511
PART 1: Solutions to Odd-Numbered Exercises and Practice Tests
Section 10.7 Graphs of Polar Equations
When graphing polar equations:
1. Test for symmetry
(a) 0 = ,rr/2: Replace (r, 0) by (r, ,n" - 0) or (-r, -0).
(b) Polar axis: Replace (r, 0) by (r, -0) or (-r, ~ - 0).
(c) Pole: Replace (r, 0) by (r, !r + 0) or (-r, 0).
(d) r = f(sin 0) is symmetric with respect to the line 0 =
(e) r = f(cos 0) is symmetric with respect to the polar axis.
2. Find the 0 values for which [r[ is maximum.
3. Find the 0 values for which r = 0.
4. Know the different types of polar graphs.
(b) Rose Curves, n >- 2
(a) Lima~ons
r=a+bcosO
r = a cos nO
r = a sin nO
r = a + bsin 0
(d) Lenmiscates
(c) Circles
r~ = a2cos 20
r = a cos 0
r~ = a2 sin 20
r = a sin O
r=a
You should be able to graph polar equations of the form r = f(O) with your graphing utility. If your utility
does not have a polar mode, use
x = f(t) cos t
y =f(t) sint
in parametric mode.
Solutions to Odd-Numbered Exercises
3. r = 3 cos 0 is a circle.
1. r = 3 cos 20 is a rose curve.
7. r= 10 + 4cos0
: - = 10+4cos(-
- r = 10 + 4 cos 0 Not an equivalent equation
r = 10 + 4 cos(’tr - 0)
r = 10 + 4(cos ’n’cos 0 + sin ~rsin 0)
r = 10 - 4 cos 0 Not an equivalent equation
Polar axis: r = 10 + 4 cos(- 0)
r = 10 + 4 cos 0 Equivalent equation
Pole: - r = 10 + 4 cos 0
Not an equivalent equation
r = 10 + 4 cos(~" + 0)
r = 10 + 4(coscr cos O - sin ~r sin 0)
Not an equivalent equation
r = 10 - 4 cos 0
Answer: Symmetric with respect to polar axis.
5. r = 6 sin 20 is a rose curve.
512,
PART l: Solutions to Odd-Numbered Exercises and Practice Tests
6
1 + sin 0
11. r = 6sin0
6
0=-~: r= 1 +sin(’n’- 0)
2"
-r = 6 sin(-0)
r = 6 sin 0
6
1 + sin ,n-cos 0 - cos ~rsin 0
6
1 + sin 0
Equivalent equation
Polar axis: r = 6 sin(- 0)
r = -6 sin 0
Not an equivalent equation
Equivalent equation
Polar r
axis:
- r = 6 sin(~- - 0)
6
1 + sin(- 0)
6
1 - sin 0
Not an equivalent equation
-r = 6(sin "rr cos 0 - cos "n" sin 0)
- r = 6 sin 0
Not an equivalent equation
Pole:
6
1 + sin(~ - 0)
6
1 + sin 0
Not an equivalent equation
r = 6 sin(’rr + 0)
r = - 6 sin 0
Not an equivalent equation
Not an equivalent equation
The
pole:
-r=
6
1 + sin 0
~r
Answer: Symmetric with respect to 0 = ~.
Not an equivalent equation
6
1 + sin(~" + 0)
6
1 - sin 0
Not an equivalent equation
Answer: Symmetric with respect to 0 = ~r
2"
13. r=4sec0csc0
0
2"
-r = 4 sec(- 0) csc(- 0)
-r = --4 sec 0csc 0
r = 4 sec 0csc 0
Equivalent equation
Polar axis: -r = 4 sec(’rr - O) csc(~r - 0)
-r = 4(-see 0) csc 0
r = 4 sec 0 csc 0
Equivalent equation
Pole:
- r = 6 sin 0
r = 4 sec(’n" + 0) csc(’rr + 0)
r = 4(-sec 0)(-csc 0)
r = 4 sec 0 csc 0
Equivalent equation
Answer: Symmetric with respect to 0 = ~-/2, pole axis, and pole
513
PART l: Solutions to Odd-Numberdd Exercises and Practice Tests
15. ? = 25sin20
(- r)2 = 25 sin(2(- 0))
rz = -25 sin 20
Not an equivalent equation
r~ = 25 sin(20r - 0))
r~ = 25 sin(2~ - 20)
r~ = 25(sin 2qr cos 20 - cos 27r sin 20)
rz = - 25 sin 20
Polar axis: r~ = 25 sin(2(- 0))
ta = - 25 sin 20
Not an equivalent equation
Not an equivalent equation
(-r)2 = 25 sin(2(~ - 0))
r~ = -25 sin 20
Not an equivalent equation
(-r)z = 25 sin(20)
Pol~:
r~ = 25 sin 20
Equivalent equation
Answer: Symmetric with respect to pole.
17. Irl = 110(1 - sin 0)1
19.
[cos 301 = 1
= 1011 - sin 01 -< 10(2) = 20
cos 30 = 5:1
qr 2~r
11 - sin 0[ = 2
I -sin0=2 or 1-sin0=-2
sin 0 = -1
3qr
0=
2
= 14cos 301 = 4 [cos 301 -<4
sin 0 = 3
¯ " 2~
Maximum: Irl = 4when 0 = 0, =,--, ~r
33
Not possible
3~"
Maximum: Irl = 20.when 0 = ~-.
r = 0 when 1 - sin 0 = 0
r=0 when cos 30=0
~r ~r 5~r
0=T.
sin0= 1
2
21. Circle: r = 5
2
2S. r = 3sin0
6
Symmetric with respect to 0 = ~
2
Circle with radius of ~2
2
123
514
3(1 - cos 0)
Cardioid
PART 1: Solutions to Odd-Numbered Exercises and Practice Tests
29. r= 3-4cosO
31. r=6+sinO
Convex lima~on
Limaqon
2
2
4
33. r = 5cos 30
Rose curve
35. r = 7sin20
Rose curve
_~
39. - 10~" <- 0 < lO~r
0
37. r=2
Symmetric with respect
~r
to0=-2
Spiral
4 petals
41. 0 <_ 0_< "rr
43. 0_< 0-<2~r
47.0< O< ~
2
49. -2~ < 0 < 2~"
4
45. 0-< O<2~r
4
515
PART l." Solutions to Odd-Numbered Exercises and Practice Tests
55. r = 3- 2cos0,0 <_ 0 < 2"tr
53. 0< 0<2~-
51.0<0<~
-12
57. r=2+sin0, 0~0<2~r
3
61. rz = 4sin20, 0_<0<~
63. r=2-secO
x = - 1 is an asymptote.
(Use rI = 4s~and r2 = -- 4S~.)
4
©.
-.
2
65. r=0
y = 2 is an asymptote.
ll’rr
67. False. If 0 = --6-’-’ r = 4
71. (a)
(b)
.,(
(
"
69. False. It has 5 petals
(C)
,a
.,(.3
516
PART l: Solutions to Odd-Numbered Exercises and Practice Tests
73. Use the result of Exercise 72.
(a) Rotation: ~b 2
Original graph: r = f(sin 0)
Rotat~l graph: r=f(sin(O-~))=f(-cosO)
(b) Rotation: ~ = ,rr
Original graph: r = f(sin 0)
Rotated graph: r = f(sin(0 - ~r)) = f(-sin 0)
(c) Rotation: ~b =
Original graph: r =f(sin 0)
Rotated graph: r=f(sin(O-~-))=f(cosO)
(b) r = 2 sin[2(0- ~)]
= 2 sin (20 - ~)
= -2 sin 20
= -4 sin 0cos 0
= 4 sin(0 - ~) cos(0 - ~)
= 2 sin(20 - ~)
= sin 20 - v/~ cos 20
(d) r = 2 sin[2(O -
(c) r=2sin 2 O-
= 2 sin(20 - 2~r)
= 2 sin 20
= 4 sin t9 cos 0
= 4 sin(0 - ~) cos(0 - ~)
= ~eos20 - sin20
77. r = 2 + k cos O
k=0
k=l
Circle
Convex limaqo9
k=2
k=3
Cardioid
Limaqon with inner loop
517
PART l: Solutions to Odd-Numbered Exercises and Practice Tests
81. a~ = 150. ak+~ = ak - 18
79. at =2, a8 =23
as=at-j8- l)d
a2= 150- 18= 132
23 =2+7d :=~ d= 3
a3= 132- 18= 114
a~ = 2, a2 = 5, a3 = 8, a,~ = 11, a5 = 14
a,~= 114- 18=96
a.,=2-r(n- 1)3 =3n- [
a5=96- 18=78
d = - 18. a,, = 150 + (n - 1)(-18) = 168 - 18n
83. ~ 4n = 420"21"
2 = 840
n=1
85.,,---~-~tn + 5)
=
2
+ 5(120) 87..~0\~/z~’-__
= 7860
1 -- --1
8
7
Section 10.8 Polar Equations of Conics
[] The graph of a polar equation of the form
ep
ep
[
5:esin0
1 + ecos 0
is a conic, where e > 0 is the eccena:icity and IP[ is the distance between the focus (pole) and the directrix.
(a) If e < 1, the graph is an ellipse.
(b) If e = 1. the graph is a parabola.
(c) If e > 1. the graph is a hyperbola.
[] Guidelines for finding polar equations of conics:
ep
(a) Horizontal direca’ix above the pole: r =
l + esin0
ep
(b) Horizontal directrix below the pole: r - 1 - e sin 0
ep
(c) Vertical directrix to the right of the pole: r = 1 + e Cos 0
(d) Vertical direcldx to the left of the pole: r =
l - e cos 0
Solutions to Odd-Numbered Exercises
5. Matches (b).
(Parabola e = 1)
7. Matches (d).
(Hyperbola e = 2)