Stewart Precalc 6e Section 8.1: Polar Coordinates

1 Concepts - Coordinate Systems · Level 1

We can describe the location of a point in the plane using different ______ systems. The point \(P\) shown in the figure has rectangular coordinates \((\ ,\ )\) and polar coordinates \((\ ,\ )\).

2 Concepts - Conversion Formulas · Level 1

Let \(P\) be a point in the plane. (a) If \(P\) has polar coordinates \((r, \theta)\) then it has rectangular coordinates \((x, y)\) where \(x = \) ____ and \(y = \) ____. (b) If \(P\) has rectangular coordinates \((x, y)\) then it has polar coordinates \((r, \theta)\) where \(r^2 = \) ____ and \(\tan \theta = \) ____.

3 Skills - Plotting Polar Points · Level 1

Plot the point that has the given polar coordinates: \(\left(4, \dfrac{\pi}{4}\right)\).

4 Skills - Plotting Polar Points · Level 1

Plot the point that has the given polar coordinates: \((1, 0)\).

5 Skills - Plotting Polar Points · Level 1

Plot the point that has the given polar coordinates: \((6, -(7 \pi)/6)\).

6 Skills - Plotting Polar Points · Level 1

Plot the point that has the given polar coordinates: \((3, -(2 \pi)/3)\).

7 Skills - Plotting Polar Points · Level 2

Plot the point that has the given polar coordinates: \((-2, (4 \pi)/3)\).

8 Skills - Plotting Polar Points · Level 2

Plot the point that has the given polar coordinates: \((-5, -(17 \pi)/6)\).

9 Skills - Equivalent Representations · Level 2

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r < 0\) and the other with \(r > 0\): \(\left(3, \dfrac{\pi}{2}\right)\).

10 Skills - Equivalent Representations · Level 2

Plot the point and give two other polar coordinate representations, one with \(r < 0\) and one with \(r > 0\): \((2, (3 \pi)/4)\).

11 Skills - Equivalent Representations · Level 2

Plot the point and give two other polar coordinate representations, one with \(r < 0\) and one with \(r > 0\): \((-1, (7 \pi)/6)\).

12 Skills - Equivalent Representations · Level 2

Plot the point and give two other polar coordinate representations, one with \(r < 0\) and one with \(r > 0\): \(\left(-2, -\dfrac{\pi}{3}\right)\).

13 Skills - Equivalent Representations · Level 2

Plot the point and give two other polar coordinate representations, one with \(r < 0\) and one with \(r > 0\): \((-5, 0)\).

14 Skills - Equivalent Representations · Level 2

Plot the point and give two other polar coordinate representations, one with \(r < 0\) and one with \(r > 0\): \((3, 1)\).

15 Skills - Identifying Points · Level 2

Determine which point in the figure, \(P\), \(Q\), \(R\), or \(S\), has the given polar coordinates: \((4, (3 \pi)/4)\).

16 Skills - Identifying Points · Level 2

Determine which point in the figure, \(P\), \(Q\), \(R\), or \(S\), has the given polar coordinates: \((4, -(3 \pi)/4)\).

17 Skills - Identifying Points · Level 3

Determine which point in the figure, \(P\), \(Q\), \(R\), or \(S\), has the given polar coordinates: \(\left(-4, -\dfrac{\pi}{4}\right)\).

18 Skills - Identifying Points · Level 3

Determine which point in the figure, \(P\), \(Q\), \(R\), or \(S\), has the given polar coordinates: \((-4, (13 \pi)/4)\).

19 Skills - Identifying Points · Level 3

Determine which point in the figure, \(P\), \(Q\), \(R\), or \(S\), has the given polar coordinates: \((4, -(23 \pi)/4)\).

20 Skills - Identifying Points · Level 3

Determine which point in the figure, \(P\), \(Q\), \(R\), or \(S\), has the given polar coordinates: \((-4, (23 \pi)/4)\).

21 Skills - Identifying Points · Level 3

Determine which point in the figure, \(P\), \(Q\), \(R\), or \(S\), has the given polar coordinates: \((-4, (101 \pi)/4)\).

22 Skills - Identifying Points · Level 3

Determine which point in the figure, \(P\), \(Q\), \(R\), or \(S\), has the given polar coordinates: \((4, (103 \pi)/4)\).

23 Skills - Rectangular to Polar (Graph) · Level 2

A point is graphed in rectangular form. Find polar coordinates for the point, with \(r > 0\) and \(0 < \theta < 2 \pi\).

24 Skills - Rectangular to Polar (Graph) · Level 2

A point is graphed in rectangular form. Find polar coordinates for the point, with \(r > 0\) and \(0 < \theta < 2 \pi\).

25 Skills - Polar to Rectangular (Graph) · Level 2

A point is graphed in polar form. Find its rectangular coordinates.

26 Skills - Polar to Rectangular (Graph) · Level 2

A point is graphed in polar form. Find its rectangular coordinates.

27 Skills - Polar to Rectangular · Level 2

Find the rectangular coordinates for the point whose polar coordinates are given: \(\left(4, \dfrac{\pi}{6}\right)\).

28 Skills - Polar to Rectangular · Level 2

Find the rectangular coordinates: \((6, (2 \pi)/3)\).

29 Skills - Polar to Rectangular · Level 2

Find the rectangular coordinates: \(\left(\sqrt{2}, -\dfrac{\pi}{4}\right)\).

30 Skills - Polar to Rectangular · Level 2

Find the rectangular coordinates: \((-1, (5 \pi)/2)\).

31 Skills - Polar to Rectangular · Level 2

Find the rectangular coordinates: \((5, 5 \pi)\).

32 Skills - Polar to Rectangular · Level 1

Find the rectangular coordinates: \((0, 13 \pi)\).

33 Skills - Polar to Rectangular · Level 2

Find the rectangular coordinates: \((6 \sqrt{2}, (11 \pi)/6)\).

34 Skills - Polar to Rectangular · Level 2

Find the rectangular coordinates: \((\sqrt{3}, -(5 \pi)/3)\).

35 Skills - Rectangular to Polar · Level 2

Convert the rectangular coordinates to polar coordinates with \(r > 0\) and \(0 \leq \theta < 2 \pi\): \((-1, 1)\).

36 Skills - Rectangular to Polar · Level 2

Convert to polar with \(r > 0\) and \(0 \leq \theta < 2 \pi\): \((3 \sqrt{3}, -3)\).

37 Skills - Rectangular to Polar · Level 2

Convert to polar with \(r > 0\) and \(0 \leq \theta < 2 \pi\): \((\sqrt{8}, \sqrt{8})\).

38 Skills - Rectangular to Polar · Level 2

Convert to polar with \(r > 0\) and \(0 \leq \theta < 2 \pi\): \((-\sqrt{6}, -\sqrt{2})\).

39 Skills - Rectangular to Polar · Level 2

Convert to polar with \(r > 0\) and \(0 \leq \theta < 2 \pi\): \((3, 4)\).

40 Skills - Rectangular to Polar · Level 2

Convert to polar with \(r > 0\) and \(0 \leq \theta < 2 \pi\): \((1, -2)\).

41 Skills - Rectangular to Polar · Level 1

Convert to polar with \(r > 0\) and \(0 \leq \theta < 2 \pi\): \((-6, 0)\).

42 Skills - Rectangular to Polar · Level 1

Convert to polar with \(r > 0\) and \(0 \leq \theta < 2 \pi\): \((0, -\sqrt{3})\).

43 Skills - Equation to Polar · Level 2

Convert the equation to polar form: \(x = y\).

44 Skills - Equation to Polar · Level 1

Convert the equation to polar form: \(x^2 + y^2 = 9\).

45 Skills - Equation to Polar · Level 2

Convert the equation to polar form: \(y = x^2\).

46 Skills - Equation to Polar · Level 2

Convert the equation to polar form: \(y = 5\).

47 Skills - Equation to Polar · Level 2

Convert the equation to polar form: \(x = 4\).

48 Skills - Equation to Polar · Level 3

Convert the equation to polar form: \(x^2 - y^2 = 1\).

49 Skills - Polar Equation to Rectangular · Level 1

Convert the polar equation to rectangular coordinates: \(r = 7\).

50 Skills - Polar Equation to Rectangular · Level 2

Convert the polar equation to rectangular coordinates: \(\theta = -\dfrac{\pi}{2}\).

51 Skills - Polar Equation to Rectangular · Level 3

Convert the polar equation to rectangular coordinates: \(r = 1 + \cos \theta\).

52 Skills - Polar Equation to Rectangular · Level 3

Convert the polar equation to rectangular coordinates: \(r = 3(1 - \sin \theta)\).

53 Skills - Polar Equation to Rectangular · Level 3

Convert the polar equation to rectangular coordinates: \(r = 1 + 2 \sin \theta\).

54 Skills - Polar Equation to Rectangular · Level 3

Convert the polar equation to rectangular coordinates: \(r = 2 - \cos \theta\).

55 Skills - Polar Equation to Rectangular · Level 2

Convert the polar equation to rectangular coordinates: \(r = 1/(\sin \theta - \cos \theta)\).

56 Skills - Polar Equation to Rectangular · Level 3

Convert the polar equation to rectangular coordinates: \(r = 1/(1 + \sin \theta)\).

57 Skills - Polar Equation to Rectangular · Level 3

Convert the polar equation to rectangular coordinates: \(r = 4/(1 + 2 \sin \theta)\).

58 Skills - Polar Equation to Rectangular · Level 3

Convert the polar equation to rectangular coordinates: \(r = 2/(1 - \cos \theta)\).

59 Skills - Polar Equation to Rectangular · Level 3

Convert the polar equation to rectangular coordinates: \(r^2 = \tan \theta\).

60 Skills - Polar Equation to Rectangular · Level 3

Convert the polar equation to rectangular coordinates: \(r^2 = \sin(2 \theta)\).

61 Skills - Polar Equation to Rectangular · Level 2

Convert the polar equation to rectangular coordinates: \(\sec \theta = 2\).

62 Skills - Polar Equation to Rectangular · Level 2

Convert the polar equation to rectangular coordinates: \(\cos(2 \theta) = 1\).

63 Discovery - Distance Formula in Polar · Level 3

The Distance Formula in Polar Coordinates. (a) Use the Law of Cosines to prove that the distance between the polar points \((r_1, \theta_1)\) and \((r_2, \theta_2)\) is \(d = \sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta_2 - \theta_1)}\). (b) Find the distance between the points whose polar coordinates are \((3, (3 \pi)/4)\) and \((1, (7 \pi)/6)\), using the formula from part (a). (c) Now convert the points in part (b) to rectangular coordinates. Find the distance between them using the usual Distance Formula. Do you get the same answer?

64 Example - Plotting Polar Points · Level 1

Plot the points whose polar coordinates are given. (a) \((1, (3 \pi)/4)\) (b) \(\left(3, -\dfrac{\pi}{6}\right)\) (c) \((3, 3 \pi)\) (d) \(\left(-4, \dfrac{\pi}{4}\right)\)

65 Example - Equivalent Polar Representations · Level 2

(a) Graph the point with polar coordinates \(P\left(2, \dfrac{\pi}{3}\right)\). (b) Find two other polar coordinate representations of \(P\) with \(r > 0\) and two with \(r < 0\).

66 Example - Polar to Rectangular Conversion · Level 2

Find rectangular coordinates for the point that has polar coordinates \((4, (2 \pi)/3)\).

67 Example - Rectangular to Polar Conversion · Level 2

Find polar coordinates for the point that has rectangular coordinates \((2, -2)\).

68 Example - Converting Rectangular Equation to Polar · Level 2

Express the equation \(x^2 = 4 y\) in polar coordinates.

69 Example - Converting Polar Equation to Rectangular · Level 3

Express the polar equation in rectangular coordinates. If possible, determine the graph of the equation from its rectangular form. (a) \(r = 5 \sec \theta\) (b) \(r = 2 \sin \theta\) (c) \(r = 2 + 2 \cos \theta\)

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