and

of sin 6 is 277, soI, and sketch the35.

SECTION 8.2 Polar Equations and Graphs 597

8.2 Assess Your Understanding

nent about one

graph is the unilans to obtain the

'Are You Prepared?'

turners are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

1. It the rectangular coordinates of a point are (4, -6), thepoint symmetric to it with respect to the origin is .(pp. 17-19)

1 The difference formula for cosine is cos(a - ft) = .(p. 473)

4. Is the sine function even, odd, or neither? (pp. 398-399)

_. (pp. 380-381). 577

5. sin — =4

3. The standard equation of a circle with center at (—2, 5) andradius 3 is . (pp. 44-49)

Concepts and Vocabulary

7. An equation whose variables are polar coordinates is called

27T6. cos — : , (pp. 380-381)

8. Using polar coordinates (r, 9), the circle x2 + y2 = 2xtakes the form .

9. A polar equation is symmetric with respect to the pole if anequivalent equation results when r is replaced by .

10. True or False: The tests for symmetry in polar coordinatesare necessary, but not sufficient.

11. True or False: The graph of a cardioid never passes throughthe pole.

12. True or False: All polar equations have a symmetric feature.

kill Building

Problems 13-28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation.rify your graph using a graphing utility.

)wer semicircle

: unit circle, doesresponding value: of functions, thefunctions in polar

' for equations ir

rves. Finally, aboutd the tremendousble in the descrip-stry. From then on

r = 4 14. r = 2 V 15. 6 = 16. 6 =

17. r sin 0 = 4 18. r cos 0 = 4 V 19. r cos (9 = -2 20. r sin 0 = -2

21. r = 2 cos 6 22. r = 2 sin 9 23. r = -4 sin 0 24. r = -4 cos 9

25. r sec 9 = 4 26. r esc 9 = 8 27. r esc 0 = -2 28. r sec 9 = -4

nProblems 29-36, match each of the graphs (A) through (H) to one of the following polar equations.

.r = 2 30. 9 = — 31. r = 1 cos 0 32. r cos 0 = 24

r = 1 + cos 0 34. r = 2 sin 0 35. 9 = — 36. r sin 6 = 24

yi xi n y\| o-1! ah ijl

e=3f • e = J e=3f H 1 »=3-f - , e = f e=3f e = |/ ,„•• ,. \/ V/ X X X \ X

«-T /O 2 ~~a = 0 » - v ft" \ 'o = 0 » - •" ff" 2 ""9 - 0 « - •" 0*V 2 4 H = 0

Q _ S T T .Q-^?7 6=^ 6 = 7 — 9 = 5 — 0-^ 6=^ H - 7 —

e=3f e=3f e=3f e=3f

(A) (B) (C) (D)

598 CHAPTERS Polar Coordinates; Vectors

=

e = 0

(E) (F) (G)

In Problems 37-42, match each of the graphs (A) through (F) to one of the following polar equations.

37. r = 4 38. r = 3 cos 0 39. r = 3 sin 0

40. r sin 0 = 3 41. r cos 9 = 3 42. r = 2 + sin 0

5 2 3

-7.5 7.5

-5

(A)

4

-3

-7.5 7.5

-2.3-4 -5

(D) (E)

/« Problems 43-66, identify and graph each polar equation. Verify your graph using a graphing utility.

44. r = 1 + sin 6 45. r = 3 - 3 sin 0\. r = 2 + 2 cos 0

47. r = 2 + sin 0

\. r = 3 cos(20)

\. r2 = 9 cos(20)

63. r = 1 - cos 0

48. r = 2 - cos 0

52. r = 1 - 2 sin £

56. r = 2 sin(30)

60. r2 = sin(20)

64. r = 3 + cos 0

49. r = 4 - 2 cos 0

53. r = 2 - 3 cos 0

57. r = 4sin(50)

61. r = 2e

65. r = 1 - 3 cos 0

46. r = 2 - 2 cos 0

50. r = 4 + 2 sin 0

54. r = 2 + 4 cos 0

58. r = 3 cos(40)

62. r = 39

66. r = 4 cos(30)

Applications and ExtensionsIn Problems 67-70, the polar equation for each graph is either r = a + b cos 9 or r = a + b sin 0, a > 0, b > 0. Select the comaequation and find the values of a and b.67. 68.

e = ir

69.

e = -rrA

In Problems "/

71. r = -

73. r = —

75. r = 0, (

77. /- = csc0

79. r = tan 6

81. Show thatal line athe pole i

83. Show thacircle ofcoordinat

85. Show thaia circle ccoordinat

Discussion87. Explain v

Replace requivalent

respect to

(a) Showthis ne

(b) Show Ithis ne

'Are You PI- (-4,6)

I

SECTION 8.2 Polar Equations and Graphs 599

70.

9 = 7 T

liProblems 71-80, graph each polar equation. Verify your graph using a graphing utility.

I L r =

13, r =

i - cos ei

(parabola)

(ellipse)3 - 2 cos 6

15. r = 6, 6 s 0 (spiral of Archimedes)

77. r = esc 6» - 2, 0 < 0 < TT (conchoid)

77 7T• r = tan 0, — — < 0 < — (kappa curve)

81. Show that the graph of the equation r sin 6 = a is a horizon-tal line a units above the pole if a > 0 and a\s belowthe pole if a < 0.

83. Show that the graph of the equation r = 2a sin 6, a > 0, is acircle of radius a with center at (0, a) in rectangularcoordinates.

85. Show that the graph of the equation r = 2a cos 6, a > 0, isa circle of radius a with center at (a, 0) in rectangularcoordinates.

Discussion and Writing

7. Explain why the following test for symmetry is valid:Replace r by —r and 9 by -0 in a polar equation. If anequivalent equation results, the graph is symmetric with

respect to the line 6 = — (y-axis).

(a) Show that the test on page 587 fails for r2 = cos 6, yetthis new test works.

(b) Show that the test on page 587 works for r2 = sin 6, yetthis new test fails.

74. r =1 - cos

(parabola)

76. r = - (reciprocal spiral)6

78. r = sin 9 tan d (cissoid)

e80. r = cos -

82. Show that the graph of the equation r cos 6 = a is a verticalline a units to the right of the pole if a > 0 and \a\s tothe left of the pole if a < 0.

84. Show that the graph of the equation r = —2a sin 9, a > 0, isa circle of radius a with center at (0, —a) in rectangularcoordinates.

86. Show that the graph of the equation r = -la cos 6, a > 0,is a circle of radius a with center at (—a, 0) in rectangularcoordinates.

88. Develop a new test for symmetry with respect to the pole.(a) Find a polar equation for which this new test fails, yet

the test on page 587 works.(b) Find a polar equation for which the test on page 587

fails, yet the new test works.

89. Write down two different tests for symmetry with respect tothe polar axis. Find examples in which one test works andthe other fails. Which test do you prefer to use? Justify youranswer.

'Are You Prepared?' Answers

1, (-4, 6) 2. cos a cos /3 + sin a sin /3 3. (x + 2)2 + (y - 5)2 = 9 4. odd 5.-V~2

6.-