Download Section 8-2 Review

PreCalculus 2 Polar Graphs and the Complex Plane 9-3
I. Graphing Points: Complex Numbers
Complex Plane - Real axis(x) and Imaginary axis(y)
A. 2 + 3i
B. 4 - 5i
C. -3 + 5i
z  x  yi is a complex number. The magnitude or modulus of z . Denoted by
z , is defined as the distance from the origin to the point (x,y). That is z  x 2  y 2
Magnitude of z is called the absolute value of z . The conjugate z  x  yi
Therefore: z  zz
II. Complex Numbers: Rectangular Form vs. Polar Form
Draw the triangle including x, y, r , and !
If r  0 and 0    2 , the complex number z  x  yi may be written in polar form
as z  x  yi  (r cos  )  (r sin  )i  rcis  is called the argument of z .
Remember: r 
x2  y2
tan  
y
x
Plot the complex number in the complex plane and write it in polar form.
Express the argument in degrees:
1  i
 3i
5  7i
Write each complex number in Rectangular Form
6(cos 30° + i sin 30°)
5
5 

2  cos
 i sin

6
6 

III. Product-Quotient Rule of Complex Numbers:
Product Theorem
Quotient Theorem
(r1cis1 )(r2 cis 2 )  r1r2 cis 1   2 
r1cis1 r1
 cis 1   2 
r2 cis 2 r2
Example: Find the product and quotient for each and write it in polar form.
z  8cis20 
w  4cis10 
find z  w
find
z
w
z  1 i
w  1  3i
find z  w
z
w
IV. Power Rule: De Moivre’s Theorem
A. Theorem
[ r( cos θ + i sin θ)] n = rn (cos nθ + i sin nθ)
B. Examples: Write each expression in the standard form a+bi
1. [ 2 (cos 135° + i sin 135°)]4
2. (1 + i 3 )3
V. The “Nth” Root Theorem
nth Root Theorem
Put in polar form :
n
r
k 

Z k  n r (c is
  2 k
n
)
and k  0, 1, 2, ..., n  1
and n  the number of distinct roots
1. The complex cube roots of 8  8i
2. Find the complex 4th roots of
3 i