UNIT TWO
POLAR COORDINATES AND
COMPLEX NUMBERS
MATH 611B
15 HOURS
Revised Jan 9, 03
1
SCO: By the end of grade
12, students will be
expected to:
C97 construct and examine
graphs in the polar
plane
Elaborations - Instructional Strategies/Suggestions
Polar Coordinates (9.1)
Terms students should become familiar with are:
< pole
< polar axis
< polar equation
Student groups should be able to graph points and polar equations
containing either “r” or “2 ”.
Students should recognize that the polar coordinates are not specific to
only one point much like angles in standard position can be defined by
more than one standard angle.
If a point P has polar coordinates (r,2), then P can also be represented
as (r, 2 + 2Bk) or (r, 2 + (2k + 1)B), where “k” is any integer.
Student groups should recognize the distance formula for a Polar Plane
P1 P2 =
r12 + r2 2 − 2 r1r2 cos(θ 2 − θ 1 )
as the cosine formula in the rectangular plane.
c =
a 2 + b 2 − 2 ab cos θ
Note: Students will be expected to research the applications and
advantages of the Polar Plane.
Note: Students will be expected to get a table of values( manually or
using the TI-83), draw the graph on polar graph paper and check the
result on the TI-83.
2
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Polar Coordinates Worksheet 9.1
Research/Presentation
Research the applications and advantages of polar coordinates
and, as a group, present your group’s findings to the class.
Research/Presentation
Write a short paper on the life of Jakob Amsler-Laffon and his
contributions to engineering.
Pencil/Paper
Graph each point in the polar plane:
a) (3,30°)
b) (!2,90°)
c) (2,5B/6)
d) (4,!60°)
e) (!3,!B/4)
f) (2,270°)
Pencil/Paper
Name three other pairs of polar coordinates for each point:
a) (3,B/2)
b) (2,65°)
Pencil/Paper
Graph each polar equation:
a) r = 2
b) 2 = 2B/3
c) r = !1
d) 2 = !150°
Group Activity
Find the distance between points (3,150°) and (5,100°).
3
Polar Coordinates (9.1)
1. Explain why a point in the polar plane can’t be labelled using a unique ordered pair (r, 2).
2. Explain how to graph (r, 2) if r < 0 and 2 > 0.
3. Name two values of 2 such that (!4, 2) represents the same point as (4, 120°).
4. Graph each point:
a) A(1, 135°) b) B(2.5, !B/6) c) C(!3, !120°)
e) E(2, 30°)
f) F(1,B/2)
d) D(!2, 13B/6)
g) G(1/2, 3B/4)
h) H(5/2, !210°)
i) I(2, !90°)
5. Name four different pairs of polar coordinates for each point:
a) (!2, B/6)
b) (1.5, 180°) c) (!1, B/3)
d) (4, 315°)
6. Graph each polar equation:
a) r = 1
b) 2 = !B/3
c) r = 3.5
d) r = 1.5
e) 2 = 5B/4
f) 2 = !150°
7. Find the distance between the points with the given polar coordinates:
a) P1 (1, B/6) and P2 (5,3B/4)
b) P1 (1.3, !47°) and P2 (!3.6, !62°)
8. When designing web-sites with circular graphics, it is often convenient to use polar coordinates. If the
origin is at the centre of the screen, what are the polar equations of the lines that cut the region into the
eight congruent slices shown.
4
5
SCO: By the end of grade
12, students will be
expected to:
Elaborations - Instructional Strategies/Suggestions
Graphs of Polar Equations (9.2)
C97 construct and examine
graphs in the polar
plane
The classical curves using polar coordinates are described in the table on
p.564. There are three major categories:
< rose - of which a lemniscate is a special case.
< limacon - of which a cardioid is a special case.
< spiral
Example: Graph r = 2 cos 32.
The general equation for a rose is r = a cos n2;
where n is even, then there are 2n petals
where n is odd, then there is n petals
General Equations:
< rose - r = a cos n2 or r = a sin n2
< lemniscate of Bernoulli- r2 = a2 cos 22 or r = a2 sin 22
< limacon of Pascal - r = a + b cos 2 or r = a + b sin 2
< cardioid - r = a + a cos 2 or r = a + a sin 2
< spiral of Archimedes - r = a 2 ( 2 must be in radians)
Extension:
< Cissoid of Diocles - r = 2a tan 2 sin 2
6
Worthwhile Tasks for Instruction and/or Assessment
Research/Presentation
Research and present, as a group, to the class on the life and
times of one of the following. Include in the presentation a
description of the classical curve they were responsible for
developing:
a) Etienne Pascal
b) Jacob Bernoulli
c) Archimedes
Group Activity
Identify the type of curve each represents and graph each of
the following polar equations:
a) r = 2 cos 32
b) r = 32
c) r2 = 4 cos 22
d) r = 3 + 3 sin 2
e) r = 5 + 2 cos 2
Communication
Write an equation for a rose with 4 petals and describe your
equation and graph to the class.
7
Suggested Resources
Polar Equation Worksheets (9.2)
Note:
The locus of a cardioid is a point on
the circumference of a circle that is
rolling around the circumference of a
circle of equal radius.
Polar Equation Worksheet (9.2) Part 1
1. Graph the following classical curves. Inductively state the pattern that evolves.
a) r = cos 2
b) r = 2 cos 2
c) r = 3 cos 2
d) What effect does “a” have on the graph?
e) Make a conjecture as to how the r = sin 2 graph compares with the r = cos 2 graph.
2. Verify your conjecture by graphing the following:
a) r = sin 2
b) r = 2 sin 2
c) r = 3 sin 2
3. Graph the following classical curves. Inductively state the pattern that evolves.
a) r = 2 cos 2
b) r = 2 cos 22
c) r = 2 cos 32
d) What effect does “n” have on the graph?
e) Make a prognostication as to what r = 2 cos 52 would look like. Verify your prognostication by
graphing.
4. Graph the following classical curves. Inductively state the pattern that evolves.
a) r = 1 + 2 cos 2
b) r = 2 + 2 cos 2
c) r = 3 + 2 cos 2
d) r = 4 + 2 cos 2
e) What effect does “a” have on the graph?
f) Which of these curves is a cardioid?
g) Predict what the graph of r = 1 + 2 sin 2 would look like. Verify your prediction by graphing.
h) Take part (c) above and investigate what effect changing the sign of “a” and/or “b” has on the basic
graph. Check your work by trying a few more examples. Finally make a statement about the effects of
the size and sign of these coefficients on the graph and convey this to the class.
5. Graph the following classical curves:
a) r2 = 4 cos 22
b) r2 = 4 sin 22
c) What is the name for this particular classical curve?
6. Graph the following classical curves. Inductively state the pattern that evolves
a) r = 2
b) r = 22
c) r = 32
d) r = !2
e) r = !22
f) r = !32
g) What effect does the “a” coefficient have on the graph?
8
Polar Equation Worksheet (9.2) Part 2
1. Write a polar equation whose graph is a rose.
2. Determine the maximum value of r in the equation r = 3 + 5 sin 2. What is the minimum value of r?
3. Identify the type of curve each represents, then graph.
a) r = 1 + sin 2
b) 2 ! 3 sin 2
c) r = cos 22
d) r = 1.5 2
e) r = !3 sin 2
f) r = 3 + 3 cos 2
g)r = 3 2
h) r2 = 4 cos 22
i) r = 2 sin 32
j) r = !2 sin 32
k) r = 52/2
l) r = !5 + 3 cos 2
m) r = !2 !2 sin 2
n) r2 = 9 sin 22
o) r = 9 sin 22
p) r = sin 42
q) r = 2 + 2 cos 2
4. Graph each system of polar equations. Solve the system using algebra and trigonometry. Assume
0 # 2 < 2 B.
a) r = 3
r = 2 + cos 2
b) r = 1 + cos 2
r = 1 ! cos 2
c) r = 2 sin 2
r = 2 sin 22
5. Graph each system using a graphing calculator. Find the points of intersection. Round coordinates to
the nearest tenth. Assume 0 # 2 < 2B.
a) r = 1
b) r = 3 + 3 sin 2
r = 2 cos 22
r=2
c) r = 2 + 2 cos 2
r = 3 + sin 2
9
SCO: By the end of grade
12, students will be
expected to:
Elaborations - Instructional Strategies/Suggestions
Polar and Rectangular Coordinates (9.3)
A27 translate between
polar and rectangular
coordinates
The trigonometric definitions:
x
r
y
sin θ =
r
cos θ =
⇒ x = r cos θ
⇒ y = r sin θ
can be used to convert from polar to rectangular coordinates.
To convert from rectangular to polar coordinates use:
r =
x2 + y2
tan θ =
y
x
⇒ θ = tan −1
Another way of expressing tan
y
x
−1
y
y
is arctan
x
x
Note: At the end of the unit is an explanation of how to use the TI-83 for
converting degrees to radians and vice versa. As well, the explanation
shows how to convert from polar coordinates to rectangular coordinates
and vice versa (p.38).
10
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Polar and Rectangular Coordinates (9.3)
Pencil/Paper
Find the polar coordinates of the following points in
rectangular form. Use 0 # 2 < 2B and r $ 0.
a) (8,15)
b) (!3,4)
Pencil/Paper
Write each point using rectangular coordinates:
a) (2,120°)
b) (!3,B/2)
Pencil/Paper
Write each polar equation in rectangular form:
a) r = 3 cos 2
b) r = !4
c) 2 = B/4
Pencil/Paper
Write each rectangular equation in polar form:
a) x = 4
b) y = !2
c) x2 + y2 = 16
d) x2 + y2 = 6y
Group Discussion
Write r =
tan θ
in rectangular form.
cos θ
When converted to rectangular form we get “ y = x2".
11
Polar and Rectangular Coordinates
Worksheet (9.3)
Polar and Rectangular Coordinates (9.3)
1. Write the polar coordinates of the points in the graphs shown.
2. Explain why you have to consider what quadrant a point lies in when converting from rectangular to
polar coordinates.
3. Find the polar coordinates of each point with the given rectangular coordinates.
Use 0 # 2 < 2B and r $ 0.
a) ( − 2 ,
2)
b) (!2, !5)
c) (2, !2)
d) (0,1)
e) (3, 8)
f) (4, !7)
4. Find the rectangular coordinates of each point with the given polar coordinates.
b) (2.5, 250°) c) (3, B/2)
d) (4, 210°)
a) (!2, 4B/3)
5. Write each rectangular equation in polar form.
a) y = 2
b) x2 + y2 = 16
c) x = !7
f) x2 + y2 = 2y g) x2 ! y2 = 1
d) y = 5
h) x2 + (y !2)2 = 4
6. Write each polar equation in rectangular form.
a) r = 6
b) r = ! sec 2
c) r = 2
f) r = 2 csc 2
g) r = 3 cos 2
e) x2 + y2 = 25
d) r = !3
e) 2 = B/3
h) r2 sin 22 = 8 i) r( cos 2 + 2 sin 2) = 4
7. A surveyor identifies a landmark at the point with polar coordinates (325, 70°). What are the
rectangular coordinates of this point?
12
13
SCO: By the end of grade
12, students will be
expected to:
Elaborations - Instructional Strategies/Suggestions
Simplifying Complex Numbers (9.5)
B43 simplify and perform
operations on complex
numbers
Students should review complex numbers in rectangular form.
A complex number of the form a + bi , where a is called the real part
and b is called the imaginary part.
1. If b = 0, the complex number is a real number (part of 2).
2. If b
… 0, the complex number is an imaginary number.
3. If a = 0 & b
(part of 2).
… 0, the complex number is a pure imaginary number.
Note: All previous sets of numbers are subsumed by the set of complex
numbers.
Note: Complex numbers
were covered in Math
521A, Unit 2
Math Power 11 p.185
Students will be expected to be able to perform operations on complex
numbers. They should as well be exposed to the concept of complex
conjugates.
Note to Teachers: At the end of the unit is a short essay on “The
Development of Number Systems” which may be useful (p.41).
14
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Simplifying Complex Numbers (9.5)
Simplifying Complex Numbers
Worksheet (9.5)
Pencil/Paper
Simplify:
a) i
3
12
b) i
c) ( 3 + 5i ) + ( −7 + i )
d) ( −3 + 2i )
2
2i
1 − 5i
f) ( 2 − 5i )( 4 + i )
e)
Journal
Write to explain how to simplify any integral power of “i ”.
15
Simplifying Complex Numbers Worksheet (9.5)
1. Read pages 41 and 42 in the workbook.
2. Describe how to simplify any integral power of i.
3. Draw a Venn diagram to show the relationship between real, pure imaginary and complex numbers.
4. Explain why it is useful to multiply by the conjugate of the denominator over itself when simplifying a
fraction containing complex numbers.
5. Write a quadratic equation that has two complex conjugate solutions.
6. Simplify.
a) i!6
c) (2 + 3i) + (!6 + i)
d) (2.3 + 4.1i) ! (!1.2 ! 6.3i)
e) (2 + 4i) + (!1 + 5i)
f) (!2 ! i)2
g) i6
h) i19
j) i9 + i!5
k) (3 + 2i) + (!4 + 6i)
l) (7 ! 4i) + (2 !3i)
m) (1/2 + i) ! (2 ! i)
n) (!3 !i) ! (4 ! 5i)
o) (2 + i)(4 + 3i)
p) (1 + 4i)2
q) (1 +
r) ( 2 +
b) i10 + i2
i) i1776
−3 )( −1 +
−12 )
s)
2 +i
1 + 2i
t)
7i )( −2 −
3 − 2i
−4 − i
u)
5i )
5−i
5+i
7. Write a quadratic equation with solutions i and !i.
8. Write a quadratic equation with solutions 2 + i and 2 ! i.
9. Simplify.
a) (2 ! i)(3 + 2i)(1 ! 4i)
2 −
d)
3+
2i
6i
b) (!1 !3i)(2 + 2i)(1 ! 2i)
1
+ 3i
2
c)
1 − 2i
(1 + i ) 2
f)
( −3 + 2i ) 2
3+i
e)
(2 + i ) 2
16
17
SCO: By the end of grade
12, students will be
expected to:
A26 translate between
polar and rectangular
coordinates on the
complex plane
C88 represent complex
numbers in a variety
of ways
Note: Replacing 2 with B
yields Euler’s Equation;
πi
e = cos π + i sin π
Elaborations - Instructional Strategies/Suggestions
Complex Numbers in Polar Form (9.6)
Students will be expected to convert complex numbers from rectangular
to polar form and vice versa.
Students should be familiar with the following concepts:
For complex numbers in rectangular form:
< Argand Plane
< real axis, imaginary axis
< absolute value of a complex number.
If z = a + bi, then z =
a 2 + b2
This absolute value represents the distance from zero on the complex
plane.
For complex numbers in polar form:
< modulus, r (absolute value of the complex number)
< argument, 2 (amplitude of the complex number or the angle between r
and the zero line)
Euler’s Formula states that e
θi
= cos θ + i sin θ
< z = r (cos θ + i sin θ ) = rcisθ = reθi
eπ i = −1
eπ i + 1 = 0
Note: If a complex number
is in rectangular form, then
plot it on a rectangular
coordinate plane.
If it is in polar form, graph
it on a polar coordinate
plane.
Ex 4 p.588: Express !3 + 4i in polar
form.
Using the TI-83: enter !3 + 4i math < < CPX 7: < Polar enter
enter
or 5 cis 2.21
To convert to rectangular form press math < < CPX 6: < Rect enter ...
18
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Complex Numbers in Polar Form (9.6)
Journal
Write to explain how to find the absolute value of a complex
number.
Pencil/Paper
Solve for x and y: 2x + y + 3xi + 5yi = 7 + 4i
Pencil/Paper
Graph !3 ! 2i on the Argand Plane and find its absolute
value.
Pencil/Paper
Express 4 ! 3i in polar form.
Pencil/Paper
Graph 3(cos 2 + i sin 2) and express in rectangular form.
Group Research/Group Presentation
Write a short paper on the contributions of Jean Robert
Argand to mathematics.
19
Complex Numbers in Polar Form
Worksheet 9.6
Complex Numbers in Polar Form Worksheet (9.6)
1. Explain how to find the absolute value of a complex number.
2. Write the polar form of i.
3. Solve each equation for x and y, where x and y are real numbers.
a) 2x + y + yi = 5 + 4i
b) 1 + (x + y)i = y + 3xi
4. Graph each number in the complex plane and find its absolute value.
b) 1 +
a) !2 ! i
e) −1 +
2i
d) !1 ! 5i
c) 2 + 3i
5i
5. Express each complex number in polar form.
b) −1 −
a) 4 + 5i
c) !4 + i
3i
d) !2 + 4i
e) −4 2
6. Graph each complex number. Then express it in rectangular form.
a) 4 (cos
d) 2 (cos
π
3
+ i sin
π
3
)
4π
4π
+ i sin
)
3
3
b)
3
(cos 2π + i sin 2π )
2
e) 2 (cos
5π
5π
+ i sin
)
4
4
c) 3(cos
π
4
+ i sin
π
4
)
f) 5(cos 0 + i sin 0)
7. A series circuit contains two sources of impedance, one of 10(cos 0.7 + j sin 0.7 ) ohms and the
other of 16(cos 0.5 + j sin 0.5) . (Note: j is used by engineers in place of i.)
a) Convert these complex number to rectangular form.
b) Add your answers from part a to find the total impedance in the circuit.
c) Convert the total impedance back to polar form.
20
21
SCO: By the end of grade
12, students will be
expected to:
Elaborations - Instructional Strategies/Suggestions
Products and Quotients of Complex Numbers in Polar Form (9.7)
Product of complex numbers in polar form:
B42 multiply and divide
complex numbers in
polar form
r1 (cos θ1 + i sin θ 2 ) ⋅ r2 (cos θ 2 + i sin θ 2 )
= r1eiθ1 ⋅ r2 eiθ 2
= r1r2 ei (θ1 +θ 2 )
= r1r2 [cos(θ1 + θ 2 ) + i sin(θ1 + θ 2 )]
= r1r2 cis (θ1 + θ 2 )
The modulus (r1r2) of the product of 2 complex numbers is the
product of their modulii.
The argument or amplitude ( 21 + 22) of the product of 2 complex
numbers is the sum of their arguments.
r1cisθ1 ⋅ r2 cisθ 2
Thus
Ex.
= r1r2 cis (θ1 + θ 2 )
4cis45o ⋅ 5cis15o
= 20cis60o
Quotient of complex numbers in polar form:
r1 (cos θ1 + i sin θ1 )
r2 (cos θ 2 + i sin θ 2 )
=
r1 i (θ1 −θ 2 )
e
r2
=
r1
[cos(θ1 − θ 2 ) + i sin(θ1 − θ 2 )]
r2
=
r1
cis (θ1 − θ 2 )
r2
The modulus of the quotient of 2 complex numbers is the quotient of
their modulii. The argument is the difference of their arguments.
22
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Products and Quotients of Complex Numbers in Polar
Form (9.7)
Pencil/Paper
Find each product or quotient. Convert the answer into
rectangular form.
a) 4 cis 30° @ 3 cis 20°
b) 8 cis 70° @ 2 cis 40°
10cis90o
5cis25o
12 cis40o
d)
2 cis60o
c)
e) 4 cis
π
⋅ 9 cis
3
2π
25cis
3
f)
π
5cis
6
π
4
23
Products and Quotients of Complex
Numbers in Polar Form Worksheet
(9.7)
Products and Quotients of Complex Numbers in Polar Form Worksheet (9.7)
1. Explain how to find the quotient of two complex numbers in polar form.
2. List which operations with complex numbers you think are easier in rectangular form and which you
think are easier in polar form. Defend your choices with examples.
3. Find each product or quotient. Express the result in rectangular form.
2π
2π
+ i sin
)
6
6
3
3
1
π
π
5π
5π
b)
+ i sin
(cos + i sin ) ⋅ 6(cos
)
2
3
3
6
6
a) 3(cos
π
+ i sin
π
) ÷ 4(cos
4. Determine the voltage in a circuit when there is a current of 2 (cos
impedance of 3 cos(
π
3
+ j sin
π
3
11π
11π
+ j sin
) amps and an
6
6
) ohms.
5. Find each product or quotient. Express the result in rectangular form.
π
π
3π
3π
+ i sin
) ÷ 2 (cos + i sin )
4
4
4
4
3π
3π
b) 5(cos π + i sin π ) ⋅ 2 (cos
+ i sin
)
4
4
7π
7π
π
π
c) 3(cos
+ i sin
) ÷ (cos + i sin )
3
3
2
2
7π
7π
2
3π
3π
d) 2 (cos
+ i sin
) ÷
(cos
+ i sin
)
4
4
2
4
4
e) 4[cos( −2 ) + i sin( − 2 )] ÷ (cos 3.6 + i sin 3.6)
3π
3π
π
π
f) 2 (cos
+ i sin
) ⋅ 2 (cos + i sin )
4
4
2
2
a) 6(cos
6. If z1 = 4 (cos
1
π
π
z
5π
5π
+ i sin
) and z2 =
(cos + i sin ) , find 1 and express the
2
3
3
z2
3
3
result in rectangular form.
24
25
SCO: By the end of grade
12, students will be
expected to:
Elaborations - Instructional Strategies/Suggestions
Powers and Roots of Complex Numbers (9.8)
De Moivre’s Theorem
B44 derive and apply De
Moivre’s Theorem for
powers and roots
z n = [r (cos θ + i sin θ )]n
= [ reiθ ]n
= r n einθ
= r n [cos nθ + i sin nθ ]
= r n cis nθ
Looking at examples 1&2 on p.599 & 600 we see the answer to be
4096. It is important for students to see the process by which the answer
was arrived at but they should also be aware that the calculator can do
the work as well.
A useful application of De Moivre’s Theorem is in finding the roots of a
complex number. The theorem can be re-written as:
⎛ θ + 2 kπ ⎞
z p = [ rcisθ ] p = [ rcis (θ + 2kπ )] p = r p cis ⎜
⎟
p ⎠
⎝
1
1
1
1
where k = 0,1,...,p!1
For instance we would normally think of the answer of
in fact there are two other cube roots
( −1 +
3i ) and ( −1 −
3
8 to be 2 but
3i ) .
Refer to p.43 in the workbook for a detailed solution.
In the essay, “The Development of Number Systems” ,at the end of
the unit we will solve this example and others in detail.
26
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Powers and Roots of Complex Numbers (9.8)
Powers and Roots of Complex
Numbers Worksheet (9.8)
Pencil/Paper/Technology
Find ( 3 − 5i ) . Express the result in both rectangular and
polar form.
4
Using the TI-83:
note the mode
Technology
Use the TI-83 to find all roots for x4 ! 1 = 0.
Presentation
Present to the class a summary of the life and contributions of
Benoit Mandlebrot.
Research/Presentation
Investigate the life of Abraham De Moivre and his
contributions to mathematics.
27
Powers and Roots of Complex Numbers Worksheet (9.8)
1. Evaluate the product (1 + i)(1 + i)(1 + i)(1 + i)(1 + i) by traditional multiplication. Compare the results
with the results using De Moivre’s Theorem on (1 + i)5. Which method do you prefer?
2. Explain how to use De Moivre’s Theorem to find the reciprocal of a complex number in polar form.
3. Find each power. Express the result in rectangular form.
a) ( 3 − i )
d) [ 2 (cos
3
π
4
b) (3 ! 5i)4
+ i sin
π
4
c) [ 3(cos
)]5
6
+ i sin
π
6
)]3
f) (1 +
e) (!2 + 2i)3
h) (2 + 3i)!2
g) (3 !6i)4
π
3i ) 4
i) Raise 2 + 4i to the fourth power.
4. Find all roots of and write answers in a + bi form.
a)the cube roots of !8
b) the fifth roots of 32
c) the fourth roots of 16i
5. Find each principal root. Express the result in the form a + bi with a and b rounded to the nearest
hundredth.
a)
1
i6
d) ( −2 +
b) ( −2 −
1
4
i)
e) ( 4 −
1
i) 3
1
i) 3
2π
2π 5
c) [ 32 (cos
+ i sin
)]
3
3
1
f) Find the principal square root of i.
6. Solve each equation. Then graph the roots in the complex plane.
b) 2x3 + 4 + 2i = 0
c) x3 ! 1 = 0
a) x4 + i = 0
d) 3x4 + 48 = 0
e) x4 ! (1 + i) = 0
f) 2 x
4
+ 2 + 2 3i = 0
7. Use a graphing calculator to find all of the indicated roots.
b) sixth roots of 2 + 4i
a) fifth roots of 10 ! 9i
c) eighth roots of 36 + 20i
8. Gloria works for an advertising firm. She must incorporate a hexagon design into a logo for one of the
ads she is working on. She can locate the vertices of a regular hexagon by graphing the solutions to the
equation x6 ! 1 = 0 in the complex plane. What are the solutions to this equation?
28
Polar Earth Map (9.1)
29
Hodographs (9.1)
A hodograph is a plot representing the vertical distribution of horizontal winds using polar coordinates.
A hodograph is obtained using data from a radiosonde balloon. By plotting the end points of the wind
vectors (wind speed and direction) at various altitudes and connecting these points in order of increasing
height. Interpretation of a hodograph can help in forecasting the subsequent evolution of thunderstorms.
30
Polar Planimeter (9.1)
Planimeters don’t calculate in the sense of allowing a user to enter some numbers and producing a result.
They do, however, allow the user to calculate the area of any closed shape. They are, in essence, an
integration machine. Visit http://www.hpmuseum.org/planim.htm for more details.
31
15° Polar Graph Paper (9.2)
90
90
120
120
60
60
150
150
30
30
180
180
0
210
210
330
240
0
330
240
300
90
120
90
120
60
150
180
0
210
60
150
30
30
180
330
240
300
270
270
0
210
330
300
240
270
300
270
32
Polar Graph Paper (9.2)
33
Polar Graph Paper ( 9.2)
π
2π
3
2
π
3
5π
6
π
6
π
7π
6
11π
6
4π
3
3π
2
34
5π
3
35
Polar Graph Paper ( 9.2)
π
2
2π
3
π
π
3
5π
6
2
2π
3
π
3
5π
6
π
6
π
6
π
π
7π
6
7π
6
11π
6
4π
3
3π
2
11π
6
5π
3
4π
3
π
2π
3
2
5π
3
3π
2
π
π
2π
3
3
5π
6
2
π
3
5π
6
π
6
π
π
6
π
7π
6
7π
6
11π
6
4π
3
3π
2
5π
3
11π
6
4π
3
36
5π
3
Spiral Graph Example (9.2)
37
Use of TI-83 (9.3)
Convert degrees to radians
Convert 50° to radians.
Have calculator in radian mode.
Press 50 2nd Angle 1:° enter
Convert radians to degrees
Convert 2.3 to degrees.
Have calculator in degree mode.
Press 2.3 2nd Angle 3:° enter
Convert rectangular to polar coordinates
Convert ( !8,!12) to polar coordinates. Press 2nd Angle 5:R < Pr (!8,!12) enter
Press 2nd Angle 6:R < P2 (!8,!12) enter
This illustrates the limitations of calculators. The answer should be 236.31
If the calculator was in radian mode the answer would be given in radians.
The answer in polar coordinates is (14.4, 236.3) or (14.4, 4.1).
Convert polar to rectangular coordinates
Convert (2,80°) to rectangular coordinates.
is in degrees.
Calculator must be in degree mode when angle in question
Press 2nd Angle 7:P < Rx (2, 80) enter
Press 2nd Angle 8:P < Ry (2, 80) enter
The answer in rectangular coordinates is (.35, 1.97).
38
Number Systems (9.5)
39
Graphic of Powers of Complex Numbers (9.8)
40
The Development of Number Systems
The Egyptians invented the number system we use today replacing the Roman numeral system
making computations much simpler. Once people started borrowing gardening implements from their
neighbours, the need arose for negative numbers. When a hunter tried to share with his 5 friends the 3
geese he had bagged , then a need for fractions was born (either that or he didn’t tell some of his friends
about his good fortune). Once algebra came into play, problems like x2 ! 5 = 0 needed to be dealt with
using irrational numbers. These number systems taken together completely filled the number line (the
real number line ú).
All seemed right with the world until the Renaissance when some troublemaker looked for a
solution to x2 + 1 = 0. This created a problem because any real number when squared gives a result
greater than or equal to zero. To a chorus of protests from many of the prominent mathematicians of the
day, the number i = −1 was defined. These objecting mathematicians coined the phrase
“
imaginary number ” to voice their opposition. An important property of this number is that when squared
it yields a negative result; i
2
= −1 .
It was evident that numbers like 2i, 5 ! 4i, etc. were very useful, but there was no way of
representing these numbers on the real number line. The solution to this dilemma was put forth by a
Swiss mathematician, Jean Robert Argand. He placed an imaginary number line at right angles to the real
number line. This is called the Argand Plane.
For all real numbers a,b the number a + bi is a complex number. The letter, C , is used to
represent the set of complex numbers. We can represent a complex number z = a + bi as a vector on the
Cartesian coordinate plane or the Argand plane.
T
he conjugate of a complex number z = a + bi is z = a − bi .
Thus 4 ! 3i is the conjugate of 4 + 3i.
The modulus(magnitude), r, or absolute value of a complex number z = a + bi is
r = z =
a 2 + b2 .
This is a measure of the length of the vector z = a + bi.
41
The argument, 2, or amplitude of a complex number z = a + bi is
Arg ( a + bi ) = θ = tan −1
This is the angle the vector makes with the positive side of the horizontal axis.
From trigonometry we know that:
cos θ =
a
r
and sin θ =
a = r cos θ
or
b
r
or
b = r sin θ .
Thus z = a + bi = rcos 2 + irsin 2 = r(cos 2 + isin 2) = rcis 2
Multiplication of complex numbers
The modulus of the product of two complex numbers is the product of their modulii.
The argument of the product of two complex numbers is the sum of their arguments.
r1e iθ1 ⋅ r2 e iθ 2 = r1r2 ei (θ1 +θ 2 ) = r1cisθ1 ⋅ r2 cisθ 2 = r1r2 cis (θ1 + θ 2 )
Ex. 4cis 45° @ 5cis 15° = 20cis 60°
Division of complex numbers
The modulus of the quotient of two complex numbers is the quotient of their modulii.
The argument of the quotient of two complex numbers is the difference of their modulii.
r1cisθ 1
r
= 1 cis(θ 1 − θ 2 )
r2 cisθ 2
r2
8cis75o
Ex.
= 4cis45o
o
2cis30
42
bI
F
G
Ha J
K
De Moivre’s Theorem
z n = [ rcisθ ]n = r n cisnθ ; it is useful in finding the pth root of a complex number.
It states that:
⎛θ ⎞
z p = [ rcisθ ] p = r p cis ⎜ ⎟
⎝ p⎠
1
1
1
Without De Moivre’s Theorem problems like; find the cube roots of 8; will not yield all possible roots. A
pth root problem should yield p roots. To find all possible roots we can use the fact that:
sin θ = sin(θ + 2π ) = sin(θ + 4π ) = . . . = sin(θ + 2 kπ )
cos θ = cos(θ + 2π ) = cos(θ + 4π ) =. . . = cos(θ + 2 kπ )
1
p
z = [ rcisθ ]
1
p
= [rcis (θ + 2kπ )]
Therefore
1
p
θ + 2 kπ
= r cis (
p
1
p
⎛ θ + 2kπ
) or r (cos ⎜
p
⎝
1
p
⎞
⎛ θ + 2kπ
⎟ + i sin ⎜
p
⎠
⎝
⎞
⎟)
⎠
Evaluating this formula for k = 0,1,2,...,p!1 will yield the p roots.
Ex. Solve
1
83
for all roots and represent them on the Argand plane.
Solution: 8 means 8 + 0i where a = 8 and b = 0. Thus z = r = 8
and
θ = tan −1
θ = 0
Using r = 8,
2 = 0, p = 3, k = 0,1,2
⎛ θ + 2 kπ
z = r (cos ⎜
n
⎝
1
p
1
3
1
p
⎞
⎛ θ + 2 kπ
⎟ + i sin ⎜
n
⎠
⎝
⎞
⎟)
⎠
1
3
8 = 8 (cos 0 + i sin 0) = 2 k = 0
2π
2π
+ i sin ) = −1 + 3i k = 1
3
3
1
1
4π
4π
8 3 = 8 3 (cos
+ i sin ) = −1 − 3i k = 2
3
3
1
1
8 3 = 8 3 (cos
43
b
a
Verify:
23 = 8
( −1 −
3i ) 3 = 8
( −1 +
3i ) 3 = 8
T
o
previous screens, use r = 8, 2 = 0, p = 3, k = 0,1,2
get
X1T = r(to the desired root) cos T
Y1T = r(to the desired root) sin T
Tmin = 2/p = 0/3 = 0
Tmax = Tmin + 2B = 2B
Tstep = 2B/p = 2B/3
Press Trace and <
44
the
Ex 3 p.601 AMC
Find
3
8i
or the cube roots of 8i. Check your work by graphing on the TI-83.
Solution
Convert 8i to polar form.
r = 8, 2 = B/2
Using DeMoivre’s Theorem
8i = 8[cos
⎡ ⎛ θ + 2 kπ
z = r ⎢ cos ⎜
p
⎣ ⎝
1
p
1
p
π
2
+ i sin
⎡ ⎛π
⎢ ⎜ 2 + 2kπ
8 = 8 ⎢ cos ⎜
3
⎢ ⎜
⎢⎣ ⎝
1
3
⎞
⎛π
⎟
⎜ 2 + 2kπ
⎟ + i sin ⎜
3
⎟
⎜
⎠
⎝
⎞⎤
⎟⎥
⎟⎥
⎟⎥
⎠ ⎥⎦
when k = 0
⎡
⎛π ⎞
⎛ π ⎞⎤
8 = 8 ⎢ cos ⎜ ⎟ + i sin ⎜ ⎟ ⎥ = 3 + i
⎝6⎠
⎝ 6 ⎠⎦
⎣
when k = 1
⎡
⎛ 5π
8 = 8 ⎢ cos ⎜
⎝ 6
⎣
⎞
⎛ 5π ⎞ ⎤
⎟ + i sin ⎜
⎟⎥ = − 3 + i
⎠
⎝ 6 ⎠⎦
when k = 2
⎡
⎛ 3π
8 = 8 ⎢ cos ⎜
⎝ 2
⎣
⎞
⎛ 3π
⎟ + i sin ⎜
⎠
⎝ 2
1
3
1
3
1
3
1
3
1
3
1
3
⎞⎤
⎟ ⎥ = −2i
⎠⎦
Using the TI-83 (use the steps on p.602).
45
2
]
⎞
⎛ θ + 2kπ
⎟ + i sin ⎜
p
⎠
⎝
r = 8, 2 = B/2, p = 3, k = 0,1,2
1
3
π
⎞⎤
⎟⎥
⎠⎦
To get the previous screens, use r = 8, 2 = B/2, p = 3, k = 0,1,2
X1T = r(to the desired root) cos T
Y1T = r(to the desired root) sin T
Tmin = 2/p = B/2/3 = B/6
Tmax = Tmin + 2B = 13B/6
Tstep = 2B/p = 2B/3
46
Ex 4 p.601 AMC
Find the three cube roots of !2 !2i and check your work by graphing on the TI-83.
Solution
Convert !2 !2i to polar form.
z
1
p
1
p
r=2 2
, 2 = 5B/4
5π
5π ⎤
⎡
−2 − 2i = 8 ⎢ cos
+ i sin
4
4 ⎥⎦
⎣
F
Fθ + 2π IJ)
θ + 2π I
+ i sinG
G
J
Hp K Hp K
F5π + 2kπ I
F5π
G4
J
G4
= ( 2 2 ) (cosG
+ i sinG
J
3
G
J
H
K G
H
= r (cos
( −2 −
1
2i ) 3
when k = 0
when k = 1
when k = 2
1
3
+ 2 kπ
3
I
J
)
J
J
K
( −2 −
1
2i ) 3
=
2 (cos
5π
5π
+ i sin
) = 0.37 + 1.37i
12
12
( −2 −
1
2i ) 3
=
2 (cos
13π
13π
+ i sin
) = −1.37 − 0.37i
12
12
( −2 −
1
2i ) 3
=
2 (cos
21π
21π
+ i sin
) = 1−i
12
12
Using the TI-83( and steps on p.602)
47
To get the previous screens, use
r = 2 2 ,θ =
5π
, p = 3, k = 0,1, 2
4
X1T = r(to the desired root) cos T
Y1T = r(to the desired root) sin T
Tmin = 2/p = 5B/4/3 = 5B/12
Tmax = Tmin + 2B = 29B/12
Tstep = 2B/p = 2B/3
48
Ex 5 p.603 AMC
Solve for all roots x5 ! 32 = 0.
Solution:
x5 = 32 This reads as: find the fifth roots of 32.
32 = 32 + 0i = 32(cos 0 + isin 0)
r = 32, 2 = 0, p = 5, k = 0,1,2,3,4
⎛ θ + 2 kπ ⎞
⎛ θ + 2 kπ ⎞
z = r (cos ⎜
⎟ + i sin ⎜
⎟)
n
n
⎝
⎠
⎝
⎠
1
n
1
n
1
1
32 5 = 32 n (cos 0 + i sin 0) = 2 k = 0
2π
5
1
1
4π
32 5 = 32 5 (cos
5
1
1
6π
32 5 = 32 5 (cos
5
1
1
8π
32 5 = 32 5 (cos
5
1
1
32 5 = 32 5 (cos
2π
) = .62 + 1.90i k = 1
5
4π
+ i sin ) = −1.62 + 1.18i k = 2
5
6π
+ i sin ) = −1.62 − 1.18i k = 3
5
8π
+ i sin ) = .62 − 1.90i k = 4
5
+ i sin
Pre
ss Trace and <
cursor through the roots
The points are vertices of a regular pentagon and are concyclic in nature. That is they are equally spaced
on a circle.
49
X1T = r(to the desired root) cos T
Y1T = r(to the desired root) sin T
Tmin = 2/p = 0/3 = 0
Tmax = Tmin + 2B = 2B
Tstep = 2B/p = 2B/5
Fundamental Theorem of Algebra(extension)
One of the most important uses of complex numbers is in solving equations in engineering and the
…
sciences of the type: a 0 x n + a1 x n − 1 + . . . a n − 1 x + a n = 0 where a0 0 and a1,...,an are complex
numbers. The FTA states that the above equation has at least one complex root.
50