Polar Equations and Their Graphs - Chapter 8 Chapter 8

Learning Objectives

1. Graph polar curves by plotting points: circles and lemniscates.

2. Graph polar roses.

3. Graph cardioids and limacons.

Graphing curves in the polar coordinate plane takes some time and practice to master. In this section we outline two methods. One method is to simply make a table of coordinates and then plot the resulting points. A second method is to use a familiar reference graph in rectangular coordinates as a guide to trace the polar curve.

Objective: Graph polar equations by plotting points.

When working with equations, we typically write one variable in terms of another. In the case of polar equations, r is usually written in terms of θ. Written in this form, θ is the independent variable and r is the dependent variable. Find points to plot by choosing values for θ and then substitute to find corresponding values for r.

Example: Graph r=6 cosθ .

Solution: Converting this equation to rectangular form we have,

( )

² ²

6 cos 3 9

r= θ ⇔ x− +y =

In this form, we recognize that it is the equation of a circle centered at

( )

3, 0 with a radius of 3. Furthermore, we know the graph will have the same shape using any positional system.

Therefore, to graph the circle using polar coordinates we begin by choosing a number of angles, in degrees or radians, and then substitute into the equation to find the corresponding r-values.

θ r=6 cosθ

( )

^r^,^θ

0° r=6 cos 0° = 6

(

^{6, 0°}

)

30° r=6 cos 30° ≈5.2

(

^{5.2, 30°}

)

45° r=6 cos 45° ≈4.2

(

^{4.2, 45°}

)

60° r=6 cos 60° = 3

(

^{3, 60°}

)

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90° r=6 cos 90° = 0

(

^{0, 90°}

)

These five points allow us to trace out the entire upper half of the graph. Now continuing with angles beyond 90°,

θ r=6 cosθ

( )

^r^,^θ

120° r=6 cos120° = − 3

(

⁻^3,120^°

)

135° r=6 cos135° ≈ −4.2

(

⁻^4.2,135^°

)

150° r=6 cos150° ≈ −5.2

(

⁻^5.2,150^°

)

180° r=6 cos180° = − 6

(

⁻^6,180^°

)

At 180°, and beyond, the points begin to repeat. Also notice that the r is negative, these points allow us to trace the bottom half of the graph. Plot the points and use them to draw a sketch of the graph.

Answer:

Try this! Graph r=4 sinθ. Answer:

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In general, the polar equation of a circle passing through the pole has the form, sin

r=a θ or r=acosθ [ Circle Interactives ]

Next, we look at a more complicated polar equation,

( )

2 sin 2

r =a θ or ^r^{= ±} ^a^{sin 2}

( )

^θ

For each angle θ in the domain we will obtain two values for r; one positive and one negative.

Example: Graph ^r² ⁼^{25sin 2}

( )

^θ

Solution: Begin by applying the square root property and then substitute some angles that are in the domain. Note, angles that yield negative radicands are not in the domain.

θ ^r ^{= ±} ^{25sin 2}

( )

^θ

^{( )}

^r^,^θ

0° ^r ^{= ±} ^{25sin 2 0}

(

^{⋅ ° =}

)

⁰

⁽

^{0, 0°}

⁾

30° ^r^{= ±} ^{25sin 2 30}

(

^{⋅ ° = ±}

)

^4.7

⁽

^±^{4.7, 30}^°

⁾

45° ^r^{= ±} ^{25sin 2 45}

(

^{⋅ ° = ±}

)

⁵

⁽

^±^{5, 45}^°

⁾

60° ^r^{= ±} ^{25sin 2 60}

(

^{⋅ ° ≈ ±}

)

^4.7

⁽

^±^{4.7, 60}^°

⁾

90° ^r^{= ±} ^{25sin 2 90}

(

^{⋅ ° =}

)

⁰

⁽

^{0, 90°}

⁾

Values for θ between 180° and 360° are not defined.

Answer:

631

Try this! Graph ^r² ⁼^{4 cos 2}

( )

^θ

Answer:

In general, these ribbon-shaped curves are called lemniscates and have the form,

( )

2 sin 2

r =a θ or ^r² =^a^{cos 2}

( )

θ [ Lemniscate Interactives ]

The word “lemniscate” comes the Latin word “lēmniscātus” meaning “decorated with ribbons.”

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Objective: Graph polar roses.

In this objective we introduce another method for tracing a polar graph using a reference graph in the rectangular coordinate plane.

Example: Graph ^r⁼^{5 cos 2}

( )

^θ

Solution: Referencing the graph of ^y⁼^{5 cos 2}

( )

^x in the rectangular coordinate plane can aid in determining points to plot in the polar coordinate plane. Here we can interpret the graph where the y-values correspond to r and the x-values correspond to the angle θ.

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Continuing from π/4 to π/2, the r-values will be negative and the trace is as follows:

The trace continues in a similar manner.

Tracing beyond π radians we have,

633

After 2π radians, the points on the polar graph begin to repeat and the graph is complete.

Answer:

Try this! Graph ^r⁼^{5sin 2}

( )

^θ ^.

Answer:

634

Example: Graph ^r⁼^{5sin 3}

( )

^θ

Solution: Here we use ^y⁼^{5sin 3}

( )

^θ as a guide to graph the polar curve.

In this case, the points on the polar graph begin to repeat beyond π radians.

Answer:

Try this! Graph ^r⁼^{5 cos 3}

( )

^θ ^.

Answer:

635

These graphs are called polar roses (sometimes referred to as a rhodonea curves), and were named by Italian mathematician Guido Grandi between 1723 and 1728, because they resembled roses. In general, any polar rose has the form

( )

sin

r =a nθ or ^r⁼^a^cos

( )

ⁿ^θ

[ Polar Rose Interactive ]

If n is an odd integer, a polar rose will have n leaves, while if n is an even, the rose will have 2n leaves. You are encouraged to explore polar roses for any real number n.

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Objective: Graph cardioids and limacons.

A limaçon or limacon, pronounced “liməsɒn,” is a polar curve of the form, cos

r= +a b θ or r= +a bsinθ

The word comes from the Latin word “limax,” meaning "snail." We begin with the special case where a=b, which is called a cardioid.

Example: Graph r= +3 3sinθ

Solution: We begin with the graph of y= +3 3sinx and use it as a guide to trace out the polar graph.

Note that there are no negative r values and when θ =3 / 2π we have r=0 resulting in a cusp, a pointed end where the curves meet, at that point on the graph.

Answer:

636

Try this! Graph r= +2 2 cosθ . Answer:

Next, we consider the graph of a limacon when b<a. When this is the case, the value for r will always be positive.

Example: Graph r= +3 2 sinθ

Solution: Use y= +3 2 sinx as a guide to trace the curve in the polar coordinate plane.

637

Note that the r-values are always and smooth, no cusp, at every point.

Answer:

Try this! Graph r= +4 2 cosθ . Answer:

638

When b>a the value for r will attain negative values and an inner loop will form.

Example: Graph r= +2 4 sinθ

Solution: Use y= +2 4 sinx as a guide to plot the polar curve.

Take care to study the transition from positive to negative r-values when the angle is between π and 2π radians.

Try this! Graph r= +2 3cosθ. Answer:

In general, limacons have the general form, cos

r= +a b θ or r= +a bsinθ [ Limacon Interactive ]

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Key Takeaways ***** ***** *****

• We can graph in the polar coordinate plane by plotting points. In other words, choose some values for θ, substitute and find the corresponding r-values. The more points we find the more accurate the graph will be.

• Rather than making a table of values we can use a reference graph to aid in tracing the graph of a polar equation.

• The shape of a graph using polar coordinates is the same as the shape of the graph using rectangular coordinates.

• Familiarize yourself with the following common polar graphs.

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Section Exercises Part A

Make a table with angles that are multiples of 30° and use the points to sketch the graph.

Convert to a polar equation and then graph by making a table of angles that are multiples of 45°.

11. x²+y² = 16

Part B Graph.

21. ^r⁼^{4 cos 2}

( )

^θ

22. ^r⁼^{3sin 2}

( )

^θ

23. ^r⁼^{5sin 2}

( )

^θ

24. ^r⁼^{2 cos 2}

( )

^θ

25. ^r⁼^{4 sin 3}

( )

^θ

26. ^r⁼^{6 cos 3}

( )

^θ

27. ^r⁼^{2 cos 3}

( )

^θ

28. ^r⁼^{3sin 3}

( )

^θ

29. ^r⁼^{5sin 4}

( )

^θ

30. ^r⁼^{5 cos 4}

( )

^θ

31. ^r⁼^{4 sin 5}

( )

^θ

32. ^r⁼^{4 cos 5}

( )

^θ

Part C Graph.

33. r= +3 3cosθ 34. r= +1 sinθ 35. r= +2 2 sinθ 36. r= +1 cosθ 37. r= +4 2 sinθ 38. r= +3 cosθ

39. r= +3 2 cosθ 40. r= +4 2 sinθ 41. r= +2 3sinθ 42. r= +2 4 cosθ 43. r= +1 3cosθ 44. r= +1 4 sinθ

Answers to Section Exercises Part A

1. 2.

642

10.

643

11.

12.

13.

14.

15.

16.

17.

18.

644

19. 20.

Part B 21.

22.

23.

24.

25.

26.

645

27.

28.

29.

30.

31.

32.

Part C

33. 34.

646

35.

36.

37.

38.

39.

40.

41.

42.

647

43. 44.

648

Trigonometry

Chapter 8 - Complex Numbers and Polar Graphs

In document Chapter 8 Chapter 8 - Complex Numbers and Polar Graphs (Page 63-83)