Exercises: 10.1 Polar Coordinates

Bimal Kunwor; Donna Densmore; Jared Eusea; Yi Zhen

Chapter 10: Polar Coordinates and Complex Numbers

Exercises: 10.1 Polar Coordinates

SKILLS

Practice each skill in the Homework Problems listed.

Plot points in polar coordinates #1–8
Write polar coordinates for points #9–16
Convert Cartesian coordinates to polar #17–24
Convert Polar coordinates to Cartesian #25–32
Write alternate versions of polar coordinates #33–38
Sketch a region described by polar inequalities #39–44
Write polar inequalities to describe a region #45–50
Convert an equation to Cartesian coordinates #51–64
Convert an equation to polar coordinates #65–72

Suggested Homework

Problems: #2, 8, 10, 20, 22, 26, 32, 44, 50, 56, 64, 68, 70

Exercises for 10.1 Polar Coordinates

Exercise Group

For Problems 1–8, use the grid below to plot the points whose polar coordinates are given. polar grid

1.

$(2, \frac{2 π}{3})$

2.

$(3, \frac{5 π}{4})$

3.

$(3, \frac{3 π}{2})$

4.

$(4, π)$

5.

$(- 4, \frac{π}{6})$

6.

$(- 1, \frac{π}{2})$

7.

$(- 1, \frac{7 π}{4})$

8.

$(- 2, \frac{5 π}{3})$

Exercise Group

For Problems 9–16, give polar coordinates for each point shown below, with $r \geq 0$ and $0 \leq θ \leq 2 π .$
polar grid

9.

$A$

10.

$B$

11.

$C$

12.

$D$

13.

$E$

14.

$F$

15.

$G$

16.

$H$

Exercise Group

For Problems 17–24, convert the polar coordinates to Cartesian coordinates.

17.

$(6, \frac{2 π}{3})$

18.

$(5, \frac{7 π}{6})$

19.

$(- 3, \frac{3 π}{4})$

20.

$(- 4, \frac{5 π}{3})$

21.

$(2.4, 3.6)$

22.

$(1.7, 5.2)$

23.

$(- 2, 1.5)$

24.

$(- 3, 4.5)$

Exercise Group

For Problems 25–32, convert the Cartesian coordinates to polar coordinates with $r \geq 0$ and $0 \leq θ \leq 2 π .$ Give exact values for $r$ and $θ .$

25.

$(7, 7)$

26.

$(- 7, - 7)$

27.

$(\sqrt{6}, - \sqrt{2})$

28.

$(- \sqrt{3}, 3)$

29.

$(- 3, - 2)$

30.

$(1, 4)$

31.

$(- 2, 0)$

32.

$(0, - 5)$

Exercise Group

In Problems 33–38, polar coordinates are given.

Convert to polar coordinates with $r \leq 0, 0 \leq θ < 2 π .$
Convert to polar coordinates with $r \geq 0, - 2 π < θ \leq 0 .$

33.

$(2, \frac{5 π}{6})$

34.

$(1, \frac{4 π}{3})$

35.

$(3, π)$

36.

$(4, 0)$

37.

$(2.3, 5.2)$

38.

$(1.2, 1.2)$

Exercise Group

For Problems 39–44, sketch the region described by the inequalities.

39.

$r \leq 3$

40.

$2 \leq r \leq 3$

41.

$\frac{π}{6} \leq θ \leq \frac{π}{3}$

42.

$0 \leq θ \leq \frac{π}{2}$

43.

$r \geq 4, \frac{π}{2} \leq θ \leq \frac{3 π}{4}$

44.

$2 \leq r \leq 3, \frac{5 π}{3} \leq θ \leq \frac{11 π}{6}$

Exercise Group

For Problems 45–50, write inequalities to describe the region.

45.

polar region

46.

polar region

47.

polar region

48.

polar region

49.

polar region

50.

polar region

Exercise Group

For Problems 51–64, convert the equation into Cartesian coordinates.

51.

$r^{2} = 2$

52.

$r = 2$

53.

$r = 4 \cos θ$

54.

$r = \sin θ$

55.

$r = \csc θ$

56.

$r = 2 \sec θ$

57.

$\tan θ = 2$

58.

$r^{2} = \tan θ$

59.

$r \sec θ = 3$

60.

$θ = \frac{π}{3}$

61.

$r = \frac{2}{1 + \sin θ}$

62.

$r = \frac{4}{4 - \cos θ}$

63.

$2 r \cos θ + r \sin θ = 1$

64.

$\tan θ = r \cos θ - 2$

Exercise Group

For Problems 65–72, convert the equation into polar coordinates.

65.

$x = 2$

66.

$y^{2} = 2 x - x^{2}$

67.

$2 x y = 1$

68.

$y = - x$

69.

$y^{2} = 4 x$

70.

$x^{2} - y^{2} = 1$

71.

$x^{2} + y^{2} = 4 \sqrt{x^{2} + y^{2}}$

72.

$(x^{2} + y^{2})^{3} = 9 x^{2} y^{2}$

73.

Use the law of cosines to prove the distance formula in polar coordinates:

$d = \sqrt{r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos (θ_{2} - θ_{1})}$

74.

Show that the graph of $r = a \cos θ + b \sin θ$ is a circle. Find its center and radius.

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