Stewart 9e Section 10.3: Polar Coordinates

1 Polar Coordinates - Multiple Representations · Level 1

Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with \(r > 0\) and one with \(r < 0\).

(a) \(\left(1, \dfrac{\pi}{4}\right)\)

Answer for 1(a)

(b) \(\left(-2, 3 \dfrac{\pi}{2}\right)\)

Answer for 1(b)

(c) \(\left(3, -\dfrac{\pi}{3}\right)\)

Answer for 1(c)

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2 Polar Coordinates - Multiple Representations · Level 1

Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with \(r > 0\) and one with \(r < 0\).

(a) \(\left(2, 5 \dfrac{\pi}{6}\right)\)

Answer for 2(a)

(b) \(\left(1, -2 \dfrac{\pi}{3}\right)\)

Answer for 2(b)

(c) \(\left(-1, 5 \dfrac{\pi}{4}\right)\)

Answer for 2(c)

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3 Polar to Cartesian Conversion · Level 1

Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point.

(a) \(\left(2, 3 \dfrac{\pi}{2}\right)\)

Answer for 3(a)

(b) \(\left(\sqrt{2}, \dfrac{\pi}{4}\right)\)

Answer for 3(b)

(c) \(\left(-1, -\dfrac{\pi}{6}\right)\)

Answer for 3(c)

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4 Polar to Cartesian Conversion · Level 1

Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point.

(a) \(\left(4, 4 \dfrac{\pi}{3}\right)\)

Answer for 4(a)

(b) \(\left(-2, 3 \dfrac{\pi}{4}\right)\)

Answer for 4(b)

(c) \(\left(-3, -\dfrac{\pi}{3}\right)\)

Answer for 4(c)

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5 Cartesian to Polar Conversion · Level 2

The Cartesian coordinates of a point are given. (i) Find polar coordinates \((r, \theta)\) of the point, where \(r > 0\) and \(0 \leq \theta < 2 \pi\). (ii) Find polar coordinates \((r, \theta)\) of the point, where \(r < 0\) and \(0 \leq \theta < 2 \pi\).

(a) \((-4, 4)\)

Answer for 5(a)

(b) \((3, 3 \sqrt{3})\)

Answer for 5(b)

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6 Cartesian to Polar Conversion · Level 2

The Cartesian coordinates of a point are given. (i) Find polar coordinates \((r, \theta)\) of the point, where \(r > 0\) and \(0 \leq \theta < 2 \pi\). (ii) Find polar coordinates \((r, \theta)\) of the point, where \(r < 0\) and \(0 \leq \theta < 2 \pi\).

(a) \((\sqrt{3}, -1)\)

Answer for 6(a)

(b) \((-6, 0)\)

Answer for 6(b)

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7 Polar Region Sketching · Level 1

Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. \(r \geq 1\)

8 Polar Region Sketching · Level 1

Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. \(0 \leq r < 2\), \(\pi \leq \theta \leq 3 \dfrac{\pi}{2}\)

9 Polar Region Sketching · Level 1

Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. \(r \geq 0\), \(\dfrac{\pi}{4} \leq \theta \leq 3 \dfrac{\pi}{4}\)

10 Polar Region Sketching · Level 1

Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. \(1 \leq r \leq 3\), \(\dfrac{\pi}{6} < \theta < 5 \dfrac{\pi}{6}\)

11 Polar Region Sketching · Level 2

Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. \(2 < r < 3\), \(5 \dfrac{\pi}{3} \leq \theta \leq 7 \dfrac{\pi}{3}\)

12 Polar Region Sketching · Level 1

Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. \(r \geq 1\), \(\pi \leq \theta \leq 2 \pi\)

13 Distance Between Polar Points · Level 2

Find the distance between the points with polar coordinates \(\left(4, 4 \dfrac{\pi}{3}\right)\) and \(\left(6, 5 \dfrac{\pi}{3}\right)\).

14 Distance Formula in Polar Coordinates · Level 2

Find a formula for the distance between the points with polar coordinates \((r_1, \theta_1)\) and \((r_2, \theta_2)\).

15 Polar to Cartesian Equation · Level 1

Identify the curve by finding a Cartesian equation for the curve. \(r^2 = 5\)

16 Polar to Cartesian Equation · Level 1

Identify the curve by finding a Cartesian equation for the curve. \(r = 4 \sec \theta\)

17 Polar to Cartesian Equation · Level 2

Identify the curve by finding a Cartesian equation for the curve. \(r = 5 \cos \theta\)

18 Polar to Cartesian Equation · Level 1

Identify the curve by finding a Cartesian equation for the curve. \(\theta = \dfrac{\pi}{3}\)

19 Polar to Cartesian Equation · Level 2

Identify the curve by finding a Cartesian equation for the curve. \(r^2 \cos 2 \theta = 1\)

20 Polar to Cartesian Equation · Level 2

Identify the curve by finding a Cartesian equation for the curve. \(r^2 \sin 2 \theta = 1\)

21 Cartesian to Polar Equation · Level 1

Find a polar equation for the curve represented by the given Cartesian equation. \(y = 2\)

22 Cartesian to Polar Equation · Level 1

Find a polar equation for the curve represented by the given Cartesian equation. \(y = x\)

23 Cartesian to Polar Equation · Level 2

Find a polar equation for the curve represented by the given Cartesian equation. \(y = 1 + 3 x\)

24 Cartesian to Polar Equation · Level 2

Find a polar equation for the curve represented by the given Cartesian equation. \(4 y^2 = x\)

25 Cartesian to Polar Equation · Level 2

Find a polar equation for the curve represented by the given Cartesian equation. \(x^2 + y^2 = 2 c x\)

26 Cartesian to Polar Equation · Level 2

Find a polar equation for the curve represented by the given Cartesian equation. \(x^2 - y^2 = 4\)

27 Choose Polar or Cartesian Form · Level 1

For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve.

(a) A line through the origin that makes an angle of \(\dfrac{\pi}{6}\) with the positive \(x\)-axis.

Answer for 27(a)

(b) A vertical line through the point \((3, 3)\).

Answer for 27(b)

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28 Choose Polar or Cartesian Form · Level 1

For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve.

(a) A circle with radius 5 and center \((2, 3)\).

Answer for 28(a)

(b) A circle centered at the origin with radius 4.

Answer for 28(b)

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29 Sketch Polar Curve · Level 1

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = -2 \sin \theta\)

30 Sketch Polar Curve · Level 1

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = 1 - \cos \theta\)

31 Sketch Polar Curve · Level 1

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = 2 (1 + \cos \theta)\)

32 Sketch Polar Curve · Level 2

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = 1 + 2 \cos \theta\)

33 Sketch Polar Curve · Level 1

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = \theta\), \(\theta \geq 0\)

34 Sketch Polar Curve · Level 2

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = \theta^2\), \(-2 \pi \leq \theta \leq 2 \pi\)

35 Sketch Polar Curve · Level 2

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = 3 \cos 3 \theta\)

36 Sketch Polar Curve · Level 2

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = -\sin 5 \theta\)

37 Sketch Polar Curve · Level 2

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = 2 \cos 4 \theta\)

38 Sketch Polar Curve · Level 2

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = 2 \sin 6 \theta\)

39 Sketch Polar Curve · Level 2

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = 1 + 3 \cos \theta\)

40 Sketch Polar Curve · Level 2

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = 1 + 5 \sin \theta\)

41 Sketch Polar Curve · Level 3

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r^2 = 9 \sin 2 \theta\)

42 Sketch Polar Curve · Level 3

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r^2 = \cos 4 \theta\)

43 Sketch Polar Curve · Level 2

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = 2 + \sin 3 \theta\)

44 Sketch Polar Curve · Level 3

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r^2 \theta = 1\)

45 Sketch Polar Curve · Level 2

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = \sin\left(\dfrac{\theta}{2}\right)\)

46 Sketch Polar Curve · Level 2

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r = \cos\left(\dfrac{\theta}{3}\right)\)

47 Polar Curve from Cartesian Graph · Level 2

The figure shows a graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. Use it to sketch the corresponding polar curve.

48 Polar Curve from Cartesian Graph · Level 2

The figure shows a graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. Use it to sketch the corresponding polar curve.

49 Polar Asymptotes - Conchoid · Level 3

Show that the polar curve \(r = 4 + 2 \sec \theta\) (called a conchoid) has the line \(x = 2\) as a vertical asymptote by showing that \(\operatorname*{lim}\limits_{r \rightarrow \pm \infty} x = 2\). Use this fact to help sketch the conchoid.

50 Polar Asymptotes - Conchoid · Level 3

Show that the curve \(r = 2 - \csc \theta\) (also a conchoid) has the line \(y = -1\) as a horizontal asymptote by showing that \(\operatorname*{lim}\limits_{r \rightarrow \pm \infty} y = -1\). Use this fact to help sketch the conchoid.

51 Polar Asymptotes - Cissoid · Level 3

Show that the curve \(r = \sin \theta \tan \theta\) (called a cissoid of Diocles) has the line \(x = 1\) as a vertical asymptote. Show also that the curve lies entirely within the vertical strip \(0 \leq x < 1\). Use these facts to help sketch the cissoid.

52 Polar Curve Sketch · Level 3

Sketch the curve \((x^2 + y^2)^3 = 4 x^2 y^2\).

53 Limacon Properties · Level 3

(a) In Example 11 the graphs suggest that the limacon \(r = 1 + c \sin \theta\) has an inner loop when \(|c| > 1\). Prove that this is true, and find the values of \(\theta\) that correspond to the inner loop.

Answer for 53(a)

(b) From Figure 19 it appears that the limacon loses its dimple when \(c = \dfrac{1}{2}\). Prove this.

Answer for 53(b)

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54 Match Polar Equations with Graphs · Level 2

Match the polar equations with the graphs labeled I-VI. Give reasons for your choices. (Don't use a graphing device.)

(a) \(r = \ln \theta\), \(1 \leq \theta \leq 6 \pi\)

Answer for 54(a)

(b) \(r = \theta^2\), \(0 \leq \theta \leq 8 \pi\)

Answer for 54(b)

(c) \(r = \cos 3 \theta\)

Answer for 54(c)

(d) \(r = 2 + \cos 3 \theta\)

Answer for 54(d)

(e) \(r = \cos\left(\dfrac{\theta}{2}\right)\)

Answer for 54(e)

(f) \(r = 2 + \cos\left(3 \dfrac{\theta}{2}\right)\)

Answer for 54(f)

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55 Slope of Tangent to Polar Curve · Level 2

Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\). \(r = 2 \cos \theta\), \(\theta = \dfrac{\pi}{3}\)

56 Slope of Tangent to Polar Curve · Level 3

Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\). \(r = 2 + \sin 3 \theta\), \(\theta = \dfrac{\pi}{4}\)

57 Slope of Tangent to Polar Curve · Level 2

Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\). \(r = \dfrac{1}{\theta}\), \(\theta = \pi\)

58 Slope of Tangent to Polar Curve · Level 2

Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\). \(r = \cos\left(\dfrac{\theta}{3}\right)\), \(\theta = \pi\)

59 Slope of Tangent to Polar Curve · Level 2

Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\). \(r = \cos 2 \theta\), \(\theta = \dfrac{\pi}{4}\)

60 Slope of Tangent to Polar Curve · Level 2

Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\). \(r = 1 + 2 \cos \theta\), \(\theta = \dfrac{\pi}{3}\)

61 Horizontal and Vertical Tangents · Level 2

Find the points on the given curve where the tangent line is horizontal or vertical. \(r = 3 \cos \theta\)

62 Horizontal and Vertical Tangents · Level 3

Find the points on the given curve where the tangent line is horizontal or vertical. \(r = 1 - \sin \theta\)

63 Horizontal and Vertical Tangents · Level 3

Find the points on the given curve where the tangent line is horizontal or vertical. \(r = 1 + \cos \theta\)

64 Horizontal and Vertical Tangents · Level 3

Find the points on the given curve where the tangent line is horizontal or vertical. \(r = e^\theta\)

65 Polar Equation of a Circle · Level 2

Show that the polar equation \(r = a \sin \theta + b \cos \theta\), where \(a b \neq 0\), represents a circle, and find its center and radius.

66 Orthogonal Polar Curves · Level 3

Show that the curves \(r = a \sin \theta\) and \(r = a \cos \theta\) intersect at right angles.

67 Graphing Device - Special Polar Curves · Level 2

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

68 Graphing Device - Special Polar Curves · Level 2

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

69 Graphing Device - Special Polar Curves · Level 2

Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. \(r = e^{\sin \theta} - 2 \cos(4 \theta)\) (butterfly curve)

70 Graphing Device - Special Polar Curves · Level 2

Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. \(r = |\tan \theta|^|\cot \theta|\) (valentine curve)

71 Graphing Device - Special Polar Curves · Level 2

Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. \(r = 1 + \cos^{999} \theta\) (Pac-Man curve)

72 Graphing Device - Special Polar Curves · Level 2

Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. \(r = 2 + \cos\left(9 \dfrac{\theta}{4}\right)\)

73 Rotation of Polar Graphs · Level 2

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 \(r = f(\theta)\)?

74 Optimization on Polar Curve · Level 3

Use a graph to estimate the \(y\)-coordinate of the highest points on the curve \(r = \sin 2 \theta\). Then use calculus to find the exact value.

75 Family of Polar Curves · Level 2

Investigate the family of curves with polar equations \(r = 1 + c \cos \theta\), where \(c\) is a real number. How does the shape change as \(c\) changes?

76 Family of Polar Curves · Level 3

Investigate the family of polar curves \(r = 1 + \cos^n \theta\) where \(n\) is a positive integer. How does the shape change as \(n\) increases? What happens as \(n\) becomes large? Explain the shape for large \(n\) by considering the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates.

77 Tangent-Radial Angle in Polar Curves · Level 3

Let \(P\) be any point (except the origin) on the curve \(r = f(\theta)\). If \(\psi\) is the angle between the tangent line at \(P\) and the radial line \(O P\), show that \(\tan \psi = r / \left(d \dfrac{r}{d} \theta\right)\) [Hint: Observe that \(\psi = \phi - \theta\) in the figure.]

78 Constant-Angle Polar Curves · Level 3

(a) Use Exercise 77 to show that the angle between the tangent line and the radial line is \(\psi = \dfrac{\pi}{4}\) at every point on the curve \(r = e^\theta\).

Answer for 78(a)

(b) Illustrate part (a) by graphing the curve and the tangent lines at the points where \(\theta = 0\) and \(\dfrac{\pi}{2}\).

Answer for 78(b)

(c) Prove that any polar curve \(r = f(\theta)\) with the property that the angle \(\psi\) between the radial line and the tangent line is a constant must be of the form \(r = C e^{k \theta}\), where \(C\) and \(k\) are constants.

Answer for 78(c)

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79 Example - Tangent Lines to Polar Curves · Level 3

(a) For the cardioid \(r = 1 + \sin \theta\), find the slope of the tangent line when \(\theta = \dfrac{\pi}{3}\).

Answer for 79(a)

(b) Find the points on the cardioid where the tangent line is horizontal or vertical.

Answer for 79(b)

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80 Example - Graphing Polar Curves · Level 2

Graph the curve \(r = \sin\left(8 \dfrac{\theta}{5}\right)\).

81 Example - Family of Limacons · Level 2

Investigate the family of polar curves given by \(r = 1 + c \sin \theta\). How does the shape change as \(c\) changes? (These curves are called limacons.)

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