displays symmetry with respect to the lineyet it fails all the three symmetry tests. A graphing calculator will immediately illustrate the graph’s reflective quality.

Next, we find the zeros and maximum, and plot the reflecting points to verify any symmetry. Settingresults inbeing undefined. What does this mean? How couldbe undefined? The angleis undefined for any value ofTherefore,is undefined because there is no value offor whichConsequently, the graph does not pass through the pole. Perhaps the graph does cross the polar axis, but not at the pole. We can investigate other intercepts by calculating when

So, there is at least one polar axis intercept at

Next, as the maximum value of the sine function is 1 when we will substitute
into the equation and solve forThus,

Make a table of the coordinates similar to (Figure).

 4 2.5 1.4 1 1.4 2.5 4 5.5 6.6 7 6.6 5.5 4

The graph is shown in (Figure).

#### 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 is likely symmetric with respect to the line evaluating more points helps to verify that the graph is correct.

### Try It

Sketch the graph of

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 byandwhereandThe graph of the inner-loop limaçon passes through the pole twice: once for the outer loop, and once for the inner loop. See (Figure) for the graphs.

### Sketching the Graph of an Inner-Loop Limaçon

Sketch the graph of

Testing for symmetry, we find that the graph of the equation is symmetric about the polar axis. Next, finding the zeros reveals that when
The maximumis found whenor whenThus, the maximum is found at the point (7, 0).

Even though we have found symmetry, the zero, and the maximum, plotting more points will help to define the shape, and then a pattern will emerge.

See (Figure).

 7 6.3 4.5 2 −0.5 −2.3 −3 −2.3 −0.5 2 4.5 6.3 7

As expected, the values begin to repeat afterThe graph is shown in (Figure).

#### Investigating Lemniscates

The lemniscate is a polar curve resembling the infinity symbolor 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 byandwhereThe formulais symmetric with respect to the pole. The formulais symmetric with respect to the pole, the lineand the polar axis. See (Figure) for the graphs.

### Sketching the Graph of a Lemniscate

Sketch the graph of

The equation exhibits symmetry with respect to the linethe polar axis, and the pole.

Let’s find the zeros. It should be routine by now, but we will approach this equation a little differently by making the substitution

So, the pointis a zero of the equation.

Now let’s find the maximum value. Since the maximum ofwhenthe maximumwhenThus,

We have a maximum at (2, 0). Since this graph is symmetric with respect to the pole, the line and the polar axis, we only need to plot points in the first quadrant.

Make a table similar to (Figure).

 0 2 0 0

Plot the points on the graph, such as the one shown in (Figure).

#### Analysis

Making a substitution such asis 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 ofThis is because there are no real square roots for these values ofIn other words, the corresponding r-values of
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 byandwhereIfis even, the curve haspetals. Ifis odd, the curve haspetals. See (Figure).

### Sketching the Graph of a Rose Curve (n Even)

Sketch the graph of

Testing for symmetry, we find again that the symmetry tests do not tell the whole story. The graph is not only symmetric with respect to the polar axis, but also with respect to the lineand the pole.

Now we will find the zeros. First make the substitution

The zero isThe pointis on the curve.

Next, we find the maximumWe know that the maximum value ofwhenThus,

The pointis on the curve.

The graph of the rose curve has unique properties, which are revealed in (Figure).

 0 2 0 −2 0 2 0 −2

Aswhenit makes sense to divide values in the table byunits. A definite pattern emerges. Look at the range of r-values: 2, 0, −2, 0, 2, 0, −2, and so on. This represents the development of the curve one petal at a time. Starting ateach petal extends out a distance ofand then turns back to zerotimes for a total of eight petals. See the graph in (Figure).

#### Analysis

When these curves are drawn, it is best to plot the points in order, as in the (Figure). 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

Sketch the graph of

The graph is a rose curve,even

### Sketching the Graph of a Rose Curve (n Odd)

Sketch the graph of

The graph of the equation shows symmetry with respect to the lineNext, find the zeros and maximum. We will want to make the substitution

The maximum value is calculated at the angle where is a maximum. Therefore,

Thus, the maximum value of the polar equation is 2. This is the length of each petal. As the curve forodd yields the same number of petals asthere will be five petals on the graph. See (Figure).

Create a table of values similar to (Figure).

 0 0 1 −1.73 2 −1.73 1 0

### Try It

Sketch the graph of

Rose curve,odd

#### 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
forAsincreases,
increases at a constant rate in an ever-widening, never-ending, spiraling path. See (Figure).

### How To

Given an Archimedes’ spiral oversketch the graph.

1. Make a table of values forandover the given domain.
2. Plot the points and sketch the graph.

### Sketching the Graph of an Archimedes’ Spiral

Sketch the graph ofover

Asis equal tothe plot of the Archimedes’ spiral begins at the pole at the point (0, 0). While the graph hints of symmetry, there is no formal symmetry with regard to passing the symmetry tests. Further, there is no maximum value, unless the domain is restricted.

Create a table such as (Figure).

 0.785 1.57 3.14 4.71 5.50 6.28

Notice that the r-values are just the decimal form of the angle measured in radians. We can see them on a graph in (Figure).

#### Analysis

The domain of this polar curve isIn general, however, the domain of this function isGraphing the equation of the Archimedes’ spiral is rather simple, although the image makes it seem like it would be complex.

### Try It

Sketch the graph ofover the interval

### Summary of Curves

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

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

### Key Concepts

• It is easier to graph polar equations if we can test the equations for symmetry with respect to the linethe polar axis, or the pole.
• There are three symmetry tests that indicate whether the graph of a polar equation will exhibit symmetry. If an equation fails a symmetry test, the graph may or may not exhibit symmetry. See (Figure).
• Polar equations may be graphed by making a table of values forand
• The maximum value of a polar equation is found by substituting the valuethat leads to the maximum value of the trigonometric expression.
• The zeros of a polar equation are found by settingand solving forSee (Figure).
• Some formulas that produce the graph of a circle in polar coordinates are given byandSee (Figure).
• The formulas that produce the graphs of a cardioid are given byandforandSee (Figure).
• The formulas that produce the graphs of a one-loop limaçon are given byandforSee (Figure).
• The formulas that produce the graphs of an inner-loop limaçon are given byandfor and See (Figure).
• The formulas that produce the graphs of a lemniscates are given byandwhereSee (Figure).
• The formulas that produce the graphs of rose curves are given byandwhereifis even, there arepetals, and ifis odd, there arepetals. See (Figure) and (Figure).
• The formula that produces the graph of an Archimedes’ spiral is given bySee (Figure).

### Section Exercises

#### Verbal

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

Symmetry with respect to the polar axis is similar to symmetry about the-axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the lineis similar to symmetry about the-axis.

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

What are the steps to follow when graphing polar equations?

Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, limaçon, lemniscate, etc., then plot points at and sketch the graph.

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

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

The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation.

#### Graphical

For the following exercises, test the equation for symmetry.

symmetric with respect to the polar axis, symmetric with respect to the line symmetric with respect to the pole

no symmetry

no symmetry

symmetric with respect to the pole

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

cardioid

one-loop/dimpled limaçon

inner loop/two-loop limaçon

inner loop/two-loop limaçon

inner loop/two-loop limaçon

lemniscate

lemniscate

rose curve

rose curve

Archimedes’ spiral

Archimedes’ spiral

#### Technology

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

a cissoid

, a hippopede

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

They are both spirals, but not quite the same.

Both graphs are curves with 2 loops. The equation with a coefficient of has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 toto get a better picture.

On a graphing utility, graphonandDescribe the effect of increasing the width of the domain.

When the width of the domain is increased, more petals of the flower are visible.

On a graphing utility, graph and sketchon

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

The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve’s distance from the pole.

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

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

The graphs are spirals. The smaller the coefficient, the tighter the spiral.

#### Extensions

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

and at sinceis squared

### Glossary

Archimedes’ spiral
a polar curve given byWhen multiplied by a constant, the equation appears asAsthe curve continues to widen in a spiral path over the domain.
cardioid
a member of the limaçon family of curves, named for its resemblance to a heart; its equation is given as andwhere
convex limaҫon
a type of one-loop limaçon represented byandsuch that
dimpled limaҫon
a type of one-loop limaçon represented by and such that
inner-loop limaçon
a polar curve similar to the cardioid, but with an inner loop; passes through the pole twice; represented by andwhere
lemniscate
a polar curve resembling a figure 8 and given by the equationand
one-loop limaҫon
a polar curve represented by and such that and may be dimpled or convex; does not pass through the pole
polar equation
an equation describing a curve on the polar grid.
rose curve
a polar equation resembling a flower, given by the equations and when is even there are petals, and the curve is highly symmetrical; whenis odd there are petals.