Polar Coordinates And Complex Numbers Worksheets With Answers - Math 611b Page 26
ADVERTISEMENT
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50SCO: By the end of grade
Elaborations - Instructional Strategies/Suggestions
12, students will be
Powers and Roots of Complex Numbers (9.8)
expected to:
De Moivre’s Theorem
B44 derive and apply De
θ
θ
=
+
Moivre’s Theorem for
n
n
z
[ (cos
r
i
sin )]
powers and roots
θ
=
i
n
[
re
]
θ
=
n in
r e
θ
θ
=
+
n
r
[cos
n
i
sin
n
]
θ
=
n
r cis n
Looking at examples 1&2 on p.599 & 600 we see the answer to be
4096. It is important for students to see the process by which the answer
was arrived at but they should also be aware that the calculator can do
the work as well.
A useful application of De Moivre’s Theorem is in finding the roots of a
complex number. The theorem can be re-written as:
θ
π
1
1
1
1
⎛
+
⎞
2
k
θ
θ
π
=
=
+
=
p
p
p
p
⎜
⎟
z
[
rcis
]
[
rcis
(
2
k
)]
r cis
⎝
⎠
p
!
where k = 0,1,...,p
1
3
8
For instance we would normally think of the answer of
to be 2 but
in fact there are two other cube roots
− +
− −
(
1
3
i and
)
(
1
3
i
)
.
Refer to p.43 in the workbook for a detailed solution.
In the essay, “The Development of Number Systems” ,at the end of
the unit we will solve this example and others in detail.
26
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education