Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 9.3
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Complex Plane & DeMoivre's Theorem
Complex Numbers: Plot on the complex plane, and convert to & from polar format
1) Plot the each of the following complex numbers on the graph, clearly labeling which point
corresponds to which complex number. Then convert each complex number to the polar form for a
complex number.
1. 3 + 3i
2. -2 – i
3. -3
4. -2i
2) Plot each of the following polar coordinates, then convert the polar coordinates above into
rectangular form.

1. 2 cos 45  i sin 45


2. 4 cos120  i sin120
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
Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 9.3
Page 2 / 3
Complex Numbers: Addition
3) Add the following complex numbers, in the format that's given (i.e., don’t convert between formats,
but instead just do the addition)
2  3i    5  8i 
4) Add the following complex numbers, in the format that's given (i.e., don’t convert between formats,
but instead just do the addition)
7  6i    3  4i 
Complex Numbers: Multiplication
5) Evaluate the following expression:  3  3i  4  8i 
6) Evaluate the following expression:




2 cos 45  i sin 45  5 cos 30  i sin30 



7) Evaluate the following expression:




2 2 cos 35  i sin35   10 cos 70  i sin70 



8) Try multiplying  3  3i  4  8i  by converting to polar form first.
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Algebra &Trigonometry
Sullivan & Sullivan, Fourth Edition
§ 9.3
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Complex Numbers: Division
4  7i
 3  2i
9) Evaluate the following expression:
10) Evaluate the following expression:

8  cos 20

 i sin 20 
12 cos 75  i sin 75
Complex Numbers: Powers – De Moivre's Theorem
11) Using the multiplication rule, if z  r  cos   i sin  , what is
z2?
z3?
zn?
(this last one is DeMoivre's Theorem)
12) Use De Moivre’s Theorem to write the following in polar form, then convert it to standard (rectangular) form.


2 cos 20  i sin20 


3
13) Use De Moivre’s Theorem to write the following in polar form, then convert it to standard (rectangular) standard
form.
 

 
 i sin  
 4  cos
10
10  
 
5
RegularCoordinates
Rectangular
x , y 
Polar
r ,  
Complex #s
x yi
r  cos   i sin 
r 
Vectors
x, y  xiy j
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