Complex Numbers Meet Trigonometry
Complex numbers can be graphed as two‐dimensional points in the plane, very
similarly to vectors.
The x‐axis is referred to as the “Real Axis”
The y‐axis is the “Imaginary Axis”
When you plot a complex number in the plane, the real part and the imaginary
part both tell you where to plot the “head” of the complex vector.
Real part = horizontal component
Im. Part = vertical component
For example, the complex number
represented by the vector Z1 is
Z1 = 3 + 2i
ex) What are the complex numbers represented by the other vectors?
Z2 =
Z3 =
Z4 =
Z5 =
Polar Form of a Complex Number
Since we’re likening complex numbers to vectors, we can represent complex
numbers in a form which uses the magnitude of the complex vector and the angle
it makes with the positive real axis (remember, this is the positive x‐axis).
The form z = a + bi is referred to as the rectangular form of a complex number.
Rewriting the number using its magnitude and direction is called its polar form.
For the rectangular complex number z = a + bi , to convert it to polar form:
1st Get the magnitude of the number Æ r = a2 + b2
2nd Get the direction angle for the complex vector
imaginary part
(will usually require TAN−1 )
tanθ =
real part
Once you have your r and θ values you can
rewrite the complex number as
z = r ( cosθ + i sinθ )
ex) Convert the complex number 1 − 3 i into its polar form.
(IT ALWAYS HELPS TO GRAPH THE COMPLEX NUMBER FIRST)
ex) Convert the complex number −6 + 6 i into its polar form.
Products and Quotients of Complex Numbers in Polar Form
For a pair of complex numbers IN POLAR FORM:
z1 = r1 ( cosθ1 + i sinθ1 ) and z2 = r2 ( cosθ2 + i sinθ2 )
Their Product Æ
Their Quotient Æ
z1 z2 = r1r2 ( cos(θ1 + θ2 ) + i sin(θ1 + θ2 ) )
z1 r1
= ( cos(θ1 − θ2 ) + i sin(θ1 − θ2 ) ) (as long as z2 ≠ 0 )
z2 r2
ex) Find the product and the quotient (leave your answer in polar form).
2π
2π
π
π

z1 = cos
a)
+ i sin
and z2 = 2  cos + i sin 
3
3
6
6

b)
z1 = 4 ( cos80D + i sin80D ) and z2 = 3( cos200D + i sin200D )
DeMoivre’s Theorem for Powers of Complex Numbers (requires polar form)
For any complex number in polar form, z = r (cosθ + i sinθ ) , and for any positive
integer n Æ z n = r n ( cos(nθ ) + i sin(nθ ) )
(
ex) Find 1 − 3 i
)
5
using DeMoivre’s Theorem.
Write the answer in rectangular form. (HINT: we converted this one earlier)
ex) Find (1 + i)6 using DeMoivre’s Theorem.
Write the answer in rectangular form.