Stewart Precalc 6e Section 8.2: Graphs of Polar Equations

1 Concept Check · Level 1

To plot points in polar coordinates, we use a grid consisting of ___ centered at the pole and ___ emanating from ___.

2 Concept Check · Level 1

(a) To graph a polar equation \(r = f(\theta)\), we plot all the points \((r, \theta)\) that ___ the equation.

Answer for 2(a)

(b) The simplest polar equations are obtained by setting \(r\) or \(\theta\) equal to a constant. The graph of the polar equation \(r = 3\) is a ___ with radius ___ centered at the ___. The graph of the polar equation \(\theta = \dfrac{\pi}{4}\) is a ___ passing through the ___ with slope ___.

Answer for 2(b)

Enter your answer directly below each part above.

3 Skill - Matching Polar Equations · Level 2

Match the polar equation \(r = 3 \cos \theta\) with one of the graphs labeled I–VI. (Use the summary table of basic polar graphs.)

4 Skill - Matching Polar Equations · Level 2

Match the polar equation \(r = 3\) with one of the graphs labeled I–VI.

5 Skill - Matching Polar Equations · Level 2

Match the polar equation \(r = 2 + 2 \sin \theta\) with one of the graphs labeled I–VI.

6 Skill - Matching Polar Equations · Level 2

Match the polar equation \(r = 1 + 2 \cos \theta\) with one of the graphs labeled I–VI.

7 Skill - Matching Polar Equations · Level 2

Match the polar equation \(r = \sin(3 \theta)\) with one of the graphs labeled I–VI.

8 Skill - Matching Polar Equations · Level 2

Match the polar equation \(r = \sin(4 \theta)\) with one of the graphs labeled I–VI.

9 Skill - Testing Symmetry · Level 2

Test the polar equation \(r = 2 - \sin \theta\) for symmetry with respect to the polar axis, the pole, and the line \(\theta = \dfrac{\pi}{2}\).

10 Skill - Testing Symmetry · Level 2

Test the polar equation \(r = 4 + 8 \cos \theta\) for symmetry with respect to the polar axis, the pole, and the line \(\theta = \dfrac{\pi}{2}\).

11 Skill - Testing Symmetry · Level 2

Test the polar equation \(r = 3 \sec \theta\) for symmetry with respect to the polar axis, the pole, and the line \(\theta = \dfrac{\pi}{2}\).

12 Skill - Testing Symmetry · Level 2

Test the polar equation \(r = 5 \cos \theta \csc \theta\) for symmetry with respect to the polar axis, the pole, and the line \(\theta = \dfrac{\pi}{2}\).

13 Skill - Testing Symmetry · Level 2

Test the polar equation \(r = \dfrac{4}{3 - 2 \sin \theta}\) for symmetry with respect to the polar axis, the pole, and the line \(\theta = \dfrac{\pi}{2}\).

14 Skill - Testing Symmetry · Level 2

Test the polar equation \(r = \dfrac{5}{1 + 3 \cos \theta}\) for symmetry with respect to the polar axis, the pole, and the line \(\theta = \dfrac{\pi}{2}\).

15 Skill - Testing Symmetry · Level 3

Test the polar equation \(r^2 = 4 \cos(2 \theta)\) for symmetry with respect to the polar axis, the pole, and the line \(\theta = \dfrac{\pi}{2}\).

16 Skill - Testing Symmetry · Level 3

Test the polar equation \(r^2 = 9 \sin \theta\) for symmetry with respect to the polar axis, the pole, and the line \(\theta = \dfrac{\pi}{2}\).

17 Skill - Convert to Rectangular · Level 2

Sketch a graph of the polar equation \(r = 2\), and express the equation in rectangular coordinates.

18 Skill - Convert to Rectangular · Level 2

Sketch a graph of the polar equation \(r = -1\), and express the equation in rectangular coordinates.

19 Skill - Convert to Rectangular · Level 2

Sketch a graph of the polar equation \(\theta = -\dfrac{\pi}{2}\), and express the equation in rectangular coordinates.

20 Skill - Convert to Rectangular · Level 2

Sketch a graph of the polar equation \(\theta = (5 \pi)/6\), and express the equation in rectangular coordinates.

21 Skill - Convert to Rectangular · Level 2

Sketch a graph of the polar equation \(r = 6 \sin \theta\), and express the equation in rectangular coordinates.

22 Skill - Convert to Rectangular · Level 2

Sketch a graph of the polar equation \(r = \cos \theta\), and express the equation in rectangular coordinates.

23 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = -2 \cos \theta\).

24 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = 2 \sin \theta + 2 \cos \theta\).

25 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = 2 - 2 \cos \theta\).

26 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = 1 + \sin \theta\).

27 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = -3(1 + \sin \theta)\).

28 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = \cos \theta - 1\).

29 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = \sin(2 \theta)\).

30 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = 2 \cos(3 \theta)\).

31 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = -\cos(5 \theta)\).

32 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = \sin(4 \theta)\).

33 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = \sqrt{3} - 2 \sin \theta\).

34 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = 2 + \sin \theta\).

35 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = \sqrt{3} + \cos \theta\).

36 Skill - Sketching Polar Graphs · Level 3

Sketch a graph of the polar equation \(r = 1 - 2 \cos \theta\).

37 Skill - Sketching Polar Graphs · Level 4

Sketch a graph of the polar equation \(r^2 = \cos(2 \theta)\).

38 Skill - Sketching Polar Graphs · Level 4

Sketch a graph of the polar equation \(r^2 = 4 \sin(2 \theta)\).

39 Skill - Sketching Polar Graphs · Level 4

Sketch a graph of the polar equation \(r = \theta\), \(\theta \geq 0\) (spiral).

40 Skill - Sketching Polar Graphs · Level 4

Sketch a graph of the polar equation \(r \theta = 1\), \(\theta > 0\) (reciprocal spiral).

41 Skill - Sketching Polar Graphs · Level 4

Sketch a graph of the polar equation \(r = 2 + \sec \theta\) (conchoid).

42 Skill - Sketching Polar Graphs · Level 4

Sketch a graph of the polar equation \(r = \sin \theta \tan \theta\) (cissoid).

43 Skill - Graphing Device · Level 3

Use a graphing device to graph the polar equation \(r = \cos\left(\dfrac{\theta}{2}\right)\). Choose the domain of \(\theta\) to make sure you produce the entire graph.

44 Skill - Graphing Device · Level 3

Use a graphing device to graph the polar equation \(r = \sin((8 \theta)/5)\). Choose the domain of \(\theta\) to make sure you produce the entire graph.

45 Skill - Graphing Device · Level 3

Use a graphing device to graph the polar equation \(r = 1 + 2 \sin\left(\dfrac{\theta}{2}\right)\) (nephroid). Choose the domain of \(\theta\) to make sure you produce the entire graph.

46 Skill - Graphing Device · Level 3

Use a graphing device to graph the polar equation \(r = \sqrt{1 - 0.8 \sin^2 \theta}\) (hippopede). Choose the domain of \(\theta\) to make sure you produce the entire graph.

47 Skill - Family Investigation · Level 4

Graph the family of polar equations \(r = 1 + \sin(n \theta)\) for \(n = 1, 2, 3, 4\), and \(5\). How is the number of loops related to \(n\)?

48 Skill - Matching Polar Equations · Level 2

Match the polar equation with its graph (labeled A–D): \(r = \sin\left(\dfrac{\theta}{2}\right)\)

49 Skill - Matching Polar Equations · Level 2

Match the polar equation with its graph (labeled A–D): \(r = \dfrac{1}{\sqrt{\theta}}\)

50 Skill - Matching Polar Equations · Level 2

Match the polar equation with its graph (labeled A–D): \(r = \theta \sin \theta\)

51 Skill - Matching Polar Equations · Level 2

Match the polar equation with its graph (labeled A–D): \(r = 1 + 3 \cos(3 \theta)\)

52 Skill - Rectangular to Polar · Level 4

Sketch a graph of the rectangular equation \((x^2 + y^2)^3 = 4 x^2 y^2\). [Hint: First convert the equation to polar coordinates.]

53 Skill - Rectangular to Polar · Level 4

Sketch a graph of the rectangular equation \((x^2 + y^2)^3 = (x^2 - y^2)^2\). [Hint: First convert the equation to polar coordinates.]

54 Skill - Rectangular to Polar · Level 4

Sketch a graph of the rectangular equation \((x^2 + y^2)^2 = x^2 - y^2\). [Hint: First convert the equation to polar coordinates.]

55 Skill - Rectangular to Polar · Level 4

Sketch a graph of the rectangular equation \(x^2 + y^2 = (x^2 + y^2 - x)^2\). [Hint: First convert the equation to polar coordinates.]

56 Skill - Polar Circle Proof · Level 4

Show that the graph of \(r = a \cos \theta + b \sin \theta\) is a circle, and find its center and radius.

57 Skill - Graph and Convert · Level 4

(a) Graph the polar equation \(r = \tan \theta \sec \theta\) in the viewing rectangle \([-3, 3]\) by \([-1, 9]\).

Answer for 57(a)

(b) Note that your graph in part (a) looks like a parabola (see Section 2.5). Confirm this by converting the equation to rectangular coordinates.

Answer for 57(b)

Enter your answer directly below each part above.

58 Application - Satellite Orbit · Level 4

Orbit of a Satellite. Scientists and engineers often use polar equations to model the motion of satellites in earth orbit. Consider a satellite whose orbit is modeled by the equation \(r = \dfrac{22500}{4 - \cos \theta}\), where \(r\) is the distance in miles between the satellite and the center of the earth and \(\theta\) is the angle shown in the figure.

(a) On the same viewing screen, graph the circle \(r = 3960\) (to represent the earth, assumed to be a sphere of radius 3960 mi) and the polar equation of the satellite's orbit. Describe the motion of the satellite as \(\theta\) increases from \(0\) to \(2 \pi\).

Answer for 58(a)

(b) For what angle \(\theta\) is the satellite closest to the earth? Find the height of the satellite above the earth's surface for this value of \(\theta\).

Answer for 58(b)

Enter your answer directly below each part above.

59 Application - Unstable Orbit · Level 5

An Unstable Orbit. The orbit described in Exercise 59 is stable because the satellite traverses the same path over and over as \(\theta\) increases. Suppose that a meteor strikes the satellite and changes its orbit to \(r = \dfrac{22500 \left(1 - \dfrac{\theta}{40}\right)}{4 - \cos \theta}\).

(a) On the same viewing screen, graph the circle \(r = 3960\) and the new orbit equation, with \(\theta\) increasing from \(0\) to \(3 \pi\). Describe the new motion of the satellite.

Answer for 59(a)

(b) Use the TRACE feature on your graphing calculator to find the value of \(\theta\) at the moment the satellite crashes into the earth.

Answer for 59(b)

Enter your answer directly below each part above.

60 Discovery - Transformation of Polar Graphs · Level 4

A Transformation of Polar Graphs. How are the graphs of \(r = 1 + \sin\left(\theta - \dfrac{\pi}{6}\right)\) and \(r = 1 + \sin\left(\theta - \dfrac{\pi}{3}\right)\) related to the graph of \(r = 1 + \sin \theta\)? In general, how is the graph of \(r = f(\theta - \alpha)\) related to the graph of \(r = f(\theta)\)?

61 Discovery - Coordinate System Choice · Level 4

Choosing a Convenient Coordinate System. Compare the polar equation of the circle \(r = 2\) with its equation in rectangular coordinates. In which coordinate system is the equation simpler? Do the same for the equation of the four-leaved rose \(r = \sin(2 \theta)\). Which coordinate system would you choose to study these curves?

62 Discovery - Coordinate System Choice · Level 4

Choosing a Convenient Coordinate System. Compare the rectangular equation of the line \(y = 2\) with its polar equation. In which coordinate system is the equation simpler? Which coordinate system would you choose to study lines?