Chapter 11: Polar Equations and Complex Numbers
Lesson 11.1.1
11-1.
A
See diagram at right.
I,J
F,G
11-2.
C
B
D and H; F and G; I and J
H,D
11-3.
2,
a.
b.
11-4.
E
( 116! ) , ( !2, 56" ) are two ways. There are many others.
(1, 94! ) , (1, ! 74" ) are two ways. There are many others.
( !2 ) , B = ( !2, "4 ) , C = ( 3, 0 ) , D = ( 2, ! )
A = 1,
11-5.
Radius of A: r 2 = 32 + 32
(
A = 3 2,
!
4
Radius of D: r 2 = (!4)2 + (!4)2
r 2 = 18
r 2 = 32
r=3 2
r = !4 2
) , B = ( 3, ! ) , C = (1, 32! ) , D = ( !4
2,
"
4
)
11-6.
r
q
r
0
0
0.5
0.5
!
6
–0.707
7!
6
5!
4
0.71
!
4
–0.866
4!
3
0.87
!
3
–1
3!
2
1
!
2
–0.866
0.87
2!
3
–0.707
5!
3
7!
4
0.71
3!
4
–0.5
11!
6
0.5
5!
6
0
2!
CPM Educational Program © 2012
q
Chapter 11: Page 1
r = sin !
1
-1
Pre-Calculus with Trigonometry
0
!
The shape of this graph is a circle.
CPM Educational Program © 2012
Chapter 11: Page 2
Pre-Calculus with Trigonometry
11-7.
r = 3sin !
1
-1
r = " 2sin !
a.
c.
b.
In the equation y = k sin ! , k = diameter.
Review and Preview 11.1.1
11-8.
A
See diagram at right.
11-9.
(
C, D
)
(
A = ( 2, 0 ) , B = 3, – !6 , C = ( 3, ! ) , D = 1, – !2
B
)
11-10.
r
q
r
q
0
0
–1.732
7!
6
1.732
!
6
–1.414
5!
4
1.414
!
4
–1
4!
3
1
!
3
0
3!
2
0
!
2
1
5!
3
–1
2!
3
1.414
7!
4
–1.414
3!
4
1.732
11!
6
–1.732
5!
6
2
2!
–2
!
r = 2cos !
1
This is a graph of a circle of radius 1 centered at (1, 0).
CPM Educational Program © 2012
Chapter 11: Page 3
1
Pre-Calculus with Trigonometry
11-11.
a.
Circle along positive x-axis with diameter of 3 units.
b.
Circle along negative x-axis with diameter of 2 units.
11-12.
a.
!16 = 16 " !1 = 4i
b.
!8 = 4 " !1 " 2 = 2i 2
c.
5 ! !9 = 5 ! 9 " !1 = 5 ! 3i
d.
4+ !12
2
=
4+ 4 " 3" !1
2
=
4+2i 3
2
= 2+i 3
11-13.
a.
y = x 2 ! 4x + 13
b.
y = x 2 ! 4x + (4 ! 4) + 13
y = (x ! 2)2 + 9
V (2, 9)
y = x 2 ! 4x + 13
x=
!(!4)± (!4)2 ! 4(1)(13)
2(1)
x=
4 ± 16!52
2
4 ± !36
2
=
=
4 ±6i
2
x = 2 + 3i, 2 ! 3i
c.
The graph does not cross the x-axis.
11-14.
a.
See diagram at right.
b.
Distance for first boat: 10 ft/sec ! 5 seconds = 50 feet
Distance for second boat: ! 5 ft/sec ! 5 seconds = 12.5 feet
Distance for first boat: 10 ft/sec ! t seconds = 10t feet
1
2
c.
Distance for second boat:
d.
10t = 0.5t 2
1
2
v(t)
12
8
4
1st boat
2nd boat
4
8 12 16
t (seconds)
! t ft/sec ! t seconds = 0.5t 2 feet
The second boat will past the first after 20 seconds.
0 = 0.5t 2 ! 10t
0 = !0.5t(t ! 20)
t = 0, 20 seconds
11-15.
2m + m = 18
Let u = m
u 2 +u = 3 2
u( 2 + 1) = 3 2
u=
3 2
,
2 +1
m=
CPM Educational Program © 2012
( )
3 2
2 +1
2
= 3.088
Chapter 11: Page 4
Pre-Calculus with Trigonometry
11-16.
Radian mode, domain = [ !4" , 4" ] , range = [ !8, 8 ] .
11-17.
a.
cos
( 34! ) = x3
"
2
2
=
cos
( 32! ) = 5x
0=
( 34! ) = 3y
2
2
x
3
x="
b.
sin
=
y=
3 2
2
sin
( 32! ) =
x
5
1=
x=0
(!
3 2
2
, 3 22
)
y
3
3 2
2
y
5
(0, 5)
y
5
y=5
Lesson 11.1.2
11-18.
a.
A spiral graph.
b.
A circle centered on the y-axis.
c.
A circle centered on the negative x-axis.
r=
5
cos ! +sin !
1
11-19.
a.
See graph at right above. This is the line y = !x + 5 .
b.
This should be the line y = !x ! 2 .
c.
See graph at right below.
11-20.
This should be in the shape of a 4-leaf rose.
-1
r=
"2
cos ! +sin !
1
-1
r = 3 sin 2!
1
-1
11-21.
b.
r = 3 sin 4! is an 8-leaf rose and r = 3 sin 5! is an 5-leaf rose.
See graphs at right.
r = 3 sin 4!
1
-1
11-22.
Asymptotes.
r = 3sin 5!
1
CPM Educational Program © 2012
-1
Chapter 11: Page 5
Pre-Calculus with Trigonometry
Review and Preview 11.1.2
11-23.
a.
Because cos ! =
c.
x
r
.
r 2 = (r cos ! )2 + (r sin ! )2
b.
y = r sin !
d.
r sin !
r cos !
r 2 = x 2 + y2
=
tan ! =
r = x 2 + y2
y
x
y
x
tan "1 tan ! = tan "1
! = tan "1
11-24.
a.
Multiply each side by sin ! + cos ! .
b.
()
()
y
x
y
x
r cos ! + r sin ! = 5
x + y = 5, y = 5 " x
11-25.
y = 3 ! 2x
r sin " = 3 ! 2r cos "
r sin " + 2r cos " = 3
r(sin " + 2 cos " ) = 3, r =
3
2 cos " +sin "
11-26.
a.
3i(2 + 4i) = 6i + (3i)(4i) = 6i + 12i 2 = !12 + 6i
b.
(5i)2 = 5 2 i 2 = !25
c.
(2 + i)(2 ! i) = 4 ! 2i + 2i ! i 2 = 4 ! (!1) = 5
d.
(3 ! 2i)(4 + i) = 12 + 3i ! 8i ! 2i 2 = 12 ! 5i ! 2(!1) = 14 ! 5i
e.
(4 ! 5i)2 = 16 ! 20i ! 20i ! 5 2 i 2 = 16 ! 40i + 25(!1) = !9 ! 40i
11-27.
y = (x ! (4 ! i))(x ! (4 + i))
y = x 2 ! (4 ! i)x ! (4 + i)x + (4 ! i)2
y = x 2 ! 4x + ix ! 4x ! ix + 16 + i 2
y = x 2 ! 8x + 15
11-28.
3
a.
e3 ln x = eln x = x 3
c.
log 10 x = log(10 x )1/2 =
CPM Educational Program © 2012
1
2
log 10 x =
1
2
x
b.
ln e5 x = 5x
d.
10(3+log x) = 10 3 !10 log x = 1000x
Chapter 11: Page 6
Pre-Calculus with Trigonometry
CPM Educational Program © 2012
Chapter 11: Page 7
Pre-Calculus with Trigonometry
11-29.
a
(
a+
20
3
+
2a
3
21
3
+
+
4a
9
22
3
+
+
8a
27
23
3
+ . . . = 90
)
+ … = 90
a
90 = 1!2/3
a
90 = 1/3
3a = 90, a = 30
11-30.
w = 2u ! v
w = 2 !3, 2 ! 4, 1 = !6, 4 ! 4, 1
w = !6 ! 4, 4 ! 1 = !10, 3
w = (!10)2 + (3)2 = 109
11-31.
h(t) = !16t 2 + 20t + 50
h(2.5 + h) = !16(2.5 + h)2 + 20(2.5 + h) + 50
= !16(6.25 + 5h + h 2 ) + 50 + 20h + 50
0 = !16t 2 + 20t + 50
= !100 ! 80h ! 16h 2 + 100 + 20h
0 = !8t 2 + 10t + 25
0 = (4t + 5)(!2t + 5)
t = !1.2, 2.5
h(2.5) = 0
h(2.5+h)!h(2.5)
2.5+h!2.5
=
!16h 2 !60h
h
= !16h 2 ! 60h
= !16h ! 60
lim (!16h ! 60) = !60
h"0
The ball hits the ground after 2.5 seconds with a velocity of –60 feet per second.
11-32.
A circle centered at the origin with radius 4.
11-33.
A circle centered at the pole with radius 4.
CPM Educational Program © 2012
Chapter 11: Page 8
Pre-Calculus with Trigonometry
Lesson 11.1.3
11-34.
a.
(r cos ! )2 + (r sin ! )2 = 2r cos !
x 2 + y 2 = 2x
b.
r 2 cos2 ! + r 2 sin 2 ! = 2r cos !
x 2 ! 2x + y 2 = 0
r 2 = 2r cos !
r = 2 cos !
x 2 ! 2x + 1 + y 2 = 1
(x ! 1)2 + y 2 = 1
11-35.
y = x2
r sin ! = (r cos ! )2
r sin ! = r 2 cos2 !
r=
11-36.
sin !
cos2 !
r = 2 sin ! + cos !
r 2 = 2r sin ! + r cos !
x 2 + y 2 = 2y + x
11-37.
a.
See graph at right.
b.
The second graph is obtained by rotating the first graph
r = 2 sin !
r = 2sin( ! # "4 )
!
4
-1
.
1
11-38.
r = 1 + 3 sin(! " # )
11-39.
((
r = sin 2 ! "
11-40.
r=
#
3
) ) is the same graph as r = sin 2! , just rotated in angle of !3 .
sin !
cos2 !
r=
Rotated clockwise 45 o
r=
sin( " + !4 )
cos 2 ( " + !4 )
1
sin (! + " 4 )
cos2 (! + " 4 )
-1
11-41.
a.
Every point with first coordinate 3 lies on the graph.
CPM Educational Program © 2012
Chapter 11: Page 9
Pre-Calculus with Trigonometry
b.
c.
It is a line through the origin of slope 1.
It is a circle of radius 3 centered at the origin.
11-42.
a.
A line through the origin making an angle of
b.
y=
c.
!=
2!
+ 2! = 23! + 63! = 83! !!or!!y = 23!
3
" #3 + # n where n is an integer.
"!
2!
3
2!
= 3
with the positive horizontal axis.
"
3!
3
= " !3
Review and Preview 11.1.3
11-43.
r = f ! " #2
a.
)
b.
r = f ! + "2
11-44.
a.
6x ! x 2 = y 2
b.
2r cos ! + 3r sin ! = 5
r(2 cos ! + 3 sin ! ) = 5
(
6r cos " ! (r cos " )2 = (r sin " )2
(
r=
6r cos " = r 2 cos2 " + r 2 sin 2 "
)
5
2 cos ! + 3 sin !
6r cos " = r 2
r = 6 cos "
c.
x 2 + y2 = 7
d.
y = x2
r sin ! = r 2 cos2 !
r2 = 7
sin ! = r cos2 !
r= 7
r=
11-45.
a.
r = 6 sin !
r 2 = 6r sin !
x 2 + y 2 = 6y
b.
r + 3 sin ! = 0
c.
r 2 + 3r sin ! = 0
x 2 + y 2 + 3y = 0
sin !
cos2 !
r(2 cos ! + sin ! ) = 4
2r cos ! + r sin ! = 4
2x + y = 4
11-46.
CPM Educational Program © 2012
Chapter 11: Page 10
Pre-Calculus with Trigonometry
r=
sin (! + " 4 )
cos2 (! + " 4 )
r 2 cos2 (! + " 4) = r sin(! + " 4)
(r cos(! + " 4))2 = r(sin ! cos "4 + cos ! sin "4 )
( r(cos ! cos "4 # sin ! sin "4 )2 =
(
2
2
1
2
(r cos ! # r sin ! )2 =
1
2
(x # y)2 =
r(cos ! # sin ! )
(x# y)2
(x+ y)
2
2
)
2
=
2
2
2
2
2
2
r(sin ! + cos ! )
r(sin ! + cos ! )
(r sin ! + r cos ! )
(x + y)
= 2
11-47.
tan ! =
a.
opposite
adjacent
=
3t
800" 4t
b.
( 800"3t 4t )
3t
! = tan "1 ( 800"
4t )
tan "1 (tan ! ) = tan "1
t=
800
7
! 302.372!sec
" = tan #1
( ) ! 48.59°
3
7
11-48.
a.
( 3 + 2i ) ( 3 ! 2i ) = 9 ! 6i + 6i ! 4i 2
b.
(
c.
d.
3 ! 7i
a ! bi
5 !i 3
)(
= 9 ! 4(!1)
= 13
5 + i 3 = ( 5 )2 ! i 15 + i 15 ! i 2 3 " 3 = 5 ! (!1) " 3 = 8
)
11-49.
5 ! 3"2i = 15"10i = 15"10i = 15 " 10i
a.
3+2i 3"2i
2
13
13
13
9" 4i
b.
2!i " 2!i
2+i 2!1
!
c.
2i
3+ 7i
d.
3+i 2
3!i 2
=
4! 4i+i 2
4!i 2
3" 7i
3" 7i
"
=
3+i 2
3+i 2
3! 4i
5
=
3
5
2i 3"14i 2
3"149i 2
=
2i 3+14
52
=
=
3+2i 6 +2i 2
3!2i 2
!
4i
5
= 1+2i5
=
6
7
26
+
3
26
=
1
5
+
i
2 6
5
i
11-50.
CPM Educational Program © 2012
Chapter 11: Page 11
Pre-Calculus with Trigonometry
r = !5 sin "
Center at (0, !2.5) , radius = 2.5
r 2 = !5r sin "
x 2 + y 2 = !5y
x 2 + y 2 + 5y = 0
x 2 + (y + 2.5)2 = 6.25
11-51.
Vertical shift of 4 units.
Amplitude: 5!2 3 = 1
Period:
2!
b
=6
6b = 2!
b = !3
Horizontal shift of 1 unit right.
y = cos !3 (x " 1) + 4
(
)
CPM Educational Program © 2012
Chapter 11: Page 12
Pre-Calculus with Trigonometry
Lesson 11.1.4
11-52.
1.
n determines the number of petals on the flower. If n is even there will be twice as many
flowers. b acts as a stretching factor.
2.
As one set of petals shrink, the other set grows. The sums of the lengths of two alternating
petals will equal 2b. The length of the first petal is a + b.
3.
Limaçon
4.
Cardioids
5.
Circle rotated about the pole
6.
Line rotated about the pole
7.
Spiral
8.
Parabolas
9.
Ellipses and hyperbolas
Review and Preview 11.1.4
11-53.
a.
r = 8 sin !
b.
r=
3
5 sin ! +cos !
5r sin ! + r cos ! = 3
5y + x = 3
r 2 = 8r sin !
x 2 + y 2 = 8y
x 2 + y 2 " 8y = 0
x 2 + (y " 4)2 = 16
Center at (0, 4) and radius 4.
11-54.
a.
6x ! 2x 2 = 2y 2
b.
6r cos " ! 2r 2 cos2 " = 2r 2 sin 2 "
6r cos " = 2r 2 sin 2 " + 2r 2 cos2 "
5x ! 3y = !2
5r cos " ! 3r sin " = !2
r(5 cos " ! 3 sin " ) = !2
r=
6r cos " = 2r 2
3 cos " = r
!2
5 cos " ! 3 sin "
11-55.
a.
(4 + 3i) ! (2 ! 4i) = 4 + 3i ! 2 + 4i = 2 + 7i
b.
(4 ! 2i)(3 + 4i) = 12 + 16i ! 6i ! 8i 2 = 12 + 10i + 8 = 20 + 10i
c.
(3 ! 5i)2 = 9 ! 30i + 5 2 i 2 = 9 ! 30i ! 25 = !16 ! 30i
d.
1+i " 1+i
1!i 1+i
= 1+2i!1
=
1!(!1)
2i
2
=i
CPM Educational Program © 2012
Chapter 11: Page 13
Pre-Calculus with Trigonometry
11-56.
y = (x ! 3)(x ! 2 ! i)(x ! 2 + i)
y-intercept: (0, –15)
y = (x ! 3)(x 2 ! 2x + ix ! 2x + 4 ! 2i ! ix + 2i ! (!1))
y = (x ! 3)(x 2 ! 4x + 5)
y = x 3 !4x 2 + 5x ! 3x 2 + 12x ! 15
y = x 3 !7x 2 + 17x ! 15
11-57.
a.
Vertical asymptote at x = 8 ; horizontal asymptotes at y = 6 and y = !3 .
b.
See sample graph at right.
11-58.
y
4
2
f (x) = x 2 + 3x
x
f (x + h) = (x + h)2 + 3(x + h) = x 2 + 2xh + h 2 + 3x + 3h
f (x+h)! f (x)
x+h! x
=
x 2 +2 xh+h 2 + 3x+ 3h! x 2 ! 3x
h
=
2 xh+h 2 + 3h
h
= 2x + h + 3
lim( 2x + h + 3) = 2x + 3
h"0
11-59.
(
4 cos x +
!
3
) = 4 ( cos x cos ( !3 ) " sin x sin ( !3 ) )
=4
(
1
2
cos x "
3
2
sin x
)
= 2 cos x " 2 3 sin x
11-60.
c 4 ! 12c 2 ! 64 = 0
(c 2 ! 16)(c 2 + 4) = 0
11-61.
x = 2t ! 1
x + 1 = 2t
t=
x+1
2
c 2 ! 16 = 0
c 2 = 16
c = ±4
c2 + 4 = 0
c 2 = !4
c = ±2i
( x+12 ) + 1
2
f (x) = ( x+1
+1
)
2
y(x) =
CPM Educational Program © 2012
2
Chapter 11: Page 14
Pre-Calculus with Trigonometry
11-62.
x 2 + y 2 ! 6x = 0
r 2 cos2 " + r 2 sin 2 " ! 6r cos " = 0
r 2 (cos2 " + sin 2 " ) ! 6r cos " = 0
r 2 ! 6r cos " = 0
r 2 = 6r cos "
r = 6 cos "
11-63.
a.
ln 3 + ln k = ln 3k
c.
e5 ln x = eln x = x 5
5
( 12t ) = ln 12 ! ln t
b.
ln
d.
ln 12 ! ln 4 + ln(x ! 3)
ln
( 124 ) + ln(x ! 3)
ln(3(x ! 3))
ln(3x ! 9)
i
" 3 + 4i
Lesson 11.2.1
!
"2 " i
11-64.
See graph at right.
!3 + 4i = (!3)2 + 4 2 = 9 + 16 = 25 = 5
a.
b.
!2 ! i = (!2)2 + (!1)2 = 5
11-65.
C = A + B = 2 ! 5i ! 1 + i = 1 ! 4i
a.
A = (2)2 + (!5)2 = 4 + 25 = 29
b.
B =
c.
i
(!1)2
+ (1)2
"1 + i
= 1+1 = 2
!
1 " 4i
C = (1)2 + (!4)2 = 1 + 16 = 17
Triangle inequality.
2 " 5i
11-66.
a.
r = 32 + 32 = 18 = 3 2
tan ! =
!=
"
4
3
3
b.
=1
(
z = 3 2 cos "4 + i sin "4
CPM Educational Program © 2012
r = 2 2 + (!2 3)2 = 4 + 12 = 4
tan " =
" = ! #6
)
2
!2 3
=!
( ( )
1 !(Quadrant!4)
3
( ))
z = 4 cos ! #6 + i sin ! #6
Chapter 11: Page 15
Pre-Calculus with Trigonometry
11-67.
a.
( ( ) + i sin ( ) )
z = 4 cos
3!
4
3!
4
b.
( 34! ) = "4 # 12 = " 42 = "2
b = 4 sin ( 34! ) = 4 # 1 = 4 = 2 2
2
2
a = 4 cos
b=2
!
6
( !6 ) = 2 6 " 23 = 18 = 3
6 sin ( !6 ) = 2 6 " 12 = 6
2
3 2 + 6i
11-68.
a.
Because f (1) = 0 .
x=
!
6
a = 2 6 cos
2
"2 2 + 2 2i
c.
( ( ) + i sin ( ) )
z = 2 6 cos
!1± 12 ! 4(1)(1)
2(1)
=
b.
!1± !3
2
=!
11-70.
(x 4 ! 1) = (x 2 ! 1)(x 2 + 1)
1
2
±
3
2
(x ! 1)(x 2 + x + 1)
i
x2 ! 1 = 0
x2 + 1 = 0
x 2 = !1
x=± i
x2 = 1
x = ±1
Review and Preview 11.2.1
11-70.
r = 32 + (!4)2 = 25 = 5
a.
3 = 5 cos "
3
5
r = (!12)2 + (!5)2 = 169 = 13
b.
!12 = 13 cos "
= cos "
! 12
= cos "
13
" = 5.359, !0.927
!4 = 5 sin "
!
4
5
" = 3.5364
!5 = 13 sin "
= sin "
5 = sin "
! 13
" = !0.927, 5.359
" = 3.536 ! 2# = !2.747
Polar form: z = 5 ( cos(!0.927) + i sin(!0.927) )
z = 13 ( cos(3.536) + i sin(3.536) )
z = 5 ( cos(5.359) + i sin(5.359) )
11-71.
a.
( ( ) + i sin ( ) )
z = 6 cos
5!
6
z=6 "
+ 12 i = "3 3 + 3i
(
3
2
)
5!
6
CPM Educational Program © 2012
z = 13 ( cos(!2.747) + i sin(!2.747) )
b.
( ( ) + i sin ( ) )
z = 7 cos
3!
2
3!
2
z = 7(0 + ("1)i) = "7i
Chapter 11: Page 16
Pre-Calculus with Trigonometry
11-72.
a.
r 2 = cos ! sin !
r = !3 csc "
b.
r 2 r 2 = r 2 cos ! sin ! = r cos ! r sin !
r=
(x 2 + y 2 )2 = xy
!3
sin "
r sin " = !3
y = !3
11-73.
2 sin(2! ) = 2 cos !
4 sin ! cos ! = 2 cos !
cos ! = 0
! = ± "2
4 sin ! cos ! " 2 cos ! = 0
cos ! (2 sin ! " 1) = 0
( 0, ± !2 ) ,!(
2 sin ! " 1 = 0
sin ! =
r=0
1
2
! = ± #6
3, ± !6
)
r = 4 sin 2!
r= 3
1
r = 2 cos !
-1
11-74.
a.
b.
c.
x "2
x" 4
# x +2 = lim
= lim 1
x"
4
x
+2
(x"
4)(
x
+2)
x!4
x!4
x!4 ( x +2)
lim x+2x " 2 = 6+26 " 2 = 2 26" 2 = 62
x!6
lim
x 3 "8
x!2 x"2
lim
(x"2)(x 2 +2 x+ 4)
(x"2)
x!2
= lim
=
1
4 +2
=
1
4
= lim (x 2 + 2x + 4) = 12
x!2
11-75.
a.
cos a cos b ! sin a sin b = cos(a + b)
b.
20 sin(3x) cos(3x) = 10 ( 2 sin(3x) cos(3x) ) = 10 sin(2 ! 3x) = 10 sin(6x)
c.
sin(3x) cos(x) ! cos(3x) sin(x) = sin(3x ! x) = sin(2x)
d.
sin 2 (4y) ! cos2 (4y) = !(cos2 (4y) ! sin 2 (4y)) = ! cos(2 " 4y) = ! cos(8y)
11-76.
( cos ( ) + i sin ( )) " (cos ( ) # i sin ( ) ) =
!
5
!
5
!
5
!
5
( !5 ) # i cos ( !5 ) sin ( !5 ) + i cos ( !5 ) sin ( !5 ) # i 2 sin2 ( !5 ) =
cos2 ( !5 ) # i 2 sin 2 ( !5 ) = cos2 ( !5 ) + sin 2 ( !5 ) = 1
cos2
11-77.
(x + y)3 = x 3 + 3x 2 y + 3xy 2 + y 3
(2 + (!2i))3 = 2 3 + 3(2)2 (!2i) + 3(2)(!2i)2 + (!2i)3 = 8 ! 24i ! 24 + 8i = !16 ! 16i
CPM Educational Program © 2012
Chapter 11: Page 17
Pre-Calculus with Trigonometry
Lesson 11.2.2
11-78.
z1z2 = r1 (cos a + i sin a) ! r2 (cos b + i sin b)
a.
b.
c.
= r1r2 ( cos a cos b " sin a sin b + i(sin a cos b + cos a sin b) )
= r1r2 ( cos a cos b ! sin a sin b + i(sin a cos b + cos a sin b) ) = r1r2 ( cos(a + b) + i sin(a + b) )
( 6 ( cos ( ) + i sin ( ) ) )
!
6
!
6
!
3
!
3
2
( ( ) + i sin ( ) ) = 36 (
36 cos
( (
= 6 " 6 cos
1
2
+i
3
2
!
6
) = 18 + (18
11-79.
a.
z1 = 2 cos 116! + i sin 116! , z2 = 3 cos 23! + i sin 23!
(
(
)
( !6 + !6 ) ) =
)
+ !6 + i sin
(
3)i
)
) (
)
z1z2 = 6 ( cos ( 116! + 46! ) + i sin ( 116! + 46! ) ) = 6 ( cos ( !2 ) + i sin ( !2 ) ) = 6(1i) = 6i
2 ( cos11! 6+i sin11! 6 )
z1
= 3 cos 2! 3+i sin 2! 3 = 23 ( cos ( 116! " 46! ) + i sin ( 116! " 46! ) ) =
z2
(
)
2 cos 7! + i sin 7!
(6)
( 6 ) ) = 23 ( " 23 " 12 i ) = " 33 " 13 i
3(
z1z2 = 2 cos 116! + i sin 116! " 3 cos 23! + i sin 23!
b.
11-80.
a.
z1z2 = r1r2 ( cos(a + b) + i sin(a + b) )
z 2 = r ! r(cos(" + " ) + i sin(" + " ))
b.
z 2 = r 2 (cos(2" ) + i sin(2" ))
z 3 = r ! r ! r(cos(" + " + " ) + i sin(" + " + " ))
z 3 = r 3 (cos(3" ) + i sin(3" ))
z 4 = r ! r ! r ! r(cos(" + " + " + " ) + i sin(" + " + " + " ))
z 4 = r 4 (cos(4" ) + i sin(4" ))
c.
z n = r n (cos(n! ) + i sin(n! ))
11-81.
( ( )
(
z = 2 cos 6 ! "6 + i sin 6 ! "6
))
z 6 = 2 6 (cos(" ) + i sin(" ))
z 6 = 64(#1 + i ! 0) = #64
CPM Educational Program © 2012
Chapter 11: Page 18
Pre-Calculus with Trigonometry
11-82.
( ( ) + i sin ( ) )
z = 2 ! 2i
a.
7!
4
Polar form: 2 2 cos
7!
4
r = 2 2 + (!2)2 = 8 = 2 2
tan " = !1!!(Quadrant 4)
"=
b.
7#
4
( (
)
(
(2 ! 2i)8 = (2 2 )8 cos 8 " 34# + i sin 8 " 34#
) ) = 4096 ( cos ( 6# ) + i sin ( 6# ) )
= 4096(1 + i " 0) = 4096
11-83.
a.
z = r ( cos ! + i sin ! )
b.
They have the same distance.
z = r 2 cos2 ! + r 2 sin 2 !
z = r 2 (cos2 ! + sin 2 ! ) = r 2 = r
11-84.
z = 16(cos ! + i sin ! )
4
4
4
4
( (
16 ( cos (
16 ( cos (
16 ( cos (
16 cos
) + i sin ( !4 ) ) = 2 ( 12 + 12 i ) = 2 + i 2
3!
+ i sin ( 34! ) ) = 2 ( " 1 + 1 i ) = " 2 + i
4 )
2
2
5!
+ i sin ( 54! ) ) = 2 ( " 1 " 1 i ) = " 2 " i
4 )
2
2
7!
+ i sin ( 74! ) ) = 2 ( 1 " 1 i ) = 2 " i 2
4 )
2
2
!
4
2
2
Review and Preview 11.2.2
11-85.
z = cos !2 + i sin !2
(
) 22 + 22 i
2
1 ( cos 54! + i sin 54! ) = " 22 " 22 i
2
1 cos !4 + i sin !4 =
11-86.
( cos ( ) + i sin ( ) )
!
12
!
12
6
=1
6
( cos ( 6 " ) + i sin ( 6 " ) ) == cos ( ) + i sin ( ) = i
!
12
!
12
!
2
!
2
11-87.
CPM Educational Program © 2012
Chapter 11: Page 19
Pre-Calculus with Trigonometry
( ( ) + i sin ( ) )!!"!!i
!
2
i = 1 cos
!
2
n
( ( ) + i sin ( ) ) = cos ( ) + i sin ( )
= 1n cos
n!
2
n!
2
n!
2
n!
2
11-88.
( cos ( ) + i sin ( ) )
"
%
$# ( cos ( ) + i sin ( ) ) '&
( cos ( ) + i sin ( ) )
!
12
!
12
!
12
6 1/3
!
12
!
12
cos
6
!
12
2
=i
= (i)1/3
= 3i
( !6 ) + i sin ( !6 ) = 3 i
3
2
+ 12 i = 3 i
11-89.
r1 (cos a+i sin a) (cos a"i sin a)
!
r2 (cos b+i sin b) (cos b"i sin b)
=
=
r1 (cos a+i sin a) (cos b"i sin b)
!
r2 (cos b+i sin b) (cos b"i sin b)
r1 (cos a cos b+i cos a sin b+i sin a cos b+sin a sin b)
r2
1
=
r1
r2
=
r1 (cos a+i sin a)(cos b"sin b)
r2 (cos2 b+sin 2 b)
( cos(a " b) + i sin(a " b) )
11-90.
a.
Amplitude = 7 = 7
Period =
b.
2!
2
=!
Vertical shift of 11 units, y = 11 " 7 cos(! x)
10 = 11 ! 7 cos(" x)
cos(" x) =
1
7
cos!1 cos(" x) = 1.4274
" x = 1.4274
x = 0.4543 + 1.0917 = 1.546 seconds
11-91.
7 2 = 32 + b 2
cos x = !
49 = 9 + b 2
tan x = !
40 =
b2
b = 2 10
cot x = !
2 10
7
3
2 10
2 10
3
sin 2x = 2 sin x cos x = 2 " 73 " !
2 10
7
= ! 124910
11-92.
CPM Educational Program © 2012
Chapter 11: Page 20
Pre-Calculus with Trigonometry
a.
y ! 2 = !4(x ! 3)
y = !4(x ! 3) + 2
CPM Educational Program © 2012
b.
y ! 2 = !4(3.2 ! 3)
y = !4(0.2) + 2 = !0.8 + 2 = 1.2
Chapter 11: Page 21
Pre-Calculus with Trigonometry
Chapter 11 Closure
CL 11-93.
a.
r = 10 cos !
b.
r 2 = 10r cos !
3r cos ! = 2(r sin ! + 1)
3x = 2(y + 1)
3x = 2y + 2
3x " 2y = 2
x 2 + y 2 = 10x
x 2 + y 2 " 10x = 0
x 2 " 10x + 25 + y 2 = 25
(x " 5)2 + y 2 = 25
c.
r = 6 cos ! " 8 sin !
r 2 = 6r cos ! " 8r sin !
x 2 + y 2 = 6x " 8y
x 2 " 6x + y 2 + 8y = 0
x 2 " 6x + 9 + y 2 + 8y + 16 = 9 + 16
(x " 3)2 + (y + 4)2 = 25
CL 11-94.
x 2 + y 2 = 20
a.
b.
r 2 = 20
r = 20
4x + 3y = 2
4r cos ! + 3r sin ! = 2
r(4 cos ! + 3 sin ! ) = 2
r=
y= x
c.
d.
y2 = x
2
4 cos ! + 3 sin !
6y + y 2 = x 2
6y = x 2 ! y 2
(r sin ! )2 = r cos !
6r sin " = r 2 cos2 " ! r 2 sin 2 "
r 2 sin 2 ! = r cos !
6r sin " = r 2 (cos2 " ! sin 2 " )
r=
cos !
sin 2 !
6r sin " = r 2 (cos 2" )
r=
6 sin "
cos 2"
CL 11-95.
CPM Educational Program © 2012
Chapter 11: Page 22
Pre-Calculus with Trigonometry
y = x3
r sin ! = r 3 cos 3 !
r2 =
r=
sin !
cos 3 !
sin !
cos 3 !
CPM Educational Program © 2012
Chapter 11: Page 23
Pre-Calculus with Trigonometry
CL 11-96.
They can make a table and plot points. It is a 4-petal rose with each pedal having a
diameter of 3 units with each pedal drawn along each of the axes.
CL 11-97.
From problem CL 11-95, r =
sin !
cos 3 !
. Rotated 45º counter-clockwise, r =
sin (! " # 4 )
cos 3 (! " # 4 )
.
CL 11-98.
The graph forms a figure 8 along the x-axis with each petal having a diameter of a.
CL 11-99.
a.
(2 + 3i)(2 ! 3i) = 4 ! 6i + 6i ! 9i 2 = 4 + 9 = 13
b.
(5 ! 4i) ! (6 ! 5i) = 5 ! 4i ! 6 + 5i = !1 + i
c.
(4 ! 2i)2 = 16 ! 8i ! 8i + 4i 2 = 12 ! 16i
d.
e.
f.
3i " 2+ 3i
2! 3i 2+ 3i
=
6i!9
4+9
9 + 6 i
= ! 13
13
3 ! 2i = ( 3)2 + (!2)2 = 3 + 4 = 7
1+ 4i " 1+ 4i
1! 4i 1+ 4i
= 1+8i!16
=
1!(!16)
!15+8i
17
= ! 15
+ 8 i
17 17
CL 11-100.
a.
tan " =
"=
b.
( ( ) + i sin ( ) )
r = 4 2 + (!4)2 = 32 = 4 2
4 !!(Quadrant
!4
7#
4
Polar form: 4 2 cos
"=
!2
2 3
11#
6
=!
7!
4
4)
( ( ) + i sin ( ) )
r = (2 3)2 + (!2)2 = 12 + 4 = 4
tan " =
7!
4
1 !!(Quadrant
3
Polar form: 4 cos
11!
6
11!
6
4)
CL 11-101.
a.
b.
( ( ) + i sin ( ) ) = 5 ( + i ) =
2 2 ( cos ( ) + i sin ( ) ) = 2 2 ( "
5 cos
!
3
!
3
5!
4
1
2
5!
4
CPM Educational Program © 2012
3
2
5
2
+
5 3
2
1
2
"
1
2
i
)
i = "2 " 2i
Chapter 11: Page 24
Pre-Calculus with Trigonometry
CL 11-102.
a.
( 712! ) + i sin ( 712! ) $% ,!z2 = 3 "# cos ( !4 ) + i sin ( !4 ) $%
z1z2 = 18 ( cos ( 712! + 312! ) + i sin ( 712! + 312! ) )
z1z2 = 18 ( cos ( 56! ) + i sin ( 56! ) )
z1 = 6 "# cos
(
b.
z1z2 = 18 &
3
2
z1
z2
z1
z2
= 2 cos
7!
12
z1
z2
=2
+i
1
2
) = &9
3 + 9i
( ( " ) + i sin (
= 2 ( cos ( ) + i sin ( ) )
(
!
3
1
2
+i
3
2
3!
12
7!
12
"
3!
12
))
!
3
) = 1+
3i
CL 11-103.
r = 2 2 + (!2)2 = 2 2
tan " =
"=
!2 !!(Quadrant
2
4)
7#
4
z6
z6
z6
CL 11-104.
z = i = cos
Roots:
3
(
( ( ) + i sin ( ) ) w
= (2 2 ) ( cos ( 6 " ) + i sin ( 6 " ) )
= 512 ( cos (
) + i sin ( ) )
= 512 ( cos ( ) + i sin ( ) ) = 512i
7!
4
Polar form: z = 2 2 cos
7!
4
7!
4
6
7!
4
21!
2
!
2
21!
2
!
2
( !2 ) + i sin ( !2 )
)
1 cos !6 + i sin !6 =
3
2
+ 12 i, 3 1 cos 56! + i sin 56! = "
3
2
+ 12 i, 3 1 cos 32! + i sin
3!
2
= "i
CL 11-105.
Horizontal velocity: 100 cos 30º = 86.603
Vertical velocity: 100 sin 30º = 50, t = 0!50
= 1.5625
!32
Ball was in the air: 1.5625 ! 2 = 3.125 seconds
The ball travels: 86.603 ! 3.125 = 270.6 feet
CL 11-106.
w = 2u ! 3v = 2 !3, 2 ! 3 4, !1 = !6, 4 ! 12, !3 = !6 ! 12, 4 ! (!3) = !18, 7
w = (!18)2 + 7 2 = 373 " 19.313
CPM Educational Program © 2012
Chapter 11: Page 25
Pre-Calculus with Trigonometry