Download Slide 6- 1 Homework, Page 548

Homework, Page 548
(a) Complete the table for the equation and (b) plot the points.
1. r  3cos 2
θ
r
0 π/4 π/2
3 0 –3
π
3
3π/4
0
5π/4
0
3π/2
–3
7π/4
0
y




x













Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 1
Homework, Page 548
Draw the graph of the rose curve. State the smallest θ-interval
(0 ≤ θ ≤ k) that will produce a complete graph.
5. r  3cos 2
y




x













The smallest  -interval that will produce the entire
graph is 0    2 .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 2
Homework, Page 548
Match the equation with its graph without using your calculator.
9. Does the graph of r  2  2sin  or r  2  2cos  appear
in the figure? Explain.
r  0   2  2sin 0  2
r  0   2  2cos 0  0
 2   2  2sin  2  4
r    2  2cos   2
2
2
r 

Points  0,0  and 2, 
2
are on the graph, so the
graph is of r  2  2cos 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slide 6- 3
Homework, Page 548
Use the polar symmetry tests to determine if the graph is
symmetric about the x-axis, the y-axis, or the origin..
13.
r  3  3sin 
r  3  3sin 
r  3  3sin     3  3sin   r
r  3  3sin      3  3sin    r
r  3  3sin     3  3sin     r
r  3  3sin      3  3sin    r
Symmetric about the y -axis
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 4
Homework, Page 548
Use the polar symmetry tests to determine if the graph is
symmetric about the x-axis, the y-axis, or the origin..
17.
r  5cos 2

r  5cos 2  5 cos 2   sin 2 


 
r  5  cos      sin       5  cos
r  5  cos      sin       5  cos

r  5 cos 2     sin 2     5 cos 2   sin 2   r
2
2
2
2
   sin 2     r
2
   sin 2     r
2
Symmetric about the y-axis, the x-axis, and the origin.
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Slide 6- 5
Homework, Page 548
Identify the points on 0 ≤ θ ≤ 2π where maximum r-values occur.
21. r  2  3cos
The maximum r -values occur for r  2  3cos at   0, 2 .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 6
Homework, Page 548
Analyze the graph of the polar curve.
r 3
25.
Domain:  
r 3
Range:
Continuity: Continuous
Symmetry: Symmetric about the y-axis,
the x-axis, and the origin.
Boundedness: Bounded
Maximum r-value: r  3
None
Asymptotes:
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 7
Homework, Page 548
Analyze the graph of the polar curve.
29. r  2sin 3
Domain:  
Range: 2  r  2
Continuity: Continuous
Symmetry: Symmetric about the y-axis.
Boundedness: Bounded
Maximum r-value: r  2
Asymptotes: None
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 8
Homework, Page 548
Analyze the graph of the polar curve.
33. r  4  4cos
Domain:  
Range: 0  r  8
Continuity: Continuous
Symmetry: Symmetric about the x-axis
Boundedness: Bounded
Maximum r-value: r  8
Asymptotes: None
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 9
Homework, Page 548
Analyze the graph of the polar curve.
37. r  2  5cos
Domain:  
Range: 3  r  7
Continuity: Continuous
Symmetry: Symmetric about the x-axis
Boundedness: Bounded
Maximum r-value: r  7
Asymptotes: None
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 10
Homework, Page 548
Analyze the graph of the polar curve.
r  2
41.
Domain:  
Range: 0  r  
Continuity: Continuous
Symmetry: Not symmetrical
Boundedness: Unbounded
Maximum r-value: r  
Asymptotes: None
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 11
Homework, Page 548
Find the length of each petal of the polar curve.
45. r  2  4sin 2
The large petals are 6
units long and the short
petals are  2 units long.
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Slide 6- 12
Homework, Page 548
Select the two equations whose graphs are the same curve. Then,
describe how the paths are different as θ increases from 0 to 2π.
49. r1  1  3sin  , r2  1  3sin  , r3  1  3sin 
The graphs of r1 and r2 are the same curve.
The graph of r1 starts with the outside portion
and ends with the inside portion. The graph
of r2 starts with the inside portion and ends
with the outside portion.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 13
Homework, Page 548
(a) Describe the graph of the polar equation, (b) state any
symmetry the graph possesses, and (c) state the maximum rvalue, if it exists.
2
r

2sin
2  sin 2
53.
(a) The graph of the polar equation has two large petals
and two small petals.
(b) The graph is symmetric about the origin.
(c) The maximum r-value is 3.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 14
Homework, Page 548
57. Analyze the graphs of the polar equations r  a cos n
and r  a sin n when n is an even integer.
Domain: All real numbers.
Range: a  r  a
Continuity: Continuous
Symmetry:Symmetric about the x-axis, y-axis, and origin.
Boundedness: Bounded
Maximum r-value: a
Asymptotes: None
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 15
Homework, Page 548
61. A polar curve is always bounded. Justify your
answer.
False
The spiral curve, the graph of the polar equation r = θ
is unbounded.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 16
Homework, Page 548
65. Which of the following is the maximum r-value for
r = 2 – 3 cos θ?
a.
6
r  2  3cos  1  cos  1
b.
5
3  1  3  2  3  5
3 1  3  2  3  1  5  1  5
c.
3
d.
2
e.
1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 17
Homework, Page 548



69. The graphs of r1  3sin 5 2  and r2  3sin 7 2 

may be called rose curves.
 a  Determine the smallest  -interval that will produce
a complete graph of r1; of r2 .
It took an inteval of 0    4 to complete each graph.
 b  How many petals does each graph have?
Graph r1 has 10 petals and graph r2 has 14 petals.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 18
6.6
De Moivre’s Theorem and nth Roots
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
1. Write the roots of the equation x  12  6 x in a  bi form.
2
2. Write the complex number 1  i  in standard form a  bi.
3
3. Find all real solutions to x  27  0.
Find an angle  in 0    2 which satisfies both equations.
3
1
3
and cos   
2
2
2
2
5. sin   
and cos   
2
2
4. sin  
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 20
Quick Review Solutions
1. Write the roots of the equation x  12  6 x in a  bi form.
2
3  3i, 3  3i
2. Write the complex number 1  i  in standard form a  bi.
3
2  2i
3. Find all real solutions to x  27  0. x  3
Find an angle  in 0    2 which satisfies both equations.
3
1
3
and cos   
  5 / 6
2
2
2
2
5. sin   
and cos   
  5 / 4
2
2
4. sin  
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 21
What you’ll learn about





The Complex Plane
Trigonometric Form of Complex Numbers
Multiplication and Division of Complex Numbers
Powers of Complex Numbers
Roots of Complex Numbers
… and why
The material extends your equation-solving technique to include
equations of the form zn = c, n is an integer and c is a complex
number.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 22
Complex Plane
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 23
Addition of Complex Numbers
Two complex numbers, z1  a1  b1i and z2  a2  b2i may
be added by separately adding their real and imaginary
parts z1  z2   a1  a2    b1  b2  i. This addition in the
complex plane is the same operation as vector addition
in the real Cartesian plane.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 24
Absolute Value (Modulus) of a Complex
Number
The absolute value or modulus of a complex number
z  a  bi is | z || a  bi | a  b .
In the complex plane, | a  bi | is the distance of a  bi
from the origin.
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
2
Slide 6- 25
Graph of z = a + bi
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Slide 6- 26
Trigonometric Form of a Complex Number
The trigonometric form of the complex number z  a  bi is
z  r  cos   i sin  
where a  r cos  , b  r sin  , r  a  b ,
2
2
and tan   b / a. The number r is the absolute value or
modulus of z, and  is an argument of z.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 27
Example Finding Trigonometric Form
Find the trigonometric form with 0    2 for the complex
number 1  3i.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 28
Product and Quotient of Complex Numbers
Let z1  r1  cos1  i sin 1  and z2  r2  cos 2  i sin  2  . Then
1. z1  z2  r1r2 cos 1   2   i sin 1   2   .
2.
z1 r1
 cos 1   2   i sin 1   2   , r2  0.
z2 r2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 29
Example Multiplying Complex Numbers
Express the product of z1 and z2 in standard form.






z1  4  cos  i sin  , z2  2  cos  i sin 
4
4
6
6


Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 30
Example Dividing Complex Numbers
Express the quotient of z1  z2 in standard form.






z1  6  cos  i sin  , z2  3  cos  i sin 
3
3
5
5


Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 31
A Geometric Interpretation of z2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 32
De Moivre’s Theorem
Let z  r  cos  i sin   and let n be a positive integer. Then
z   r  cos  i sin     r n  cos n  i sin n  .
n
n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 33

Example Using De Moivre’s Theorem
Find 1+ 3i

4
using De Moivre's theorem.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 34
nth Root of a Complex Number
A complex number v  a  bi is an nth root of z if
v n  z.
If z  1, then v is called an nth root of unity.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 35
Finding nth Roots of a Complex Number
If z  r  cos   i sin   , then the n distinct complex numbers
  2 k
  2 k 

r  cos
 i sin
 , where k  0,1, 2,.., n  1, are
n
n


the nth roots of the complex number z.
n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 36
Example Finding Cube Roots
Find the cube roots of 1.
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Slide 6- 37
Homework




Homework Assignment #8
Review Section: 6.6
Page 559, Exercises: 1 – 69 (EOO), 81, 83
Quiz next time
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Slide 6- 38