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Unit 6.6
- 2. 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 © 2011 Pearson, Inc. Slide 6.1 - 2
- 4. Absolute Value (Modulus) of a
Complex Number
The absolute value or modulus
of a complex number
z a bi z a bi a
b
is | | | | .
2 2
a bi a bi
In the complex plane, | | is the distance of
from the origin.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 4
- 5. Graph of z = a + bi
Copyright © 2011 Pearson, Inc. Slide 6.1 - 5
- 6. Trigonometric Form of a Complex
Number
The trigonometric form of the complex number
z a bi is
z rcos isin
where a r cos , b r sin , r a2 b2 ,
and tan b / a. The number r is the absolute
value or modulus of z, and is an argument of z.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 6
- 7. Example Finding Trigonometric
Form
Find the trigonometric form with 0 2 for the
complex number 1 3i.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 7
- 8. Example Finding Trigonometric
Form
Find the trigonometric form with 0 2 for the
complex number 1 3i.
Find r: r |1 3i | 12 32
2.
Find : tan
3
1
so
3
.
Therefore, 1 3i 2 cos
3
isin
3
.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 8
- 9. Product and Quotient of Complex
Numbers
Let z1 r1 cos1 isin1 and z2 r2 cos 2 isin 2 .
Then
1. z1 z2 r1r2 cos 1 2 isin 1 2
.
2.
z1
z2
r1
r2
cos 1 2 isin 1 2
, r2 0.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 9
- 10. Example Multiplying Complex
Numbers
Express the product of z1 and z2 in standard form.
z1 4 cos
4
isin
4
, z2 2 cos
6
isin
6
Copyright © 2011 Pearson, Inc. Slide 6.1 - 10
- 11. Example Multiplying Complex
Numbers
Express the product of z1 and z2 in standard form.
z1 4 cos
4
isin
4
, z2 2 cos
6
isin
z1 z2 r1r2 cos 1 2 isin 1 2
4 2 cos
4
6
isin
4
6
6
4 2 cos
5
12
isin
5
12
4 20.259 i0.966 1.464 5.464i
Copyright © 2011 Pearson, Inc. Slide 6.1 - 11
- 13. De Moivre’s Theorem
Let z rcos isin and let n be a positive integer.
Then
zn r cos isin
n
r n cosn isin n .
Copyright © 2011 Pearson, Inc. Slide 6.1 - 13
- 14. Example Using De Moivre’s Theorem
Find
3
2
i
1
2
4
using De Moivre's theorem.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 14
- 15. Example Using De Moivre’s Theorem
Find
3
2
i
1
2
4
using De Moivre's theorem.
The argument of z
3
2
i
1
2
is
7
6
,
and its modulus
3
2
i
1
2
3
4
1
4
1.
Hence,
z 2cos
7
6
isin
7
6
Copyright © 2011 Pearson, Inc. Slide 6.1 - 15
- 16. Example Using De Moivre’s Theorem
4
using De Moivre's theorem.
i
1
2
z4 cos 4
7
6
isin 4
7
6
cos
14
3
isin
14
3
cos
2
3
isin
2
3
1
2
i
3
2
Find
3
2
Copyright © 2011 Pearson, Inc. Slide 6.1 - 16
- 17. nth Root of a Complex Number
A complex number v a bi is an nth root of z if
vn z.
If z 1, the v is an nth root of unity.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 17
- 18. Finding nth Roots of a Complex
Number
If z rcos isin , then the n distinct
complex numbers
r n cos
2 k
n
isin
2 k
n
,
where k 0,1,2,..,n 1,
are the nth roots of the complex number z.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 18
- 19. Example Finding Cube Roots
Find the cube roots of 1.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 19
- 20. Example Finding Cube Roots
Find the cube roots of 1.
Write 1 in complex form: z 1 0i cos0 isin0
The third roots of 1 are the complex numbers
cos
0 2 k
3
isin
0 2 k
3
for k 0,1,2.
z1 cos0 isin0 1
z2 cos
2
3
isin
2
3
1
2
3
2
i
z3 cos
4
3
isin
4
3
1
2
3
2
i
Copyright © 2011 Pearson, Inc. Slide 6.1 - 20
- 21. Quick Review
1. Write the roots of the equation x2 12 6x in a bi form.
2. Write the complex number 1 i3
in standard form a bi.
3. Find all real solutions to x3 27 0.
Find an angle in 0 2 which satisfies both equations.
4. sin
1
2
and cos
3
2
5. sin
2
2
and cos
2
2
Copyright © 2011 Pearson, Inc. Slide 6.1 - 21
- 22. Quick Review Solutions
1. Write the roots of the equation x2 12 6x in a bi form.
3 3i, 3 3i
2. Write the complex number 1 i3
in standard form a bi.
2 2i
3. Find all real solutions to x3 27 0. x 3
Find an angle in 0 2 which satisfies both equations.
4. sin
1
2
and cos
3
2
5 / 6
5. sin
2
2
and cos
2
2
5 / 4
Copyright © 2011 Pearson, Inc. Slide 6.1 - 22
- 23. Chapter Test
1. Let u 2, 1 and v 4,2 . Find u v.
2. Let A (2, 1),B (3,1),C (4,2), and D (1, 5).
Find the component form and magnitude of the vector
uuur
uuur
AC
+BD
3. Given A (4,0) and B (2,1), find (a) a unit vector in
uuur
the direction of AB
and (b) a vector of magnitude 3 in
the opposite direction.
4. Given u 4,3 and v 2,5 , find (a) the direction
angles of u and v and (b) the angle between u and v.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 23
- 24. Chapter Test
5. Convert the polar coordinate ( 2.5,25o) to a rectangular
coordinate.
6. Eliminate the parameter t. x 4 t, y 8 5t, 3 t 5.
7. Find a parameterization for the line through the points
( 1, 2) and (3,4).
8. Use De Moivre's theorem to evaluate 3 cos
4
isin
4
5
.
Write your answer in (a) trigonometric form and (b) standard
form.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 24
- 25. Chapter Test
9. Convert the polar equation r 3cos 2sin to
rectangular form.
10. A 3000 pound 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.
(b) Find the component of the force perpendicular to
the street.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 25
- 26. Chapter Test Solutions
1. Let u 2, 1 and v 4,2 . Find u v. 6
2. Let A (2, 1),B (3,1),C (4,2), and D (1, 5).
Find the component form and magnitude of the vector
uuur
uuur
AC
+BD
8, 3 ; 73
3. Given A (4,0) and B (2,1), find (a) a unit vector in
uuur
the direction of AB
and (b) a vector of magnitude 3 in
the opposite direction. (a)
2
5
,
1
5
(b)
6
5
,
3
5
Copyright © 2011 Pearson, Inc. Slide 6.1 - 26
- 27. Chapter Test Solutions
4. Given u 4,3 and v 2,5 , find (a) the direction
angles of u and v and (b) the angle between u and v.
(a) tan1 3
4
0.64 tan1 5
2
1.19 (b) 0.55
5. Convert the polar coordinate ( 2.5,25o) to a
rectangular coordinate. ( 2.27, 1.06)
6. Eliminate the parameter t. x 4 t, y 8 5t,
3 t 5. y 5x 12
Copyright © 2011 Pearson, Inc. Slide 6.1 - 27
- 28. Chapter Test
7. Find a parameterization for the line through the points
( 1, 2) and (3,4). x 2t 3, y 3t 4
8. Use De Moivre's theorem to evaluate 3 cos
4
isin
4
5
.
Write your answer in (a) trigonometric form and (b) standard
form.
(a) 243 cos
5
4
isin
5
4
(b)
243 2
2
243 2
2
i
Copyright © 2011 Pearson, Inc. Slide 6.1 - 28
- 29. Chapter Test
9. Convert the polar equation r 3cos 2sin to
rectangular form. x
3
2
2
y 12
13
4
10. A 3000 pound 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. 826.91 pounds
(b) Find the component of the force perpendicular to
the street. 2883.79 pounds
Copyright © 2011 Pearson, Inc. Slide 6.1 - 29