Ch 8 Complex Numbers, Polar Equations,
and Parametric Equations
8.1 Complex Numbers
The Imaginary Unit i
i   1, and therefore i 2  1
Complex numbers
If a and b are real numbers, then any number
of the form a + bi is a complex number. In
the complex number a  bi , a is the real
part and b is the imaginary part.
Complex # s  Real# s a  bi, b  0  imaginary # s
If a  0 , then
 a  i a.
Write as the product of a real number and i,
using the definition of  a
 16
 70
 48
Solve x 2  9
Solve x 2  24  0
Solve 9 x 2  5  6 x
When working with negative radicands,
use the definition  a  i a before using
any of the other rules for radicals.
7 7 
 6   10 
 20

2
 48

24
 8   128

4
Addition and Subtraction of Complex Numbers
a  bi   c  di   a  c   b  d i
a  bi   c  di   a  c   b  d i
3  4i    2  6i 
 4  3i   6  7i 
 1  6i   8  3i   7  3i 
Multiplication of Complex Numbers
a  bi c  di   ac  bd   ad  bc i
2  3i 3  4i  
4  3i  
2
6  5i 6  5i  
a  bi a  bi  
To find the quotient of two complex numbers
in standard form, multiply numerator and
denominator by the complex conjugate of the
denominator.
3  2i

5i
3

i
Powers of i
i15
i 203
i 3
8.2 Trigonometric (Polar) Form of Complex
Numbers
The Complex Plane and Vector Representation
 Horizontal axis: real axis
 Vertical axis: imaginary axis
2  3i
Each complex number a  bi determines a
unique position vector with initial point
0,0  and terminal point a, b .
4  i 1  3i  
6  2i  4  3i  
Relationships Among x, y, r, and θ.
x  r cos 
y  r sin 
y
tan   ,  x  0
r  x2  y 2
x
Trigonometric (Polar) Form of a Complex #
The expression r cos   i sin   is the
trigonometric form (or polar form) of the
complex number x  yi . The number r is the
absolute value (or modulus) of x  yi , and
θ is the argument of x  yi .
The expression cos   i sin  is sometimes
abbreviated cis θ.
Write 2cos 300  i sin 300 in rectangular form.
Converting From Rectangular to
Trigonometric Form
1 Sketch a graph of the number x  yi in
the complex plane.
2 Find r by using the equation r  x 2  y 2
3 Find θ by using the equation
y
tan   ,  x  0, choosing the quadrant
x
indicated in Step 1.
Write  3  i in trigonometric form.
Write –3i in trigonometric form.
Write 6cos115  i sin 115 in rectangular form.
Write 5  4i in trigonometric form.
The fractal called the Julia set is shown here.
To determine if a complex number z  a  bi
is in the Julia set, perform the following
sequence of calculations.
z  1, z  1  1,
2
2
2
z
2

 1  1  1, 
2
2
If the absolute values of any of the resulting
complex numbers exceed 2, then the complex
number z is not in the Julia set. Otherwise z is
part of this set and the point (a, b) should be
shaded in the graph.
z  0  0i
z  1 1i
8.3 The Product and Quotient Theorems
1  i 3 2
3  2i  
Product Theorem
If r1 cos 1  i sin 1  and r2 cos  2  i sin  2 
are any two complex numbers, then:
r1 cos 1  i sin 1  r2 cos  2  i sin  2  
r1r2 cos1   2   i sin 1   2 
or
r1cis1 r2cis2   r1r2cis1  2 
Find 3cos 45  i sin 452cos 135  i sin 135
1 i 3
Find
 2 3  2i
Quotient Theorem
If r1 cos 1  i sin 1  and r2 cos  2  i sin  2 
are any two complex numbers, where
r2 cos  2  i sin  2   0 then:
r1 cos 1  i sin 1  r1
 cos1   2   i sin(1   2 
r2 cos  2  i sin  2  r2
or
r1 cis1  r1
 cis1   2 
r2 cis 2  r2
Find 10cis 60
5cis150
Find 9.3cis125.22.7cis 49.8
3
3 

10.42 cos  i sin 
4
4 

Find



5.21 cos  i sin 
5
5

8.4 De Moivre’s Theorem; Powers and
Roots of Complex Numbers
r cos  i sin  2 
r cos   i sin  2  r cos   i sin   
If r cos   i sin   is a complex number,
and if n is any real number, then:
n
r cos   i sin    r n cos n  i sin n 
Find 1  i 3
8
nth Root
For a positive integer n, the complex number
a  bi is an nth root of the complex number
n
x  yi if a  bi   x  yi
r cos   i sin  3  8cos135  i sin 135
nth Root Theorem
If n is any positive integer, r is a positive
real number, and θ is in degrees, then the
nonzero complex number r cos   i sin  
has exactly n distinct nth roots, given by
  360  k
n
r cos   i sin   where  
or
n
 360  k
, k  1,2,3,, n  1. If θ is in
 
n
n
  2k
 2k
radians, then  
or   
.
n
n
n
Find the two square roots of 4i.
Find all fourth roots of  8  8i 3
Find all complex number solutions of x 5  1  0
8.5 Polar Equations and Graphs
The polar coordinate system is based on a
point, called the pole, and a ray, called the
polar axis.
 Point P has rectangular
coordinates  x, y .
 Point P can also be located
by giving the directed
angle θ from the positive
x-axis to ray OP and the
directed distance r from
the pole to point P.
 The polar coordinates of
point P are r ,  .
From section 8.2,
Relationships Among x, y, r, and θ.
x  r cos 
y  r sin 
y
2
2
tan   ,  x  0
r x y
x
Plot and convert to rectangular coordinates.
P2,30
2 

Q  4, 
3 



R 5, 
4

Give 3 other pairs of polar coordinates for
P3,140
Determine 2 pairs of polar coordinates for  1,1
Polar equations
ax  by  c
x2  y 2  a2
Give polar equation and sketch graph
y  x 3
x2  y 2  4
Graphing a polar equation (cardioid)
r  1 cos 
cos θ
θ
r  1 cos 
Graphing a polar equation (rose)
r  3 cos 2
cos 2θ
θ 2θ
r  3 cos 2
Graphing a polar equation (lemniscate)
r 2  cos 2
cos 2θ
θ 2θ
r   cos 2
Graphing a polar equation (Spiral of
Archimedes)
r  2 (in radians)
θ
r  2
4
Convert r 
to rectangular
1  sin 
coordinates and graph.
8.6 Parametric Equations, Graphs, and
Applications
Parametric Equations of a Plane Curve
A plane curve is a set of points  x, y , such
that x  f t , y  g t , and f and g are both
defined on an interval I. The equations
x  f t  and y  g t  are parametric
equations with parameter t.
Let x  t 2 and y  2t  3 for t in  3,3.
Graph the set of ordered pairs  x, y .
Find a rectangular equation for x  t 2 and
y  2t  3 for t in  3,3.
Graph the plane curve defined by x  2 sin t
and y  3 cos t , t  0,2 
Give two parametric representations for the
2
equation of the parabola y   x  2   1
Graph the cycloid x  t  sin t , y  1 cos t ,
t  0,2 
Applications of Parametric Equations
If a ball is thrown with a velocity v feet/sec
at angle θ with the horizontal, its flight can
be modeled by x  v cos  t and
y  v sin  t  16t 2  h , where t in sec, h is
initial height.
Three golf balls are hit into the air at 132
ft/sec at angles of 30°, 50°, and 70°. Which
ball travels further, and which ball reaches
greatest height?
Jack Lukas launches a small rocket from a
table that is 3.36 ft above ground. Initial
velocity is 64 ft/sec, and it is launched at an
angle of 30°. Find the rectangular equation
that models its path. Determine total flight
time and horizontal distance traversed.