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```Homework, Page 562
Let u  2, 1 , v  4, 2 , and w  1, 3 . Find the expression.
1. u  v
u  v   2  4  ,  1  2   2, 3
Slide 6- 1
Homework, Page 562
Let u  2, 1 , v  4, 2 , and w  1, 3 . Find the expression.
5.
uv
u v  2  4    1 2   8  2  6
Slide 6- 2
Homework, Page 562
Let A = (2, –1), B = (3, 1), C = (–4, 2), and D = (1, –5). Find the
component form and magnitude of the vector.
9.
AC  BD
AC 
 4  2  ,  2   1 
 6,3
BD  1  3 ,  5  1  2, 6
AC  BD  6,3  2, 6  8, 3
AC  BD 
 8   3
2
2
 64  9 
73
Slide 6- 3
Homework, Page 562
Find  a  the direction angles of u and v and
 b  the angle between u and v.
13. u  4,3 , v  2,5
3
1 5
u  tan
 0.644  v  tan
 1.190
4
2
 v  u  1.190  0.644  0.546 OR
1
2
2
 29

v

2

5

5
u v  4  2  3  5  23  u  4  3
2
  cos
1
2
23
 cos
 0.5467
5  29
uv
uv
1
Slide 6- 4
Homework, Page 562
Convert the polar coordinates to rectangular coordinates.
 
2,


17.
4

2,

 4   2  cos   4  ,sin   4 
  2 
2 
  2
, 2 

 

  2   2 


2,  2

Slide 6- 5
Homework, Page 562
Rectangular coordinates of point P are given. Find all polar coordinates of P
that satisfy: (a) 0 ≤θ ≤2π (b) –π ≤ θ ≤ π (c) 0 ≤ θ ≤ 4π
21. P   2, 3
3
r  2   3  13    tan 2  0.983
    2.159 , 2    5.300 ,3    8.442 , 4    11.584
 a  P   13  cos 2.159,sin 2.159  , P  13  cos5.300,sin 5.300
2
2
1
 b  P  13  cos  0.983 ,sin  0.983  , P   13  cos 2.159,sin 2.159 
 c  P   13  cos 2.159,sin 2.159 , P  13  cos5.300,sin 5.300
P   13  cos8.442,sin8.442  , P  13  cos11.584,sin11.584 
Slide 6- 6
Homework, Page 562
Eliminate the parameter t and identify the graph.
25. x  3  5t , y  4  3t
3 x
x  3  5t , y  4  3t  5t  3  x  t 
5
 3 x 
y  4  3
  5 y  20  3  3  x   5 y  20  9  3x
 5 
29 3
3
y
 x  The graph is a line with slope  and
5 5
5
y -intercept 5.8.
Slide 6- 7
Homework, Page 562
Eliminate the parameter t and identify the graph.
x  e 2t  1, y  et
29.
 
x  e  1, y  e  x  e
2t
t
t
2
2

x

y
1
1
The graph is a parabola opening to the right.
Slide 6- 8
Homework, Page 562
Imaginary Axis
Z1
4
Refer to the complex number shown in the figure.
33. If z1 = a + bi, find a, b, and |z1|.
2
2
 25  5


3

4
 
a  3 , b  4 , z1
Real Axis
-3
a  3, b  4, z1  5
Slide 6- 9
Homework, Page 562
Write the complex number in standard form.
37.
4
4 

2.5  cos
 i sin

3
3


4
4 
5  1 5

2.5  cos
 i sin
   
3
3 
2  2 2


3
i

 2 
5 5 3
  i
4
4
Slide 6- 10
Homework, Page 562
Write the complex number in trigonometric form where 0 ≤ θ ≤ 2π.
Then write three other possible trigonometric forms for the number.
41. 3  5i
r  3   5
2
2
5
 34    tan
 1.030
3
1
3  5i  34  cos  1.030   i sin  1.030  
  34  cos  2.111  i sin  2.111     
 34  cos  5.253  i sin  5.253   2  
  34  cos  4.172   i sin  4.172      
Slide 6- 11
Homework, Page 562
Use DeMoivre’s Theorem to find the indicated power of the complex
number. Write the answer in (a) trigonometric form and (b) standard form.
45. 3  cos   i sin   

 
4
4 
 
5

5
 

 
5
5 

 i sin
3  cos 4  i sin 4    243  cos

4
4






243 2 243 2
 

i
2
2
Slide 6- 12
Homework, Page 562
Find and graph the nth roots of the complex number for the specified value
of n.
49. 3  3i, n  4
3 

r  3  3  3 2    tan
3 4



z  3 2  cos  i sin 
4
4

2
2
1
Continued on next slide.
Slide 6- 13
Homework, Page 562
49. Cont’d
z1  4 3
z2  4 3
z3  4 3
z4  4 3

  4

 


 3 2  cos  i sin 
2  cos 4  i sin 4 
16
16 

4
4 



  2
  2  4
9
9 



3
2
cos

i
sin



2  cos 4
 i sin 4
16
16


4
4




  4
  4  4
17
17 


  3 2  cos 16  i sin 16 
2  cos 4
 i sin 4


4
4




  6
  6  4
25
25 


  3 2  cos 16  i sin 16 
2  cos 4
 i sin 4


4
4




Slide 6- 14
Homework, Page 562
49. Continued

 

z1  4 18  cos  i sin 
16
16 

9
9 

z2  4 18  cos
 i sin

16
16


17
17 

4
z3  18  cos
 i sin

16
16


25
25 

4
z4  18  cos
 i sin

16
16 

y
x
Slide 6- 15
Homework, Page 562
Decide whether the graph of the polar function appears among the
four.
53. r  3sin 4
b.
Slide 6- 16
Homework, Page 562
Decide whether the graph of the polar function appears among the
four.
57. r  2  2sin
Not shown.
Slide 6- 17
Homework, Page 562
Convert the polar equation to rectangular form and identify the
graph.
r  2
2
r  2  r 2   2   x 2  y 2  4
The graph is a circle of radius 2 centered at the origin.
61.
Slide 6- 18
Homework, Page 562
Convert the rectangular equation to polar form and graph the polar
equation.
y  4
65.
y  4  r sin  4
4
r
 r  4csc
sin 
Slide 6- 19
Homework, Page 562
Analyze the graph of the polar curve.
69. r  2  5sin
Domain: All real numbers
Range: –3 ≤ r ≤ 7
Continuity: Continuous
Boundedness: Bounded
Maximum r-value: 7
Asymptotes: None
Slide 6- 20
Homework, Page 562
73.  a  Explain why r  a sec is a polar form of x  a.
Multiply both sides of r  a sec  by cos  and the equation
becomes r cos   a and r cos   x.
 b  Explain why r  a csc is a polar form of y  a.
Multiply both sides of r  a csc  by sin  and the equation
becomes r sin   a and r sin   y.
b
is a polar form
 c  Let y  mx  b. Prove that r 
sin   m cos
for the line. What is the domain of r ?
Substituting y  mx  b becomes r sin   mr cos  b
r sin  mr cos  b  r  sin   m cos   b
b
r
sin   m cos  Domain :  :    m  1
Slide 6- 21
Homework, Page 562
73.  d  Illustrate the result of part  c  by graphing the line
y  2 x  3 using the polar form from part  c  .
3
y  2x  3  r 
sin   2cos
Slide 6- 22
Homework, Page 562
77. A 3,000-lb car is parked on a street that makes an angle
of 16º With the horizontal.
(a) Find the force required to keep the car from rolling
down the hill.
F  3000 sin16  lb
(b) Find the component of the force perpendicular to the
ground.
F  3000 cos16  lb
Slide 6- 23
Homework, Page 562
81. The lowest point on a Ferris wheel of radius 40-ft is
10-ft above the ground, and the center is on the y-axis.
Find the parametric equation for Henry’s position as a
function of time t in seconds, if his starting position (t = 0)
is the point (0, 10) and the wheel turns at a rate of one
revolution every 15 sec.
2
2
2 

 15  b 
x  40sin bt  p 
 x  40sin 
t
b
15
 15 
2



y  50  40cos bt  y  50  40cos 
t
 15 
Slide 6- 24
Homework, Page 562
85. Diego releases a baseball 3.5-ft above the ground with
an initial velocity of 66-fps at an angle of 12º with the
horizontal. How many seconds after the ball is thrown
will it hit the ground? How far from Diego will the ball be
when it hits the ground?
x   vo cos  t , y  3.5   vo sin   t  16t 2
x   66cos12  t , y  3.5   66sin12  t  16t 2
The ball hits the ground about
1.06 secs after it is thrown,
68.431 ft from Diego.
Slide 6- 25
Homework, Page 562
89. A 60-ft radius Ferris wheel turns counterclockwise
one revolution every 12 sec. Sam stands at a point 80 ft to
the left of the bottom (6 o’clock) of the wheel. At the
instant Kathy is at 3 o’clock, Sam throws a ball with an
initial velocity of 100 fps and an angle of 70º with the
horizontal. He releases the ball at the same height as the
bottom of the Ferris wheel. Find the minimum distance
between the ball and Kathy. xball  80  100cos 70  t
360
yball  100sin 70  t  16t 2
p  12  b 
 30
12
xKathy   60cos30t  yKathy  60   60sin 30t 
distance 
x
ball
 xKathy    yball  yKathy 
2