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Mathematical Investigations IV
Name:
Complex Concepts-Making “i” contact
Products, Quotients, DeMoivre’s Theorem
What happens when we multiply two complex numbers? We have observed a relationship between the radii and angles of the two complex numbers and the resulting product. Let’s prove this relationship in the general case:
( r
1 cis

)

( r
2 cis

)
 r
1 r
2 cis(
  
)
Write ( r
1
cis
 )·( r
2
cis

) in polar form. ( r
1 cos

+ i
 r
1 sin

)

( r
2 cos

+ i
 r
2 sin

)
Multiply out the terms in the parentheses.
Group the real terms and the imaginary terms together.
_____________________________________
_____________________________________
Simplify, using trig identities. _____________________________________
Rewrite in cis form _____________________________________
Next, let’s divide one complex number by another and prove the analogous relationship between the radii and angles of the complex numbers and their quotient: r
Write r
2
1
cis
cis


in polar form
Rationalize the denominator
Multiply everything out
Group the real and imaginary terms
Use trig identities to simplify
Rewrite in cis form r
1 cis
 r
2 cis

 r r
2
1 cis(
  
) r
1 r
2 cos

+ i
 r cos

+ i
 r
1
2 sin sin


________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
Complex 2.1 Rev F06
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Mathematical Investigations IV
Name:
1.
2.
Fill in the blank. cis 40

cis 30

= cis _____

3cis
4

6 cis
5
= cis

2
 cis
3

= cis_____ cis



12


(5 cis 20

)( ___ cis ____) = 10 cis 70

(4 cis 50

)(___ cis ____) = -4 cis 50
 cis(70

) cis (120

) cis235

= cis ______
For z = r cis

, find the following in terms of r and
 z
2
= _______________ z
3
= _______________ z
15
= _______________ z n
= _______________
3.
Generalizing from our results from z n
, we state
DeMoivre’s Theorem:
If z = r cis

, then z n
= r n
cis (n

)
Use DeMoivre’s Theorem to simplify the following:
(2 cis 12°) 6
=________________
(4 cis 23°) 3
= _______________
(cis 35 )
6
(cis 20 )
4
= __________ cis 25
(4 cis 16°) 3
(2 cis (–10°))
3
= ____________
Complex 2.2 Rev F06
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5.
Mathematical Investigations IV
Name:
(((cis 5°) 2
cis 10°)
2
cis 20°)
2
= ___________
4. Let z = 2 cis 45°. Find an Argand diagram. z
2
, z
3
, z
4
, and z
5 in polar form. Plot z and each of these powers on
Let z = cis15°. Plot z , z
2
, z
3
, ... , z
7
on an Argand diagram. Also locate z
10
, z
30
, and z
100
. z and all of the powers listed above are all solutions to the equation z n
= 1 for some values of n . Find this value of n .
Complex 2.3 Rev F06