Algebra & Trig, Sullivan & Sullivan
Fourth Edition
Notes:§9.3
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§9.3 – Complex Plane; DeMoivre's Theorem
Representing complex numbers geometrically, & in polar format
z = x + yi
X axis becomes the 'real axis', Y axis becomes the 'imaginary axis'
Plotting these is therefore pretty straight forward.
If we want to convert to polar form, we have
z  r  cos   i  r  sin 

z  x  yi

z  r cos  i  sin   This is the polar form of a complex number
Given a complex number in polar format, if we want to get back to the normal format, we just distribute out r, then
calculate the cos/sin, and remember to keep the term with i in it separate from the other one.
Multiplying & dividing complex numbers in polar format
If
z1  r1 cos1  i sin 1  and z 2  r2 cos 2  i sin  2  , then
z1  z 2  r1  r2 cos1   2   i sin 1   2 
z1 r1
 cos1   2   i sin 1   2 
z2 r2
(assuming that z2 != 0)
Powers – De Moivre's Theorem
If
z1  r1 cos1  i sin 1  , then
z1  r1 cosn  1   i sin n  1 
n
n