Secondary II
Honors
Teacher Edition
Unit H
Northern Utah Curriculum Consortium
Project Leader
Sheri Heiter
Weber School District
Project Contributors
Ashley Martin
Bonita Richins
Craig Ashton
Davis School District
Cache School District
Cache School District
Gerald Jackman
Jeff Rawlins
Jeremy Young
Box Elder School District
Box Elder School District
Box Elder School District
Kip Motta
Marie Fitzgerald
Mike Hansen
Rich School District
Cache School District
Cache School District
Robert Hoggan
Sheena Knight
Teresa Billings
Cache School District
Weber School District
Weber School District
Wendy Barney
Helen Heiner
Susan Summerkorn
Weber School District
Davis School District
Davis School District
Lead Editor
Allen Jacobson
Davis School District
Technical Writer/Editor
Dianne Cummins
Davis School District
Unit H
Table of Contents
H.1 SOLVING MATRICES ................................................................................................................................4
Teacher Notes ..................................................................................................................................................4
Mathematics Content .......................................................................................................................................7
Solving Matrices A Develop Understanding Task 1 ........................................................................................8
Ready, Set, Go! ..............................................................................................................................................11
Solutions: .......................................................................................................................................................14
H.2 DETERMINANT AND INVERSE OF A MATRIX ..................................................................................15
Teacher Notes ................................................................................................................................................15
Mathematics Content .....................................................................................................................................18
The Determinant and Inverse of a Matrix A Develop Understanding Task 2 ................................................19
Ready, Set, Go! ..............................................................................................................................................21
Solutions: .......................................................................................................................................................24
H.3 CRYPTOGRAPHY WITH MATRICES ....................................................................................................25
Teacher Notes ................................................................................................................................................25
Mathematics Content .....................................................................................................................................26
Cryptography with Matrices A Develop Understanding Task 3 ....................................................................27
H.4 CAVALIERI’S THEOREM .......................................................................................................................29
Teacher Notes ................................................................................................................................................29
Mathematics Content .....................................................................................................................................31
Cavalieri’s Theorem A Develop Understanding Task 4 ................................................................................32
Ready, Set, Go! ..............................................................................................................................................33
Solutions: .......................................................................................................................................................35
H.5 ARE ANGLES ALWAYS DEGREES? .....................................................................................................36
Teacher Notes ................................................................................................................................................36
Mathematics Content .....................................................................................................................................38
Mathematics Content .....................................................................................................................................39
Are Angles Always Degrees? A Develop Understanding Task 5 ..................................................................40
Ready, Set, Go! ..............................................................................................................................................42
Solutions: .......................................................................................................................................................44
NUCC| Secondary II Math i
Unit H
H.6 UNIT CIRCLE ............................................................................................................................................45
Teacher Notes ................................................................................................................................................45
Mathematics Content .....................................................................................................................................47
Mathematics Content .....................................................................................................................................48
Unit Circle A Develop Understanding Task 6 ...............................................................................................49
Ready, Set, Go! ..............................................................................................................................................50
Solutions: .......................................................................................................................................................51
H.7 SIMPLIFY TRIGONOMETRIC EXPRESSIONS .....................................................................................52
Teacher Notes ................................................................................................................................................52
Mathematics Content .....................................................................................................................................59
Mathematics Content .....................................................................................................................................60
Solutions: .......................................................................................................................................................64
H.8 INVERSE OPERATIONS ..........................................................................................................................65
Teacher Notes ................................................................................................................................................65
Mathematics Content .....................................................................................................................................68
Ready, Set, Go! ..............................................................................................................................................69
H.9 PROVE ADDITION AND SUBTRACTION FORMULAS FOR SINE, COSINE, AND TANGENT ....72
Teacher Notes ................................................................................................................................................72
Mathematics Content .....................................................................................................................................77
Ready, Set, Go! ..............................................................................................................................................78
Solutions: .......................................................................................................................................................81
H.10 COMPLEX CONJUGATES .....................................................................................................................82
Teacher Notes ................................................................................................................................................82
Mathematics Content .....................................................................................................................................84
Mathematics Content .....................................................................................................................................85
Complex i’s A Develop Understanding Task 10 ............................................................................................86
Ready, Set, Go! ..............................................................................................................................................88
Solutions: .......................................................................................................................................................90
H.11 IMAGINARY AND COMPLEX NUMBERS .........................................................................................91
Teacher Notes ................................................................................................................................................91
Mathematics Content .....................................................................................................................................93
Mathematics Content .....................................................................................................................................94
Mathematics Content .....................................................................................................................................95
Can We Picture? A Develop Understanding Task 11 ....................................................................................96
NUCC| Secondary II Math ii
Unit H
Ready, Set, Go! ..............................................................................................................................................97
Solutions: .......................................................................................................................................................98
H.12 POLAR PLOT...........................................................................................................................................99
Teacher Notes ................................................................................................................................................99
Mathematics Content ...................................................................................................................................101
Polar Plot A Develop Understanding Task 12 .............................................................................................102
Ready, Set, Go! ............................................................................................................................................103
Solutions: .....................................................................................................................................................104
H.13 RECTANGULAR AND POLAR COMPLEX NUMBER .....................................................................105
Teacher Notes ..................................................................................................................................................105
Mathematics Content ...................................................................................................................................107
Mathematics Content ...................................................................................................................................108
Rectangular and Polar Complex Number A Solidify Understanding Task 13 .............................................109
Ready, Set, Go! ............................................................................................................................................110
Solutions: .....................................................................................................................................................111
H.14 POWERS (MULTIPLY/DIVIDE COMPLEX NUMBERS) .................................................................112
Teacher Notes ..............................................................................................................................................112
Mathematics Content ...................................................................................................................................114
Mathematics Content ...................................................................................................................................115
Mathematics Content ...................................................................................................................................116
Powers A Develop Understanding Task 14..................................................................................................117
Ready, Set, Go! ............................................................................................................................................118
Solutions: .....................................................................................................................................................120
NUCC| Secondary II Math iii
Unit H.1
H.1 SOLVING MATRICES
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Introduce students to the process of writing a system of equations as a matrix, then
using elementary row operations within that matrix to solve the system of equations.
Core Standards Focus A.REI.8: Represent a system of linear equations as a single matrix
equation in a vector variable.
Review: As needed, review what the students learned in Honors Secondary Mathematics 1 about
matrices.
Launch (Whole Class): As a class, review how to solve the system of equations in #1 using
graphing, substitution or elimination.
NUCC| Secondary II Math 4
Unit H.1
Explore (Individual, Small Group or Pairs): In groups, or even as a whole class, see if any of
the students could think of how to represent the system of equations as a matrix. Guide them
eventually to the correct way of representing the system of equations as a matrix.
Discuss (Whole Class or Group): Teach the students about the elementary row operations that
can be performed in a matrix and demonstrate for them how to solve the system of equation
using the matrix. As a teacher, you will have to decide how or at what point you introduce
technology in helping the students with the matrices.
NUCC| Secondary II Math 5
Unit H.1
Assignment: Ready, Set, Go!
NUCC| Secondary II Math 6
Unit H.1
Mathematics Content
Cluster Title: Solve systems of equations.
Standard (+) A.REI.8: Represent a system of linear equations as a single matrix equation in a
vector variable.
Concepts and Skills to Master


Rewrite a system of linear equations in matrix form as AX=B, where X is the vector of
variables.
Solve a system of linear equations using matrices.
Critical Background Knowledge


Element of the matrix.
Order of the matrix.
Academic Vocabulary
matrix equation, vector
Suggested Instructional Strategies


Work with systems arising from contextual situations, including those where the equation
must be rewritten to create the matrix.
Compare the process of solving a system using equations to the process of solving a system
using matrices.
Skills Based Task:
Represent and solve:
2x  y  3
Problem Task:
When asked to represent the following linear
equations as a matrix equation, a student
produced the following. Will the student’s
process result in a correct answer? Justify your
answer.
y  z 1
x  2w  0
3z  w
Some Useful Websites:
NUCC| Secondary II Math 7
Unit H.1
Solving Matrices
A Develop Understanding Task 1
Name_____________________________________
Hour___________
1. Solve the following using what you’ve learned in previous years about systems of equations.
(graphing, substitution, elimination etc.)
Mr. Jones is writing a test for his English classes. The test will have multiple-choice
questions and short answer questions. The multiple-choice questions will be worth 4 points
each, the short answer questions will be worth 5 points each and the test will be worth 100
total points. Mr. Jones would like the number of short answer questions to be one more than
the number of multiple-choice questions. How many questions of each type should Mr. Jones
put on the test?
NUCC| Secondary II Math 8
Unit H.1
2. Now discuss how the information from this problem could be represented as a matrix.
3. Go over the elementary row operations for a matrix. Write them below:
NUCC| Secondary II Math 9
Unit H.1
4. Using the elementary row operations from #3, solve the matrix that you made in #2.
NUCC| Secondary II Math 10
Unit H.1
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Represent each system of equations as a matrix and use elementary row operations to solve.
1.
3x  y  17
x  y  5
2.
3 x  9 y  24
2 x  4 y  24
3.
3x  6 y  6
 x  12 y  2
4.
3x  9 y  18
3x  9 y  30
NUCC| Secondary II Math 11
Unit H.1
Set
Represent each system of equations as a matrix, and then solve using elementary row operations.
5.
6r  s  3t  28
r  s  4t  9
6r  5s  2t  14
6. Farming: Farmer John wants to plant alfalfa, beets, and corn.
Farmer John has 500 total acres to plant.
From past experience, Farmer John knows that alfalfa costs about $20/acre to plant, beets cost
about $30/acre to plant and corn costs about $10/acre to plant. Farmer John has set aside
$10,000 for his total planting costs.
From past experience, Farmer John knows that alfalfa costs about $15/acre to harvest, beets cost
about $25cre to harvest and corn costs about $15acre to harvest. Farmer John has set aside
$8,000 for his total harvesting costs.
Find out how many acres of each crop farmer John should plant to stay within his budget.
NUCC| Secondary II Math 12
Unit H.1
Go!
7.
3x  3 y  9
5 x  3 y  15
8.
6 x  2 y  26
2 x  2 y  14
9.
x  4 y  12
11x  8 y  28
10. 6 x  4 y  20
8 x  10 y  8
11. 4 x  5 y  6 z  1
5 x  6 y  5z  11
4 x  2 y  z  26
NUCC| Secondary II Math 13
Unit H.1
Solutions:
“Go!” Answers
7.
8.
9.
10.
11.
(3, 0)
(-5, -2)
(-4, 2)
(-6, -4)
(-5, 1, -4)
NUCC| Secondary II Math 14
Unit H.2
H.2 DETERMINANT AND INVERSE OF A MATRIX
Teacher Notes
Time Frame:
Materials Needed:
Purpose: In this lesson students will learn how to find the determinant and inverse of a 2x2
matrix by hand, and how to find the inverse of larger matrices using technology.
Core Standards Focus A.REI.9: Find the inverse of a matrix if it exists and use it to solve
systems of linear equations (using technology for matrices of dimension 3x3 or greater).
Launch (Whole Class): On item 1, give the students the formula to find the determinant of a
matrix:
a b 
If A  
 ,det A  ad  bc
c d 
Explore (Individual, Small Group or Pairs): On item 2, let the students apply the determinant
formula and work through the examples.
NUCC| Secondary II Math 15
Unit H.2
On item 3, show the students the formula for the inverse of a 2x2 matrix:
A1 
1  d b 
det A  c a 
On item 4, let the students apply this formula to the examples
Discuss (Whole Class or Group): On item 5, show the students how to find the inverse of a 2x2
matrix using a doubly augmented matrix and work through an example of your choice with them.
On item 6, show the kids how to use calculators or other technology to find inverses of 3x3
matrices or larger.
Assignment: Ready, Set, Go!
NUCC| Secondary II Math 16
Unit H.2
NUCC| Secondary II Math 17
Unit H.2
Mathematics Content
Cluster Title: Solve systems of equations.
Standard (+) A.REI.9: Find the inverse of a matrix if it exists and use it to solve systems of
linear equations (using technology for matrices of dimension 3 x 3 or greater).
Concepts and Skills to Master




Use the determinant to determine whether an inverse exists.
For 2 x 2 matrices, apply the following to find the inverse: For
.
Apply the augmented matrix method, by hand for 2 x 2 matrices and using technology for 3 x
3 or greater, to find A-1.
Use matrix algebra to solve AX=B as the unique solution x=A-1B.
Critical Background Knowledge


Represent systems of equations as matrices.
Find the determinant of a matrix.
Academic Vocabulary
inverse of a matrix, identify matrix, invertible, nonsingular, determinant, A-1, I, augmented
matrix
Suggested Instructional Strategies


Investigate how to solve the matrix equations AX=B without using division. Use the
definition of the inverse of a square matrix, AA-1=In or A-1 A=In, to develop the idea of the
identity matrix and why it is necessary.
Use matrices to create and crack codes.
Skills Based Task:
Use the following system to answer the
following:

Set up a matrix equation AX=B



Find A-1.
Solve for x.
State the solutions to
the system of equations.
x yz2
x  z 1
x  y  z  4
Problem Task:
As a professional code cracker, you receive an
encoded two-digit ATM pin E  2 5 that was
encoded by multiplying the original pin number
 2 3
by the matrix K= 
 . Find the decoding key
 5 8
and use it to find the original pin number P.
Teacher Hint: PK=E
Some Useful Websites:
NUCC| Secondary II Math 18
Unit H.2
The Determinant and Inverse of a Matrix
A Develop Understanding Task 2
Name_____________________________________
Hour___________
1. As a class, discuss what the determinant of a matrix is and write down the formula for how to
find the determinant of a 2x2 matrix.
2. Practice a couple of examples of finding the determinant of a 2x2 matrix.
 2 3 
EX 1: 

4 4 
1 5 
EX 2: 

5 25
3. Talk about the inverse of a 2x2 matrix now and how to find it using the determinant of the
matrix.
4. Practice a couple of examples of finding the inverse of a 2x2 matrix.
 2 3 
EX 1: 

4 4 
1 5 
EX 2: 

5 25
NUCC| Secondary II Math 19
Unit H.2
5. There is another way to find the inverse of a 2x2 matrix. Practice using what is called a
doubly augmented matrix to find an inverse.
6. For anything larger than a 2x2 matrix, finding the inverse a matrix quite a big process, so most
people use technology to help them. Practice using technology to find inverses of matrices
that are larger than 2x2.
NUCC| Secondary II Math 20
Unit H.2
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
In exercises 1 and 2, find the determinant of the given matrix.
 4 5
1. 

1 2
1 2 
2. 

 0 4 
In exercises 3 and 4, find the inverse of the given matrix using the formula A1 
 1 2 
3. 

1 8
4.
1  d b 
det A  c a 
 7 1
 1 0 


Set
In exercises 5, find the inverse of the matrix using the augmented matrix method.
8 5
5. 

3 2 
In exercises 6 and 7, find the inverse of the matrix using technology.
8 6 2 
6.  3 6 4 
 2 4 9 
  6 4 2 


7. 12 6 4 
 8 14 12 
NUCC| Secondary II Math 21
Unit H.2
Go!
In exercises 8, 9, and 10, find the determinant of each matrix.
 0 1
8. 

 3 0 
 4 1 
10. 

 2 3
4 0
9. 

 3 4 
In exercises 11, 12, and 13, find the inverse of each matrix using the formula
1  d b 
A1 
det A  c a 
 2 1
6 
11. 
9
7
 0
12. 
 10
1
6
6
13. 

 1 2 
NUCC| Secondary II Math 22
Unit H.2
In exercises 14 and 15, find the inverse of the given matrix by using the augmented matrix method.
2 6

1 4 
14. 
 2 17 

1 9 
15. 
In exercises 16 and 17, find the inverse of the given matrix using technology.
 5 1 6 
16. 5 2 5


 3 1 5
 3 2 6 
17.  3 3 4 


 2 3 5
NUCC| Secondary II Math 23
Unit H.2
Solutions:
“Go!” Answers
8. -3
9. -16
10. -10
1

2  3 
11. 

3 2 
3 

 3

12.  5

 1
 1
 4
13. 
 1
 8

1
10 

0 
3 
4 

7

8 
 2 3

14.  1

1
 2

 9 17 
15. 

 1 2 
NUCC| Secondary II Math 24
Unit H.3
H.3 CRYPTOGRAPHY WITH MATRICES
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Students will see and learn about a cryptography application of matrices and their
inverses.
Core Standards Focus A.REI.9: Find the inverse of a matrix if it exists and use it to solve
systems of linear equations.
Launch (Whole Class): Present the steps of how to code and de-code a message using matrices
and their inverses. Choose one more example and go through it on item 6.
Explore (Individual, Small Group or Pairs): Have students code messages, send them to
another group, and have the other group de-code the message.
Discuss (Whole Class or Group): Nothing here… hum….
Assignment: Ready, Set, Go! Nothing here… hum….
NUCC | Secondary II Math 25
Unit H.3
Mathematics Content
Cluster Title: Solve systems of equations.
Standard (+) A.REI.9: Find the inverse of a matrix if it exists and use it to solve systems of
linear equations (using technology for matrices of dimension 3 x 3 or greater).
Concepts and Skills to Master




Use the determinant to determine whether an inverse exists.
For 2 x 2 matrices, apply the following to find the inverse: For
.
Apply the augmented matrix method, by hand for 2 x 2 matrices and using technology for 3 x
3 or greater, to find A-1.
Use matrix algebra to solve AX=B as the unique solution x=A-1B.
Critical Background Knowledge


Represent systems of equations as matrices.
Find the determinant of a matrix.
Academic Vocabulary
inverse of a matrix, identify matrix, invertible, nonsingular, determinant, A-1, I, augmented
matrix
Suggested Instructional Strategies


Investigate how to solve the matrix equations AX=B without using division. Use the
definition of the inverse of a square matrix, AA-1=In or A-1 A=In, to develop the idea of the
identity matrix and why it is necessary.
Use matrices to create and crack codes.
Skills Based Task:
Use the following system to answer the
following:

Set up a matrix equation AX=B



Find A-1.
Solve for x.
State the solutions to
the system of equations.
x yz2
x  z 1
x  y  z  4
Problem Task:
As a professional code cracker, you receive an
encoded two-digit ATM pin E  2 5 that was
encoded by multiplying the original pin number
 2 3
by the matrix K= 
 . Find the decoding key
 5 8
and use it to find the original pin number P.
Teacher Hint: PK=E
Some Useful Websites:
NUCC | Secondary II Math 26
Unit H.3
Cryptography with Matrices
A Develop Understanding Task 3
Name_____________________________________
Hour___________
1. Assign each letter of the alphabet to a number. You could do this in many ways, one way is
as follows.
2. Think of a message that you want to code, translate it into numbers, and put it in a matrix, for
example:
8
 27
HI_FRIEND = 8 9 27 6 18 9 14 4 = 
18

14
9
6

9

4
3. Choose a key code matrix, and multiply the message matrix by it the result is your encrypted
message. Someone would have a very difficult time de-coding it unless they knew the code
matrix.
 8 9
 58 35 


27 6  5 1 147 45 
 5 1




I choose: 

 2 3 = 108 45 
2
3
18
9










14 4 
 78 26 
NUCC | Secondary II Math 27
Unit H.3
4. To decode the message, find the inverse of the code matrix.
1
 3

 13
 5 1
13 

 2 3   2
5 



 13 13 
1
5. Multiply the encrypted message by the inverse of the code message to de-code the message.
 58
147

108

 78
35
45

45

26
1 8
 3


 13
13   27
=


5  18
 2
 13 13  14
9
6

9

4
HI_FRIEND
6. As a class practice coding and de-coding another message together below to make sure you
understand the steps of the process.
7. On a separate piece of paper, make a message, choose a code matrix and send another
individual/group your coded message. See if they can de-code it using the inverse of your code
matrix.
NUCC | Secondary II Math 28
Unit H.4
H.4 CAVALIERI’S THEOREM
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Students will discover and understand Cavaleri’s Theorem for use with sphere and
solid figures. (I put in something, as this area was missing. Does this work?)
Core Standards Focus G.GMD.2 Give an informal argument using Cavalieri’s Principle for the
formulas for the volume of a sphere and other solid figures.
Launch (Whole Class): Wikipedia posts the following definition of Cavalieri’s Theorem.
In geometry, Cavalieri's principle, sometimes called the method of indivisibles, named after
Bonaventura Cavalieri, is as follows:[1]
 2-dimensional case: Suppose two regions in a plane are included between two parallel
lines in that plane. If every line parallel to these two lines intersects both regions in line
segments of equal length, then the two regions have equal areas.
 3-dimensional case: Suppose two regions in three-space (solids) are included between
two parallel planes. If every plane parallel to these two planes intersects both regions in
cross-sections of equal area, then the two regions have equal volumes.
Today Cavalieri's principle is seen as an early step towards integral calculus, and while it is
used in some forms, such as its generalization in Fubini's theorem, results using Cavalieri's
principle can often be shown more directly via integration. In the other direction,
Cavalieri's principle grew out of the ancient Greek method of exhaustion, which used limits
but did not use infinitesimals.
| Secondary II Math
1/ http://en.wikipedia.org/wiki/Cavalieri%27s_principle#cite_note-0
29
Unit H.4
After looking at the example from the previous page, discuss the other two-dimensional shapes
and discuss how they would be proven as having the same area using Cavalieri’s Theorem. Some
examples are given below.
Cavalieri’s Theorem can also be used in threedimensional objects to prove their volume is equal. This
must be done using the area of each and every cross
section must be shown to be equal. In this case a stack of
pennies would have the same volume stacked vertically
as they would if they were offset.
Explore (Individual, Small Group or Pairs): Oops … Nothing here … hum….
Discuss (Whole Class or Group): Nothing here… hum….
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 30
Unit H.4
Mathematics Content
Cluster Title: Explain volume formulas and use them to solve problems.
Standard G.GMD.2: Give an informal argument using Cavalieri’s principle for the formulas for
the volume of a sphere and other solid figures.
Concepts and Skills to Master


Understand Cavalieri’s Principle.
Use Cavalieri’s Principle to find volumes of solid figures.
Critical Background Knowledge

Volume formulas for right prisms, pyramids, cones, and cylinders.
Academic Vocabulary
Cavalieri’s Principle, cross-sections, altitude, parallel, sphere, cone, cylinder
Suggested Instructional Strategies



Compare two stacks of pennies – one stacked vertically and the other leaning. Have students
discuss properties of volume for the two stacks. Are the volumes the same? How are they
different? Explain.
Use the idea of matching horizontal cross-sections from a cone to a cylinder and then from a
cylinder to half of a sphere.
Derive the volume of a sphere using knowledge of the volume of a cone and the volume of a
cylinder.
Skills Based Task:
Use a visual model to represent how to use
Cavalieri’s Principle to find the volume of a
sphere from the volume of a cone.
Problem Task:
Give an informal argument referencing
Cavalieri’s Principle and relating the volume of a
cone to the volume of a sphere. Justify verbally
and algebraically.
Some Useful Websites:
 Cavalieri’s Principle Applet:
http://www.matematicasvisuales.com/english/html/history/cavalieri/cavalierisphere.
html
 Cavalieri’s Principle Proof (with graphics): http://blog.zacharyabel.com/tag/cavalierisprinciple/


NUCC | Secondary II Math 31
Unit H.4
Cavalieri’s Theorem
A Develop Understanding Task 4
Name_____________________________________
Hour___________
Cavalieri’s Theorem is a theorem that connects linear measurements to the Area of two
dimensional shapes. It also compares two dimensional surfaces to the volume of three
dimensional objects. An example is the area of a triangle when the height and base stay the same,
but the angle of the sides change. See below for an example.
A
B
C
D
F
E
If AB = CD = EF, and the two horizontal lines are parallel, the formula for the area of a triangle
1
A  bh would show that all three triangles have the same area. Cavalieri’s Theorem states that
2
if all of the horizontal cross sections can be shown to have an equal length, then the areas are
also equal.
H
G
A
B
I
C
L
K
J
D
F
E
That is if GH = IJ = KL above and MN = OP = QR below for any and all horizontal lines, we
can say that the areas are equal.
M
A
N
O
B
C
P
D
R
Q
E
F
NUCC | Secondary II Math 32
Unit H.4
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
1. Describe Cavalieri’s Theorem in your own words as it relates to the area of two-dimensional
objects.
Set
2. Draw two additional figures that would have the same area as the given shape that and then
use Cavalieri’s Theorem to justify that they have the same area.
Go!
3. Draw two additional figures that would have the same area as the given shape that and then
use Cavalieri’s Theorem to justify that they have the same area.
NUCC | Secondary II Math 33
Unit H.4
4. Do the following shapes have the same area? Assume that the top and bottom segments are
congruent.
5. Draw two additional figures that would have the same area as the given shape that and then
use Cavalieri’s Theorem to justify that they have the same area.
6. Draw two additional figures that would have the same area as the given shape that and then
use Cavalieri’s Theorem to justify that they have the same area.
7. Draw two additional figures that would have the same area as the given shape that and then
use Cavalieri’s Theorem to justify that they have the same area.
NUCC | Secondary II Math 34
Unit H.4
Solutions:
NUCC | Secondary II Math 35
Unit H.5
H.5 ARE ANGLES ALWAYS DEGREES?
Teacher Notes
Time Frame:
Materials Needed: Masking tape, ruler, circular objects (discs preferred, but cylinders will also
work, if necessary) – One disc per pair of students
Purpose: Students will define a radian and convert degrees to radians and vice versa.
Core Standards Focus II.H.Trigonometric Expressions: Define trigonometric ratios and write
expressions in equivalent forms.
II.H.Trigonometric Identities: Prove trigonometric identities using definitions, the Pythagorean
Theorem, or other relationships and use the relationships to solve problems.
Launch (Whole Class): Today’s lesson involves students in creating a kind of “radian ruler”
using circular objects and tape. Each pair of students will need a circular disk and some masking
tape.
Note: To increase the impact of the lesson, make sure that each pair of students in a group has a
different size circular disk. Plastic lids, such as coffee can lids, work well. Be sure that the disks
will fit on a standard sheet of paper since students are asked to trace around the circumference of
the discs.
Explore (Pairs): The instructions for this activity should be easy for groups to follow with
minimal guidance. As groups work, circulate to verify that they are following the steps correctly
and making reasonably accurate diagrams with correct estimates. When most teams have
NUCC | Secondary II Math 36
Unit H.5
finished the front of the sheet you may want to bring the class together to summarize and
reinforce the meaning of radian.
The rest of the problems can be worked individually and then compared with their partner.
Verify that students are developing an intuitive idea of the size of a radian. Just as students
understand that it takes 12 inches to equal 1 foot, we want them to understand about how many
degrees equal 1 radian.
Discuss (Whole Class or Group): The questions below the picture on the back of the task sheet
are nice discussion problems for the close of the lesson. Have students share their answers and
thoughts about how radians and degrees compare.
If you have time, you might draw students’ attention to the radian mode on their calculators.
Students probably already know that their calculators have two modes (radian and degree) and
have needed to ensure they were in degree mode in the past, but were not sure why.
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 37
Unit H.5
Mathematics Content
Cluster Title: Similarity, right triangles, and trigonometry.
Standard H.Trigonometric Expressions: Define trigonometric ratios and write trigonometric
expressions in equivalent forms.
Concepts and Skills to Master



Show how sine, cosine, and tangent are related using trigonometric identities.
Define secant, cosecant, and cotangent in terms of sine, cosine and tangent.
Define the six trigonometric functions using the unit circle.
Critical Background Knowledge


Sine, cosine, tangent
Pythagorean Theorem.
Academic Vocabulary
sine, cosine, tangent, secant, cosecant, cotangent, unit circle
Suggested Instructional Strategies


Use special right triangles to find points on the unit circle and define trigonometric values.
Connect the co-function identities with congruent triangles whose non-right angles are
switched.
Skills Based Task:
Find a value for θ for which sin θ = cos 15° is
true.
Find cos
5
.
3
Problem Task:
Prove that sin θ = cos (90° - θ) using congruent
triangles.
Prove that (tan2 θ)(cot2 θ) = 1
Some Useful Websites:
NUCC | Secondary II Math 38
Unit H.5
Mathematics Content
Cluster Title: Similarity, right triangles, and trigonometry.
Standard H.Trigonometric Identities: Prove trigonometric identities using definitions, the
Pythagorean Theorem, or other relationships and use the relationships to solve problems.
Concepts and Skills to Master



Prove trigonometric identities based on the Pythagorean Theorem.
Simplify trigonometric expressions and solve trigonometric equations using identities.
Justify half angle and double angle formulas for trigonometric values.
Critical Background Knowledge


Pythagorean Theorem.
Trigonometric definitions
Academic Vocabulary
identity, Pythagorean Theorem
Suggested Instructional Strategies

Lead students from informal arguments of geometric representations to formal two-column
and algebraic proofs.
Skills Based Task:
Prove:
sec 2   csc 2  
Problem Task:
Develop a formula for sin (x+y+z)
1
sin   cos  
2
2
Some Useful Websites:
Algebraic proofs: http://www.themathpage.com/atrig/double-proof.htm
Geometric proofs: http://www.jesystems.com/mathisfun/desc_trigonometry.asp
NUCC | Secondary II Math 39
Unit H.5
Are Angles Always Degrees?
A Develop Understanding Task 5
Name_____________________________________
Hour___________
You have probably always measured angles in degrees, but have you ever wondered why there
are 360 of them in one whole circle? The concept of a degree was used to approximate one day
in a year. Since 360 is divisible by many numbers, it was easier to use than the actual 365 (and
¼) to represent a full circle. Many primitive calendars actually used 360 days to represent one
year. Outside of its close approximation to the number of days in a year, a degree has no further
mathematical significance. Today we will investigate a different unit of angle measurement, a
radian. The concept of radian measure as it is developed from a circle is used extensively in
advanced mathematics courses, particularly calculus.
Each group of students will receive a different size circular disk and some tape.
 Take the tape and start at any location on the circumference of your disk. Wrap the tape
all the way around the circumference of your disk until you reach the place where you
started, then cut the tape. We will call this the circumference tape. Carefully remove the
circumference tape and lay it flat on your working surface so that it will be easier to work
with. Make sure you place the tape on a location from which it can be easily removed.
 Measure the length of the radius of your disk. Start at one end of the circumference tape,
mark where one radius length ends. Continue marking at one radius intervals until you
get to the end of your circumference tape.
 Count the number of radius lengths it took to get all the way around your circle. Did it
come out to be an exact whole number? Compare the number of radius lengths you got
with the number your team members got. Where their numbers consistent with yours?
 Apparently there is a relationship between a circle’s circumference and its radius. Using
what you know about a radius length and how it relates to the circumference of a circle,
explain your answer to the bullet above.
 Place your disk on a sheet of paper and trace around it. (If your disc is too large, it’s okay
to use any of your classmates’ discs that do fit.) Carefully mark the center (P) of the
circle on your paper. Place the circumference tape back on the disk. Transfer the radius
marks onto the traced circle. Label the position for the start of the tape as A and the first
radius mark as B. Draw segments PA and PB .


If we assume the radius of the circle you just drew is one unit, what is the length of AB ?
(Note that AB is actually an arc (a part of the circumference of the circle) and not a
segment.) You have just defined a new angle measure that is based on the length of a
radius. As a result, it is more easily applied to higher mathematics than a degree, which is
based (loosely) on the number of days in a year. A RADAIN is nothing more than a
RADIUS ANGLE! In other words, if you take the length of 1 radius and lay it on the
circumference of the circle, the angle that is formed by the arc’s endpoints and the center
of the circle has a measure of 1 radian, which is equivalent to?
Assuming that the radius of your disc is one unit, what is its exact circumference?
NUCC | Secondary II Math 40
Unit H.5

Looking again at the circle you traced on your paper form today’s assignment. Let the
length of the radius of this circle be equal to 1 unit (A circle with a radius one is called
the unit circle.) How many degrees did you go around the circle in order to mark off 2
radius lengths (radians) on your paper? (Do not forget the units.) A radian is a unit of
angle measure. The central angle of a full circle is 360 degrees. How many radians is
this? What whole number of radian best approximates the central angle of a full circle?
When we measure angles in the unit circle, we measure from standard
position (the positive x-axis) as shown in the diagram.
Using the unit circle shown mark an angle with a measure of
approximately 3 radians starting from standard position. Where is the angle
located?
We want to develop a method for comparing degrees and radians. How
many degrees of angle measure are there in one half of a circle? About how
many radians of angle measure is there in one half of a circle? Exactly how many radians of
angle measure are there in one half of circle?
Write an equation that relates radian to degrees using half the circle.
Use your relationship to convert

radians into degrees.
3
How many radians are there in 200˚? Give an exact value using π in your answer.
You should now have an intuitive sense of about how many degrees are in one radian, since you
know there are about π radians in one half of a circle and about 2π radians in a whole circle.
Now, find how many degrees in one radian, accurate to three decimal places. How many radians
are in one degree, accurate to three decimal places? Which is the larger unit, a degree or a
radian? Are they very close in size, or very different?
NUCC | Secondary II Math 41
Unit H.5
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Convert the degree measure to radians. Use π to give the exact value.
1.
180˚
2. -36˚
3.
135˚
Convert the radian measure to degrees. Leave π in your answer, if necessary, to give the
exact value.
3
7
4.
5. 
6. 2
2
6
Set
7. A wheel is spinning at 500 revolutions per minute.
Remember 1 revolution equals 2π radians. How many
radians per second is that? Give exact value.
8. Andreas, the managing artist for the Mathematical
Broadcasting Company (MBC), is designing a new logo
for its on-screen mascot. Since you are working as his
assistant, he has given you the task of labeling the radian
measure for all of the marked angles around the curved
perimeter on the semicircular design he has created. He has
already labeled 0 and π for you. Label all of the distances
from 0 along the semicircle in terms of π. Write your results
as simplified fractions.
NUCC | Secondary II Math 42
Unit H.5
Go!
9. Label the dots on the unit circle if each dot is half way between the coordinate axes. Label
them in degrees and radians.
10. Label the dots on the unit circle if each dot is a third of the way between the coordinate axes.
Label them in degrees and radians.
NUCC | Secondary II Math 43
Unit H.5
Solutions:
NUCC | Secondary II Math 44
Unit H.6
H.6 UNIT CIRCLE
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Students will learn the trigonometric relationships and how they relate to special right
triangles on the unit circle.
Core Standards Focus H.Trigonometric Expressions: Define trigonometric ratios and write
expressions in equivalent forms.
H.Trigonometric Identities: Prove trigonometric identities using definitions, the Pythagorean Theorem,
or other relationships and use the relationships to solve problems.
Launch (Whole Class): Have students label the dots around the unit circle you can and teach
them the relationship on the unit circle with regards to special right triangles. You will also
introduce three new trigonometric relationships. (secant, cosecant, and cotangent). I have a
power point that can be use while teaching the unit circle or you can adapt the material to the
way you like to teach.(where is it?)
Explore (Pairs): Not much exploring with this lesson pretty much direct instruction.
Discuss (Whole Class or Group): hum…. Nothing here…..
NUCC | Secondary II Math 45
Unit H.6
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 46
Unit H.6
Mathematics Content
Cluster Title: Similarity, right triangles, and trigonometry.
Standard H.Trigonometric Expressions: Define trigonometric ratios and write trigonometric
expressions in equivalent forms.
Concepts and Skills to Master



Show how sine, cosine, and tangent are related using trigonometric identities.
Define secant, cosecant, and cotangent in terms of sine, cosine and tangent.
Define the six trigonometric functions using the unit circle.
Critical Background Knowledge


Sine, cosine, tangent
Pythagorean Theorem.
Academic Vocabulary
sine, cosine, tangent, secant, cosecant, cotangent, unit circle
Suggested Instructional Strategies


Use special right triangles to find points on the unit circle and define trigonometric values.
Connect the co-function identities with congruent triangles whose non-right angles are
switched.
Skills Based Task:
Find a value for θ for which sin θ = cos 15° is
true.
Find cos
5
.
3
Problem Task:
Prove that sin θ = cos (90° - θ) using congruent
triangles.
Prove that (tan2 θ)(cot2 θ) = 1
Some Useful Websites:
NUCC | Secondary II Math 47
Unit H.6
Mathematics Content
Cluster Title: Similarity, right triangles, and trigonometry.
Standard H.Trigonometric Identities: Prove trigonometric identities using definitions, the
Pythagorean Theorem, or other relationships and use the relationships to solve problems.
Concepts and Skills to Master



Prove trigonometric identities based on the Pythagorean Theorem.
Simplify trigonometric expressions and solve trigonometric equations using identities.
Justify half angle and double angle formulas for trigonometric values.
Critical Background Knowledge


Pythagorean Theorem.
Trigonometric definitions
Academic Vocabulary
identity, Pythagorean Theorem
Suggested Instructional Strategies

Lead students from informal arguments of geometric representations to formal two-column
and algebraic proofs.
Skills Based Task:
Prove:
sec 2   csc 2  
Problem Task:
Develop a formula for sin (x+y+z)
1
sin   cos  
2
2
Some Useful Websites:
Algebraic proofs: http://www.themathpage.com/atrig/double-proof.htm
Geometric proofs: http://www.jesystems.com/mathisfun/desc_trigonometry.asp
NUCC | Secondary II Math 48
Unit H.6
Unit Circle
A Develop Understanding Task 6
Name_____________________________________

sin 
cos
tan
Hour___________
csc
sec
cot 
0

6

4

3

2
NUCC | Secondary II Math 49
Unit H.6
Ready, Set, Go!
Name__________________________________________________
E(
(
(
(
(
,
,
(
)
)F
D(
,
)G
)
C(
,
)H
,
)J
θ
)
B(
)I
(
Point
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
,
,
,
,
)
A(
,
,
)
P(
,
)K
(
O(
,
N(
)L
degrees radians
M(
sin θ
cos θ
Hour____________
,
,
,
)
)
)
)
tan θ
csc θ
sec θ
cot θ
NUCC | Secondary II Math 50
Unit H.6
Solutions:
NUCC | Secondary II Math 51
Unit H.7
H.7 SIMPLIFY TRIGONOMETRIC EXPRESSIONS
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Students will eventually use these simplified expressions in equations to solve
trigonometric problems. This will be in a few lessons. For now, they should focus on identifying
common trends and patterns as they use their formula sheets to convert trigonometric identities
to simpler forms.
Core Standards Focus H.Trigonometric Expressions: Define trigonometric ratios and write
expressions in equivalent forms.
H.Trigonometric Identities: Prove trigonometric identities using definitions, the Pythagorean Theorem,
or other relationships and use the relationships to solve problems. (H.T.1 – I couldn’t find, so Allen
suggested these were what you are intending to use, ARE THEY WHAT YOU INTENDED?)
Launch (Whole Class): Go over the examples with students and identify what strategies are
useful in simplifying the given trigonometric expression. The following examples are worked
completely so that you can follow what has been done to simplify the trigonometric identities.
There are a couple of examples per problem type. The most common strategies to use include:
finding common denominators, factoring, and using the reciprocal identities so that other terms
will cancel. Students may use the formula sheet at your discretion.
Note: In Utah State University’s Math 1060 course, students are allowed to use the formula sheet
on all exams and the final.
Examples:
Reciprocal Identities and Quotient Identities:
If cot( x) 
5
, find tan( x) .
7
If sin( ) 
If csc( x ) 
25
25
and sec( x ) 
, find tan( x) .
7
24
If cot( x) 
3
, find csc( ) .
5
2
5 5
and sin( x) 
5
, find cos( x) .
3
NUCC | Secondary II Math 52
Unit H.7
cos( x)
sin( x)
2
cos( x)

5 5
5
3
cot( x) 
sin( x)
cos( x)
7
tan( x)  25
24
25
7
tan( x) 
24
tan( x) 
2 5
 5 5 cos( x)
3
2 5
 cos( x)
35 5
2
 cos( x)
15
Examples: Pythagorean Identities:
If tan( x)  8 and sin( x)  0 , find sin( x ) and cos( x) .
tan 2 ( x)  1  sec 2 ( x)
(8) 2  1  sec 2 ( x)
65  sec 2 ( x)
 65  sec( x)

65
 cos( x)
65
 65
 cos( x)
65
sin( x)
tan( x) 
cos( x)
sin( x)
8 
 65
65
8 65
 sin( x)
65
If sin( x) 
1
and cos( x)  0 , find cot( x) and sec( x) .
6
NUCC | Secondary II Math 53
Unit H.7
sin 2 ( x)  cos 2 ( x)  1
1
2
6
 cos 2 ( x)  1
cos 2 ( x)  35
36
cos( x)   35
6
cos( x)  35
6
cos( x)
cot( x) 
sin( x)
35
6  35
6
1
1
sec( x) 

6
cos( x) 1
6
cot( x) 
1
NUCC | Secondary II Math 54
Unit H.7
Examples of Cofunction and Even/Odd Identities


If tan( )  1.28 , find cot    
2




If sin( )  0.37 , find cos    
2


= 1.28
= -0.37


If sec( )  5.42 , find csc    
2



If tan    1.37 , find cot    
2

 
 


sec      sec       
2 
2

 
 
 


cot      cot       
2 
2

 


 sec    
2



  cot    
2

=5.42
=1.37
Explore (Pairs): hum…. Nothing here…..
Discuss (Whole Class or Group): Here are some more examples combining the identities.
1.

csc( x)sec( x)  tan( x)
1
1
sin( x)

sin( x) cos( x) cos( x)
2

1
sin ( x)

sin( x) cos( x) sin( x) cos( x)

1  sin ( x)
sin( x) cos( x)
2
1  sin 2 ( x)
csc2 ( x)  1
cos 2 ( x)

cot 2 ( x)
cos 2 ( x)

1
cos 2 ( x)
sin 2 ( x)
2
cos ( x)
sin( x) cos( x)
cos( x)

sin( x)
 cot( x)

2.

cos 2 ( x) sin 2 ( x)
1
cos 2 ( x)
 sin 2 ( x)
NUCC | Secondary II Math 55
Unit H.7
3.
cos( x)
cos( x)

sec( x)  1 sec( x)  1
cos( x)  sec( x)  1
cos( x)(sec( x)  1)

(sec( x)  1)(sec( x)  1) (sec( x)  1)(sec( x)  1)
cos( x)sec( x)  cos( x) cos( x)sec( x)  cos( x)


sec 2 ( x)  1
sec 2 ( x)  1
2cos( x)sec( x)

tan 2 ( x)
2

tan 2 ( x)

 2cot 2 ( x)
4.
cot( x)
sec( x)  tan( x)
cot( x)
sec( x)  tan( x)
sec( x)  tan( x) sec( x)  tan( x)
cot( x)sec( x)  cot( x) tan( x)

sec 2 ( x)  tan 2 ( x)
cos( x) 1
cos( x) sin( x)

sin( x) cos( x) sin( x) cos( x)

1  tan 2 ( x)  tan 2 ( x)
1

1
sin( x)
 csc( x)  1

Trigonometric equations can be solved, but often are large and unwieldy. You can use
trigonometric identities to simplify the equation, therefore making it easier to solve.
If you are using the formula sheet provided, these lessons are split into two sections. The first
assignment will use the Reciprocal Identities, Quotient Identities, Pythagorean Identities,
Even/Odd Functions, and Cofunction Identities. A list of those identities is given below.
Reciprocal Identities
Quotient Identities
1
csc x
1
cos x 
sec x
1
tan x 
cot x
tan x 
sin x 
1
sin x
1
sec x 
cos x
1
cot x 
tan x
csc x 
Pythagorean Identities
sin 2 x  cos 2 x  1
1  tan 2 x  sec 2 x
1  cot 2 x  csc 2 x
sin x
cos x
cot x 
cos x
sin x
Even/Odd Functions
sin(  x)   sin x
csc(  x)   csc x
cos(  x)  cos x
sec(  x)  sec x
tan(  x)   tan x
cot(  x)   cot x
NUCC | Secondary II Math 56
Unit H.7
Cofunction Identities


sin   x   cos x
2



cos   x   sin x
2



tan   x   cot x
2



cot   x   tan x
2



csc   x   sec x
2



sec   x   csc x
2

Assignment: Ready, Set, Go!
NUCC | Secondary II Math 57
Unit H.7
NUCC | Secondary II Math 58
Unit H.7
Mathematics Content
Cluster Title: Similarity, right triangles, and trigonometry.
Standard H.Trigonometric Expressions: Define trigonometric ratios and write trigonometric
expressions in equivalent forms.
Concepts and Skills to Master



Show how sine, cosine, and tangent are related using trigonometric identities.
Define secant, cosecant, and cotangent in terms of sine, cosine and tangent.
Define the six trigonometric functions using the unit circle.
Critical Background Knowledge


Sine, cosine, tangent
Pythagorean Theorem.
Academic Vocabulary
sine, cosine, tangent, secant, cosecant, cotangent, unit circle
Suggested Instructional Strategies


Use special right triangles to find points on the unit circle and define trigonometric values.
Connect the co-function identities with congruent triangles whose non-right angles are
switched.
Skills Based Task:
Find a value for θ for which sin θ = cos 15° is
true.
Find cos
5
.
3
Problem Task:
Prove that sin θ = cos (90° - θ) using congruent
triangles.
Prove that (tan2 θ)(cot2 θ) = 1
Some Useful Websites:
NUCC | Secondary II Math 59
Unit H.7
Mathematics Content
Cluster Title: Similarity, right triangles, and trigonometry.
Standard H.Trigonometric Identities: Prove trigonometric identities using definitions, the
Pythagorean Theorem, or other relationships and use the relationships to solve problems.
Concepts and Skills to Master



Prove trigonometric identities based on the Pythagorean Theorem.
Simplify trigonometric expressions and solve trigonometric equations using identities.
Justify half angle and double angle formulas for trigonometric values.
Critical Background Knowledge


Pythagorean Theorem.
Trigonometric definitions
Academic Vocabulary
identity, Pythagorean Theorem
Suggested Instructional Strategies

Lead students from informal arguments of geometric representations to formal two-column
and algebraic proofs.
Skills Based Task:
Prove:
sec 2   csc 2  
Problem Task:
Develop a formula for sin (x+y+z)
1
sin   cos  
2
2
Some Useful Websites:
Algebraic proofs: http://www.themathpage.com/atrig/double-proof.htm
Geometric proofs: http://www.jesystems.com/mathisfun/desc_trigonometry.asp
NUCC | Secondary II Math 60
Unit H.7
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
1. If cot( x) 
5
, find tan(x).
7
1
35
2. If cos( x)  ,sin( x) 
, find cot(x).
6
6
3. Find sec(x) and cos(x) given tan( x)  5,cos( x)  0.


4. If cos( )  0.61 , find sin    
2

5. Simplify. csc  x   cos  x  cot  x 
Set
6. If cos( x ) 
2
, find sec(x).
3
7. If sec( x )  2, tan( x )  3 , find sin(x).
8
8. Find cos(x) and tan(x) given csc( x )  , tan( x )  0.
3


9. If cot( )  1.35 , find tan    
2

NUCC | Secondary II Math 61
Unit H.7
10. Simplify.
csc( x )cos( x )  cot( x )
sec( x )cot( x )
Go!
1
11. If tan( x )  , find cot(x).
5
7
2 10
12. If csc( x )  ,cot( x ) 
, find sec(x).
3
3
13. Find sin(x) and cos(x) given cot( x )  8,csc( x )  0.
14. Find tan(x) and csc(x) given cos( x ) 


15. If tan( )  1.52 , find cot    
2

1
,sin( x )  0.
4


16. If sin( )  5.42 , find cos    
2

Simplify.
17. sec( x )cot( x )  sin( x )
18.
sec( x )csc( x )  tan( x )
sec( x )csc( x )
NUCC | Secondary II Math 62
Unit H.7
19. cot( x )  csc2 ( x)cot( x)
20.
1  cos( x )
sin( x )

tan( x )
1  cos( x )
21.
1
1

sec( x )  1 sec( x )  1
22.
sin( x )
sin( x )

csc( x )  1 csc( x )  1
23.
sin( x )
csc( x )  cot( x )
24.
sin( x )
1  sec( x )
25.
sin( x ) tan( x )
cos( x )  1
26.
sin( x )
1  sec( x )
sin( x) tan( x)
27. cos( x)  1
28. cos( x )  tan( x )sin( x )
29. tan( x )  csc( x )sec( x )
30. csc( x) tan 2 ( x)  sec2 ( x)csc( x)
NUCC | Secondary II Math 63
Unit H.7
Solutions:
“Ready, Set, Go!” Answers
7
1.
5
35
2.
35
3.
sec( x)  26,cos( x) 
4.
5.
-0.61
sin(x)
3
2
3
2
6.
7.
55
3 55
, tan( x) 
8
55
8.
cos( x) 
9.
10.
11.
-1.35
2cos(x)
5
7 10
20
 65 8 65
65 , 65
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
15
1
26

26
26
4 15
, 15
1.52
0.18
cos(x)cot(x)
cos 2 ( x)
 cot 3 ( x)
sin(x)
2cot(x)csc(x)
2 tan 2 ( x)
1 + cos(x)
sec2 ( x)(csc( x)  1)
sec(x)(csc(x)-1)
-cot(x)[cos(x) + 1]
sec(x)-1
sec(x)
-cot(x)
-csc(x)
NUCC | Secondary II Math 64
Unit H.8
H.8 INVERSE OPERATIONS
Teacher Notes
Time Frame:
Materials Needed:
Purpose: From previous chapters, the students should know that sine and cosine are cyclical
functions. That is sin(30) = sin(390) = sin (750) = …. As students start to solve these equations,
they should know that they are finding only a few of the answers and that there are really an
infinite number of solutions that could be found. They may also need a review of special right
triangles and the unit circle and what each of the x and y values of the points on the unit circle
stand for. That is cos = x, sin = y, and tan = y/x. The unit circle could be copied as the back page
of the formula sheet. Students also may or may not be familiar with radian angle measures.
Core Standards Focus hum….
Launch (Whole Class): Review some simple solving and demonstrate the idea of inverse
operations. Every mathematical operation has its inverse, add and subtract, multiply and divide,
etc. Students will be finding the inverse of trigonometric functions by using the unit circle to
identify specific values. There are several different types of equations and strategies involved.
Many are related to all of the solving that they have done before. The key is to isolate the trig
value so that it can be compared to the unit circle. Here are a few examples.
Solve by isolating:
Solve for 0  x  2
Solve for 0  x  2
2 tan( x)  3  tan( x)
4sin( x)  2sin( x)  2
tan( x)  3  0
tan( x)  3
x
 4
,
3 3
2sin( x)  2
sin( x) 
x
 3
2
2
,
4 4
Solve by using square roots:
Solve for 0  x  2
Solve for 0  x  2
4sin 2 ( x)  1  4
3cot 2 ( x)  4  7
NUCC | Secondary II Math 65
Unit H.8
4sin 2 ( x)  3
3
sin ( x) 
4
 3
sin( x) 
2
 2 4 5
x , , ,
3 3 3 3
2
3cot 2 ( x)  3
cot 2 ( x)  1
cot( x)  1
x
 3 5 7
, , ,
4 4 4 4
Solve by factoring:
Solve for 0  x  2
Solve for 0  x  2
1
cos( x)sin( x)  cos( x)
2
4cos 2 ( x)  2cos( x)  2 2 cos( x)  2
1
cos( x)sin( x)  cos( x)  0
2
1

cos( x) sin( x)    0
2

1
cos( x)  0 or sin( x) 
2
 3  5
x , , ,
2 2 6 6
4cos 2 ( x)  2cos( x)  2 2 cos( x)  2  0
2cos( x)  2cos( x)  1  2  2cos( x)  1
 2cos( x)  1 2cos( x) 
2 
1
2
or cos( x) 
2
2
2 4  7
x
, , ,
3 3 4 4
cos( x) 
Solve by rewriting as a single trig function:
Solve for 0  x  2
Solve for 0  x  2
2cos2 ( x)  sin( x)  1  0
1  cos( x)  2sin 2 ( x)
2[1  sin 2 ( x)]  sin( x)  1  0
1  cos( x)  2 1  cos 2 ( x) 
2  2sin 2 ( x)  sin( x)  1  0
1  cos( x)  2  2cos 2 ( x)
0  2sin 2 ( x)  sin( x)  1
2cos 2 ( x)  cos( x)  1  0
 2sin( x)  1sin( x)  1  0
sin( x) 
x
1
2
or sin( x)  1
 5 3
, ,
6 6 2
 2cos( x)  1cos( x)  1  0
1
or cos( x)  1
2
2 4
x
, ,0
3 3
cos( x) 
NUCC | Secondary II Math 66
Unit H.8
Explore (Individual, Small Group or Pairs): Oops … Nothing here … hum….
Discuss (Whole Class or Group): After working through the examples above, discuss how
solving these equations is related to and different from any other equation solved. Possible
discussions points could include: solving always finds the answer that you can plug in and make
the statement true, while a difference would be that these are only a few of the infinite solutions.
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 67
Unit H.8
Mathematics Content
Cluster Title: Explain volume formulas and use them to solve problems.
Standard ?.??.?: Give an informal argument using Cavalieri’s principle for the formulas for the
volume of a sphere and other solid figures. (Where is the standard?)
Concepts and Skills to Master


Understand Cavalieri’s Principle.
Use Cavalieri’s Principle to find volumes of solid figures.
Critical Background Knowledge

Volume formulas for right prisms, pyramids, cones, and cylinders.
Academic Vocabulary
Cavalieri’s Principle, cross-sections, altitude, parallel, sphere, cone, cylinder
Suggested Instructional Strategies



Compare two stacks of pennies – one stacked vertically and the other leaning. Have students
discuss properties of volume for the two stacks. Are the volumes the same? How are they
different? Explain.
Use the idea of matching horizontal cross-sections from a cone to a cylinder and then from a
cylinder to half of a sphere.
Derive the volume of a sphere using knowledge of the volume of a cone and the volume of a
cylinder.
Skills Based Task:
Use a visual model to represent how to use
Cavalieri’s Principle to find the volume of a
sphere from the volume of a cone.
Problem Task:
Give an informal argument referencing
Cavalieri’s Principle and relating the volume of a
cone to the volume of a sphere. Justify verbally
and algebraically.
Some Useful Websites:
 Cavalieri’s Principle Applet:
http://www.matematicasvisuales.com/english/html/history/cavalieri/cavalierisphere.
html
 Cavalieri’s Principle Proof (with graphics): http://blog.zacharyabel.com/tag/cavalierisprinciple/


NUCC | Secondary II Math 68
Unit H.8
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Solve for all values of 0  x  2
1. 5sin( x)  2  sin( x)
2. 3csc( x)  2csc( x)  2
3. 2sin( x)   sin( x) cos( x)
4. 2  2cos2 ( x)  sin( x)  1
Set
Solve for all values of 0  x  2
5. 5  sec2 ( x)  3
6. 11  3csc2 ( x)  7
7. sin 4 ( x)  2sin 2 ( x)  3  0
8. csc( x)  cot( x)  1
NUCC | Secondary II Math 69
Unit H.8
Go!
Solve for all values of 0  x  2
9. 2  4cos2 ( x)  1
10. 2  10sec( x)  4  9sec( x)
11. 6 tan 2 ( x)  2  4
12. csc2 ( x)  csc( x)  9  11
13. 2sin 2 ( x)  sin( x)  1
14. 1  cot 2 ( x)  csc( x)
15. tan 2 ( x)  1  sec( x)
16. cos( x)  4  sin( x)  4
17. 3sin( x)  3  3cos( x)
18. sec2 ( x) tan 2 ( x)  3sec2 ( x)  2 tan 2 ( x)  3
NUCC | Secondary II Math 70
Unit H.8
Solutions:
Answers to Honors: 8 “Go” Questions.
7 11
1.
,
6 6
 3
2.
,
4 4
3.
0,  , 2
 7 11
4.
,
,
2 6 6
 3 5 7
5.
, , ,
4 4 4 4
 2 4 5
6.
,
,
,
3 3 3 3
 3
7.
,
2 2

8.
2
 2 4 5
9.
,
,
,
3 3 3 3
2 4
10.
,
3 3
 3
11.
,
4 4
3  5
12.
, ,
2 6 6
 7 11
13.
,
,
2 6 6
 7 11
14.
,
,
2 6 6
2 4
15.
0,
,
, 2
3 3
 5
16.
,
4 4

17.
0, , 2
2
0,  , 2
18.
NUCC | Secondary II Math 71
Unit H.9
H.9 PROVE ADDITION AND SUBTRACTION
FORMULAS FOR SINE, COSINE, AND TANGENT
Teacher Notes
Time Frame:
Materials Needed:
Purpose: The addition and subtraction formulas are used to find the exact values of
trigonometric functions that are not on the unit circle. For example, the exact value of sin(15 )
can be found to be
6 2
.
4
A review of the rules of simplifying radical expressions would be useful.
5
3
2
5 3
2 5
6 3
7
4 3
Core Standards Focus ?.???.? Hum….
Launch (Whole Class): Go through the following examples and demonstrate the methods and
processes for finding the sum and difference and then applying the formulas to get the exact
values of the trigonometric functions.
Example: Find an appropriate sum or difference from values on the unit circle. There are
many options and a few are given below.
15
75
345
285
45° - 30°
135° - 120°
315° - 300°
45° + 30°
135° - 60°
300° - 225°
300° - 45°
135° + 210°
315° + 30°
225°+ 60°
315° - 30°
135° + 150°
19
12
7
12
1
12
5
12
5 

4 3
7 

4 6
3 5

4
6
3 

4 6




3 4
5 

6 4
3





4
6 4
3 

4 3
5 5

3
4

4 6
5 3

6
4


NUCC | Secondary II Math 72
Unit H.9
Introduce the identities and identify similarities and differences amongst them.
Sum/Difference Identities:
Sin(A + B) = sinA cosB + cosA sinB
Cos(A + B) = cosA cosB – sinA sinB
Sin(A – B) = sinA cosB – cosA sinB
Cos(A – B) = cosA cosB + sinA sinB
tan( A  B) 
tan A  tan B
1  tan A tan B
tan( A  B) 
tan A  tan B
1  tan A tan B
Note: I have found that it is useful at the beginning to find the same solution to a problem in
multiple ways to justify that it doesn’t matter which pair of sums or differences are chosen.
Examples: Two sets of work for the same problem.
sin(15 )
sin(15 )
sin(45  30 )
sin(135  120 )
sin(45 ) cos(30 )  cos(45 )sin(30 )
sin(135 ) cos(120 )  cos(135 )sin(120 )
2 3
2 1

2 2
2 2
6
2

4
4
6 2
4
2 1  2 3

2 2
2
2
 2  6

4
4
6 2
4
Example: Finding exact values of trigonometric values not on the unit circle (degrees).
cos(75 )
tan(255 )
tan(120  135 )
cos(45  30 )
cos(45 ) cos(30 )  sin(45 )sin(30 )
2 3
2 1

2 2
2 2
6
2

4
4
6 2
4
tan(120 )  tan(135 )
1  tan(120 ) tan(135 )
 3  (1)
1  [( 3)(1)]
 3  1 (1  3)
1  3 (1  3)
1  3
2
Example: Finding exact values of trigonometric values not on the unit circle (radians).
NUCC | Secondary II Math 73
Unit H.9
 5 
cos 

 12 
 2
cos 
 3
 2
cos 
 3



4

 
 2    
 cos    sin 
 sin  

4
 3  4
1 2
3 2

2 2
2 2
 2
6

4
4
6 2
4
 7 
tan 

 12 
 3  
tan 
 
 4 6
 3 
 
tan    tan  
 4 
6
3

   
1  tan   tan  
 4  6
1 
3
3
 3
1  [( 1) 
]
3


1
3

1
3
1
3

1 3


3  3 3  3
3 3 3 3


12  4 3 6  2 3

6
3
Example: Simplify a trigonometric expression.
NUCC | Secondary II Math 74
Unit H.9
cos  62  cos 11   sin  62  sin 11

tan 17   tan  32

1  tan 17  tan  32 
cos  A  B 
tan  A  B 
cos(62  11 )
tan(17  32 )
cos(51 )
tan( 15 )
sin  A  B 
 
 2 
tan    tan 

6
 3 
    2 
1  tan   tan 

6  3 
tan  A  B 
 3  
sin 
 
 4 3
 13 
sin 

 12 
  2 
tan  

6 3 
 5 
tan 

 6 
 3 
 
 3    
sin 
 cos    cos 
 sin  
 4 
3
 4  3
Explore (Individual, Small Group or Pairs): Oops … Nothing here … hum….
Discuss (Whole Class or Group): Nothing here… hum….
NUCC | Secondary II Math 75
Unit H.9
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 76
Unit H.9
Mathematics Content
Cluster Title: Explain volume formulas and use them to solve problems.
Standard ?.???.?: Give an informal argument using Cavalieri’s principle for the formulas for
the volume of a sphere and other solid figures. (What Standard???)
Concepts and Skills to Master


Understand Cavalieri’s Principle.
Use Cavalieri’s Principle to find volumes of solid figures.
Critical Background Knowledge

Volume formulas for right prisms, pyramids, cones, and cylinders.
Academic Vocabulary
Cavalieri’s Principle, cross-sections, altitude, parallel, sphere, cone, cylinder
Suggested Instructional Strategies



Compare two stacks of pennies – one stacked vertically and the other leaning. Have students
discuss properties of volume for the two stacks. Are the volumes the same? How are they
different? Explain.
Use the idea of matching horizontal cross-sections from a cone to a cylinder and then from a
cylinder to half of a sphere.
Derive the volume of a sphere using knowledge of the volume of a cone and the volume of a
cylinder.
Skills Based Task:
Use a visual model to represent how to use
Cavalieri’s Principle to find the volume of a
sphere from the volume of a cone.
Problem Task:
Give an informal argument referencing
Cavalieri’s Principle and relating the volume of a
cone to the volume of a sphere. Justify verbally
and algebraically.
Some Useful Websites:
 Cavalieri’s Principle Applet:
http://www.matematicasvisuales.com/english/html/history/cavalieri/cavalierisphere.
html
 Cavalieri’s Principle Proof (with graphics): http://blog.zacharyabel.com/tag/cavalierisprinciple/


NUCC | Secondary II Math 77
Unit H.9
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Find two pairs of numbers from the unit circle that add or subtract to the given number.
23
1. 105
2.
12
3.
Find the tan(360 ) by using 180 and 180 as A and B.
Set
Simplify the trigonometric expression.

 
 2 
tan    tan 

6
 3 
5.
    2 
1  tan   tan 

6  3 
4.
cos  62  cos 11   sin  62  sin 11
6.
Evaluate sin(60 ) by using 30 and 30 as A and B.
NUCC | Secondary II Math 78
Unit H.9
Go!
Evaluate exactly the following trigonometric expressions.
7.
cos(75 )
 11 
8. sin 

 12 
9.
tan(345E)
10. sin  210E
 17 
11. cos 

 12 
 
12. tan  
 12 
13. cos(285E)
 5 
14. sin  
 12 
Simplify the trigonometric expression.
15. cos(38E)cos(54E)  sin(38E) sin(54E)
 3 
 5 
tan    tan  
 2 
 4 
16.
3

   5 
1  tan   tan  
 2   4 
NUCC | Secondary II Math 79
Unit H.9
 5
17. sin 
 3

 3
 cos 

 4

 5
  cos 

 3
  3 
 sin  
  4 
18.
tan  22   tan  34

1  tan  22  tan  34 
NUCC | Secondary II Math 80
Unit H.9
Solutions:
“Ready, Set, Go!” Answers
1.
45 + 60 or 135 – 30
7 3
2 5
2.
or


6
4
3
4
00 0
3.
 0
1 0 0 1
4.
cos(51 )
 5 
5.
tan 

 6 
sin(30  30 )
sin(30 ) cos(30 )  cos(30 )sin(30 )
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
1 3
3 1

2 2
2 2
3
3

4
4
3
2
6 2
4
6 2
4
2  3
1
2
2 6
4
2 3
6 2
4
6 2
4
cos(16 )
 
tan  
4
 11 
sin 

 12 
tan(56 )
NUCC | Secondary II Math 81
Unit H.10
H.10 COMPLEX CONJUGATES
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Students will be introduced to complex conjugates by solving quadratic equations and then
will learn how to write the equations for quadratic function given the roots. Students will practice
operations with complex numbers.
Core Standards Focus H.N.CN.3 Find the conjugate of a complex number; use conjugates to find
moduli and quotients of complex numbers.
H.N.CN.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers
geometrically on the complex plane; use properties of this representation for computation. For
example, (–1 + √3i)3 = 8 because (–1 + √3i) has modulus 2 and argument 120°.
Launch (Whole Class): Today students will investigate the relationship between complex
solutions to quadratic equations and the equations that they come from and, in general, learn
more about these new numbers.
Explore (Small Group): Without further introduction, teams should be able to start immediately
with problems. Circulate and listen as they are working and discussing patterns. Ask them what
happens when they multiply or add two complex numbers that are not complex conjugates.
NUCC | Secondary II Math 82
Unit H.10
The back page asks them to find the quadratic equation corresponding to a given pair of complex
conjugate solutions. To help students you might ask, “Is there a relationship between the
solutions and the equation in standard or graphing form” or “What if you set x equal to a
solution? Some students should think of using factors. You might suggest they try that for two
integers as solutions and then see if they can extend the idea. Some additional questions, “What if
the solutions were 2 and -5? How could you get an equation? Can you use that method with
complex numbers? What if you let x equal the complex number?”
As you circulate, encourage groups that have come up with several different approaches to write
general directions about how to use each method.
Discuss (Whole Class or Group): When most groups have finished the problem task sheet, have
teams offer summary statements to see how many different methods the whole class can
generate. If the method of using factors has not been demonstrated, demonstrate setting up the
factors as another alternative.
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 83
Unit H.10
Mathematics Content
Cluster Title: Perform arithmetic operations with complex numbers.
Standard H.N.CN.3: Find the conjugate of a complex number, use conjugates to find moduli
and quotients of complex numbers
Concepts and Skills to Master



Given a complex number, determine the conjugate.
Define the modulus of a complex number as the positive square root of the sum of the
squares of the real and imaginary parts of a complex number.
Use conjugates to express quotients of complex numbers in standard form.
Critical Background Knowledge




Complex numbers
Complex plane
Rationalizing denominators
i 2  1
Academic Vocabulary
conjugate, modulus, magnitude, complex plane
Suggested Instructional Strategies


Use properties of difference of two squares to find the modulus.
Relate the modulus visually using vectors.
Skills Based Task:
Problem Task:
Write the following quotient in standard form:
Determine if the following statement is true or
false using complex conjugates: The modulus of
2  3i
.
3  5i
&
z and the modulus of z are equal. Justify your
answer with both verbal and algebraic
arguments.
Some Useful Websites:
Modulus Visual Representation: http://demonstrations.wolfram.com/ComplexNumber/
NUCC | Secondary II Math 84
Unit H.10
Mathematics Content
Cluster Title: Represent complex numbers and their operations on the complex plane.
Standard H.N.CN.5: Represent addition, subtraction, multiplication, and conjugation of
complex numbers geometrically on the complex plane; use properties of this representation for
computation. For example, ( 1  3i ) 3  8 because ( 1  3i ) has modulus 2 and argument
120°.
Concepts and Skills to Master



Represent geometrically the sum, difference, product, and conjugation of complex numbers
on the complex plane.
Show that the conjugate of a complex number in the complex plane is the reflection across
the x-axis.
Evaluate the power of a complex number, in rectangular form, using the polar form of the
complex number.
Critical Background Knowledge


Complex numbers
Complex plane
Academic Vocabulary
complex plane
Suggested Instructional Strategies


Use properties of parallelograms for addition and subtraction of complex numbers, and use
properties of similar triangles for multiplication of complex numbers.
Approach addition, subtraction, and multiplication of complex numbers as vectors by
showing that when multiplying two vectors, you add the arguments to find the resulting
argument, and multiply the moduli to find the resulting modulus.
Skills Based Task:
Problem Task:
Find the sum and product of 2 + 3i and 4 + 2i
graphically and algebraically.
Find two sets of complex numbers whose
differences are equal. Justify graphically.
Some Useful Websites:
NUCC | Secondary II Math 85
Unit H.10
Complex i’s
A Develop Understanding Task 10
Name_____________________________________
Hour___________
In this lesson, you will solve equations as well as reverse your thinking to investigate the
relationship between the complex solutions to a quadratic equation and the equation they come
from.
1.
Find the roots of each of the following quadratic functions by solving for x when y = 0.
Does the graph of either of these functions intersect the x-axis? (Graph, if helpful)
y   x  5  9
2
y  x2  4 x  9
These quadratic equations might be more complex than our current tool box at this time.
The algebraic solution to y   x  5  9  0 is 5  3i and  5  3i and y  x 2  4 x  9  0
2
is 2  i 5 . What do you notice about the complex solutions to these two equations?
Describe any patterns you see. Discuss these with your group and write down everything
you can think of.
2.
Look for patterns as you calculate the sum and the product for each pair of complex
numbers.
a.
2  i ,2  i
b.
3  5i ,3  5i
c.
4  i , 4  i
d.
1  i 3,1  i 3
e.
What complex number can you multiply 3  2i by to get a real number?
f.
What happened when you multiply  4  5i  4  5i  ?
g.
What complex number can you multiply a  bi by to get a real number?
NUCC | Secondary II Math 86
Unit H.10
What equation has these solutions?
Each of the four pairs of complex numbers in problem 2 could be the roots of a quadratic
function.
Your Task: Create a quadratic equation for each pair of complex numbers from problem 2.
Discuss the methods you use for writing the equations and write summary statements describing
your methods.
Discussion Points



How can we reverse the process of solving and work backwards?
How can we use what we know about factors and zeros?
How are the solutions related to the standard from of the equations?
NUCC | Secondary II Math 87
Unit H.10
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Simplify each expression.
1.
 6  4i    1  3i 
2.
 3  2i    5  4i 
3.
 6  2i  1  i 
4.
 2  3i 
5.
 4  3i    4  3i 
6.
 4  3i  4  3i 
2
Set
Decide which of the following equations have real roots, and which have complex roots
without completely solving them and state justify.
7.
8.
y  x2  6
y  x2  6
9.
y  x 2  3x  10
10.
y  x 2  3x  10
11.
y   x  3  4
12.
y   x  3  4
2
2
Consider this geometric sequence: i 0 , i1 , i 2 , i 3 , i 4 , i 5 ,..., i15
13. You know that i 0  1, i1  i , i 2  1 . Calculate the result for each term up to i 15 , and describe
the pattern.
14.
Use the pattern you found to calculate each of the following: i16 , i 25 , i 39 , i100
15.
What is i 4n , where n is a positive whole number?
16.
Based on your answer above simplify i 4n1 , i 4n2 , i 4n3
17.
Calculate i 396 , i 397 , i 398 , i 399
NUCC | Secondary II Math 88
Unit H.10
Go!
Find the conjugate for each complex number and find the product.
18. 5  3i
19. 1  7i
Simplify each fraction (not complex numbers allowed in the denominator).
5
4
20.
21.
2  3i
3i
22.
1  3i
2i
23.
2  5i
2  5i
NUCC | Secondary II Math 89
Unit H.10
Solutions:
NUCC | Secondary II Math 90
Unit H.11
H.11 IMAGINARY AND COMPLEX NUMBERS
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Students will be introduced to the complex plan as a way to visualize complex numbers and
complex roots for quadratic functions. They will calculate the absolute value (Modulus) of a complex
number.
Core Standards Focus H.N.CN.8 Extend polynomial identities to the complex numbers. For
example, rewrite x2 + 4 as (x + 2i)(x – 2i).
H.N.CN.3 Find the conjugate of a complex number; use conjugates to find moduli (absolute
value) and quotients of complex numbers.
H.N.CN.4 Represent complex numbers on the complex plane in rectangular and polar form
(including real and imaginary numbers), and explain why the rectangular and polar forms of a
given complex number represent the same number.
Launch (Whole Class): Pose the following question to the class: “How can you represent
imaginary and complex numbers geometrically?” Remind student that every real number can be
represented by length and direction on a number line, and that every point on a number line can
be represented by a real number. So where do the imaginary numbers go? At this point, you may
direct groups to begin working on the task sheet or gather ideas for consideration.
Some student may find it interesting to think of the complex plane in terms of 90˚ rotations.
Multiple a number by i can be represented by a counterclockwise rotation of 90 degrees. Start
with the number 1, multiple by i (rotate 90 degrees), and you are located at i. Multiply by i again
(rotate another 90 degrees) and you are located at -1, which makes sense because i 2  1 .
NUCC | Secondary II Math 91
Unit H.11
Explore (Small Group): Start groups working on the task sheet. Help direct the students, if they
get lost or stuck.
Discuss (Whole Class or Group): Lead a discussion summarizing the idea of graphing complex
number and finding the modulus. Formalize a formula if it does not come out in the discussion
z  a  bi  a2  b2 .
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 92
Unit H.11
Mathematics Content
Cluster Title: Use complex numbers in polynomial identities and equations.
Standard H.N.CN.8: Extend polynomial identities to the complex numbers. (For example, x2 +
4 as (x+2i)(x-2i).)
Concepts and Skills to Master

Express a quadratic as a product of two complex factors.
Critical Background Knowledge





Factor quadratics.
Understand that some quadratic functions have complex solutions.
Know the definition of i.
Perform operations on complex numbers.
Standard form of a complex number.
Academic Vocabulary
conjugates, complex numbers, i, factor
Suggested Instructional Strategies

Demonstrate that any binomial quadratic expression can be expressed as the difference of
two squares (e.g., x 2  16  x 2  16i 2 ).
Skills Based Task:
Problem Task:
Factor over the complex number system.
Expand the expression (x+3)(x-5i)(x+5i) two
ways:
A.  x  3 x  5i   x  5i 
x 2  16
Answer: ( x  4i )( x  4i )
x 2  10 x  34 Answer: ( x  5i )( x  5i )
B.
 x  3  x  5i  x  5i 
Compare and contrast the methods.
Some Useful Websites:
NUCC | Secondary II Math 93
Unit H.11
Mathematics Content
Cluster Title: Perform arithmetic operations with complex numbers.
Standard H.N.CN.3: Find the conjugate of a complex number, use conjugates to find moduli
and quotients of complex numbers
Concepts and Skills to Master



Given a complex number, determine the conjugate.
Define the modulus of a complex number as the positive square root of the sum of the
squares of the real and imaginary parts of a complex number.
Use conjugates to express quotients of complex numbers in standard form.
Critical Background Knowledge




Complex numbers
Complex plane
Rationalizing denominators
i 2  1
Academic Vocabulary
conjugate, modulus, magnitude, complex plane
Suggested Instructional Strategies


Use properties of difference of two squares to find the modulus.
Relate the modulus visually using vectors.
Skills Based Task:
Problem Task:
Write the following quotient in standard form:
Determine if the following statement is true or
false using complex conjugates: The modulus of
2  3i
.
3  5i
&
z and the modulus of z are equal. Justify your
answer with both verbal and algebraic
arguments.
Some Useful Websites:
Modulus Visual Representation: http://demonstrations.wolfram.com/ComplexNumber/
NUCC | Secondary II Math 94
Unit H.11
Mathematics Content
Cluster Title: Represent complex numbers and their operations on the complex plane.
Standard H.N.CN.4: Represent complex numbers on the complex plane in rectangular and
polar form (including real and imaginary numbers), and explain why the rectangular and polar
forms of a given complex number represent the same number.
Concepts and Skills to Master



Convert between the rectangular form, z  x  yi , and polar form, z  r (cos  i sin  ) , of a
complex number.
Graph complex numbers on a complex plane in both rectangular and polar form.
Justify rectangular and polar forms of a complex number as representing the same number.
Critical Background Knowledge




Complex numbers
Graphing polar coordinates
Trigonometric identities on the unit circle
Modulus
Academic Vocabulary
complex plane, rectangular form, polar form, modulus, argument
Suggested Instructional Strategies


Plot a complex number represented in rectangular form on the complex plane.
Lead students to see the relationship between (x,y) and (r,θ).
Skills Based Task:
Express the complex number z  3  i in
polar form. Plot this number on the complex
plane.
Problem Task:
Given the complex number in polar form
z  r (cos  i sin  ) , what is the polar form of
 z ? Justify your answer with both verbal and
algebraic arguments.
Some Useful Websites:
NUCC | Secondary II Math 95
Unit H.11
Can We Picture?
A Develop Understanding Task 11
Name_____________________________________
Hour___________
If the number line is filled with real numbers, how can imaginary and complex numbers be
represented geometrically? In this lesson, you will learn a way to graph complex numbers.
Avi and Tran were trying to figure out how they could represent complex numbers
geometrically. Avi decided to make a number line horizontal like the x-axis to represent the real
part as well as a vertical line like the y-axis to represent the imaginary part.
Draw a set of axes and label them as Avi described.
How could Avi and Tan graph the point to represent the complex number 3  4i ? Be prepared to
share your strategies with the class.
Use the method to plot the following complex numbers on the set of axes you created.
A - 2  5i
B - 6i
C - 5  3i
D- 4
E - 7i
F - 4  2i
On a new set of complex axes plot the points representing all of the following complex numbers
(should plot 12 points total).
3  4i ,3  4i , 3  4i , 3  4i
4  3i
5, 5,5i , 5i
What do you notice about your graph? How far from (0, 0) is each points?
On the real number line, the distance from 0 to a point on the line is defined as the absolute value
of the number. Similarly, in the complex plane (the plane defined by a set of complex axes), the
absolute value of a complex number is its distance from zero or the origin (0, 0). What did you
notice about all the distances above? The absolute value of a complex number is called the
modulus and the notation is z , where z is the complex number.
What is the absolute value of 8  6i ?
What is the modulus of 7  2i ?
What is 4  i ?
What is the modulus of a  bi ?
NUCC | Secondary II Math 96
Unit H.11
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Factor completely, use complex roots, if necessary.
x2  9
1.
2.
x2  4
3.
x 2  3x  18
4.
x2  8
Set
Graph each complex number in the complex plane. (The complex numbers are in what is
called rectangular form.) Also find the modulus of each number.
5.
6.
1  2i
7  24i
7.
3i
8.
2i
Go!
Find the reference angle (the angle from the x-axis to the line drawn to the point from the
origin). A graph will be helpful. Round the angle to the nearest degree, if necessary.
9.
10. 6  8i
2  2i
11.
 3i
12.
2  2i 3
NUCC | Secondary II Math 97
Unit H.11
Solutions:
NUCC | Secondary II Math 98
Unit H.12
H.12 POLAR PLOT
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Students will learn how to plot polar coordinates.
Core Standards Focus H.N.CN.4 Represent complex numbers on the complex plane in
rectangular and polar form (including real and imaginary numbers), and explain why the
rectangular and polar forms of a given complex number represent the same number.
Launch (Whole Class): Begin the lesson by discussing how we would plot the number 3  4i .
Describe how you can have a student move by walking three steps to the right and then 4 steps
forward to represent the complex number 3  4i . Ask the students how they would go to that
point directly. Ask them how they would describe that motion. You can use this to introduce
polar coordinates by stating that the point can be found by walking 5 steps at an angle of 53˚.
The idea of polar coordinate is to use an angle and a distance (radius).
Explore (Individual): Pass out the resource page (polar graph paper) so that students can plot
points. Polar graph paper has concentric circles fro plotting points at different radii. Mention that
although the points are written in the form  r ,  , we plot the points starting with the angle and
then move the number of units determined by r. Students need to be aware that the same point
 3 
can be found by many different polar coordinates. The point  2,  can also be found by
 4 
NUCC | Secondary II Math 99
Unit H.12
5 

7 



 2, 
 or  2,   , or  2,  . Once students understand the plotting method they should
4 
4
4 



be able to continue with the task sheet.
Discuss (Whole Class or Group): Check for understanding and make sure they can graph the
points.
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 100
Unit H.12
Mathematics Content
Cluster Title: Represent complex numbers and their operations on the complex plane.
Standard H.N.CN.4: Represent complex numbers on the complex plane in rectangular and
polar form (including real and imaginary numbers), and explain why the rectangular and polar
forms of a given complex number represent the same number.
Concepts and Skills to Master



Convert between the rectangular form, z  x  yi , and polar form, z  r (cos  i sin  ) , of a
complex number.
Graph complex numbers on a complex plane in both rectangular and polar form.
Justify rectangular and polar forms of a complex number as representing the same number.
Critical Background Knowledge




Complex numbers
Graphing polar coordinates
Trigonometric identities on the unit circle
Modulus
Academic Vocabulary
complex plane, rectangular form, polar form, modulus, argument
Suggested Instructional Strategies


Plot a complex number represented in rectangular form on the complex plane.
Lead students to see the relationship between (x,y) and (r,θ).
Skills Based Task:
Express the complex number z  3  i in
polar form. Plot this number on the complex
plane.
Problem Task:
Given the complex number in polar form
z  r (cos  i sin  ) , what is the polar form of
 z ? Justify your answer with both verbal and
algebraic arguments.
Some Useful Websites:
NUCC | Secondary II Math 101
Unit H.12
Polar Plot
A Develop Understanding Task 12
Name_____________________________________
Hour___________
For the past several years you have been graphing with standard Cartesian coordinate, which are
named for the French philosopher-mathematician, Rene Descartes. Recently, we plotted points in
the complex plane. Over the next while, we will learn about polar coordinates, a new way to
represent the position of a point in the polar plane. They are frequently used for problems that are
symmetric about a point. You will find that it is easy to rotate graphs in this coordinate system,
but difficult to translate them. This is in contrast to your prior experience with rectangular
coordinates where it was easy to translate graphs but hard to rotate them.
Use the polar graph to plots these points and label them.


 

 

A   3,  B   2,   C   1,  D   3,  E   5,0 
6
4
 4

 4

 5 
 
F   5,  G   5,0  H   3,  I   0,  J   0, 
 4 
 2
In the points you just plotted two sets of points were each
represented by polar coordinates in two different ways.
Find the two sets of points.
Find two other ways to write B in polar coordinates
Find two other ways to write C in polar coordinates
In the figure at the right, there are points marked on the
polar grid. What are the polar coordinates of each of those
points?
In the figure to the left
you will find points
plotted on a Cartesian
Grid. What are the polar
coordinates of these
points?
NUCC | Secondary II Math 102
Unit H.12
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Graph the complex numbers on the complex plane.
1.
A5
2.
B  3i
3.
C  4  2i
4.
D  4  3i
5.
E  1  i 3
6.
F  3  3i
Set
Find the polar coordinates of some of points from the Ready problems.
7.
A
8.
B
9.
E
10.
F
Graph the following polar coordinates on the polar
plane.
 
11. A   2,60 
12.
B   3, 
 4
13.
2 

C   5, 

3 

14.
D   1,90
15.


E   2,  
6

16.
 7 
F   3,

 6 

Go!
Simplify the following complex numbers.
18. 3i  2  4 i 
19.
 5i 
2
20.
 2  i  2  i 
21.
 3  2i  4  i 
22.
 4  5i 
23.
1 i
1 i
2
NUCC | Secondary II Math 103
Unit H.12
Solutions:
NUCC | Secondary II Math 104
Unit H.13
H.13 RECTANGULAR AND POLAR COMPLEX
NUMBER
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Students will graph complex number using both rectangular and polar forms.
Core Standards Focus H.N.CN.3 Find the conjugate of a complex number; use conjugates to
find moduli and quotients of complex numbers.
H.N.CN.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers
geometrically on the complex plane; use properties of this representation for computation. For
example, (–1 + √3i)3 = 8 because (–1 + √3i) has modulus 2 and argument 120°.
Launch (Whole Class): Start by reviewing that the absolute value (modulus) of a complex
number will find a real number. It is found by calculating the distance from the point to the
origin and in the polar plane this will represent.
Explore (Individual, Small Group or Pairs): Converting the polar form (also called
trigonometric form) is very similar to the work that students have done with polar coordinates
 r ,  , the complex form will be z  r  cos  i sin  . Make sure students understand that the
angle θ is the same for both the real component and the imaginary component.
NUCC | Secondary II Math 105
Unit H.13
End the lesson with finding the three cube roots of one. This problem sets up some of the work in
the next section.
Discuss (Whole Class or Group): Discuss results of the problem on the task sheet. After a
discussion of “How do you think you add, subtract, multiply and divide polar number?” Run
through some examples.
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 106
Unit H.13
Mathematics Content
Cluster Title: Perform arithmetic operations with complex numbers.
Standard H.N.CN.3: Find the conjugate of a complex number, use conjugates to find moduli
and quotients of complex numbers
Concepts and Skills to Master



Given a complex number, determine the conjugate.
Define the modulus of a complex number as the positive square root of the sum of the
squares of the real and imaginary parts of a complex number.
Use conjugates to express quotients of complex numbers in standard form.
Critical Background Knowledge




Complex numbers
Complex plane
Rationalizing denominators
i 2  1
Academic Vocabulary
conjugate, modulus, magnitude, complex plane
Suggested Instructional Strategies


Use properties of difference of two squares to find the modulus.
Relate the modulus visually using vectors.
Skills Based Task:
Problem Task:
Write the following quotient in standard form:
Determine if the following statement is true or
false using complex conjugates: The modulus of
2  3i
.
3  5i
&
z and the modulus of z are equal. Justify your
answer with both verbal and algebraic
arguments.
Some Useful Websites:
Modulus Visual Representation: http://demonstrations.wolfram.com/ComplexNumber/
NUCC | Secondary II Math 107
Unit H.13
Mathematics Content
Cluster Title: Represent complex numbers and their operations on the complex plane.
Standard H.N.CN.5: Represent addition, subtraction, multiplication, and conjugation of
complex numbers geometrically on the complex plane; use properties of this representation for
computation. For example, ( 1  3i ) 3  8 because ( 1  3i ) has modulus 2 and argument
120°.
Concepts and Skills to Master



Represent geometrically the sum, difference, product, and conjugation of complex numbers
on the complex plane.
Show that the conjugate of a complex number in the complex plane is the reflection across
the x-axis.
Evaluate the power of a complex number, in rectangular form, using the polar form of the
complex number.
Critical Background Knowledge


Complex numbers
Complex plane
Academic Vocabulary
complex plane
Suggested Instructional Strategies


Use properties of parallelograms for addition and subtraction of complex numbers, and use
properties of similar triangles for multiplication of complex numbers.
Approach addition, subtraction, and multiplication of complex numbers as vectors by
showing that when multiplying two vectors, you add the arguments to find the resulting
argument, and multiply the moduli to find the resulting modulus.
Skills Based Task:
Problem Task:
Find the sum and product of 2 + 3i and 4 + 2i
graphically and algebraically.
Find two sets of complex numbers whose
differences are equal. Justify graphically.
Some Useful Websites:
NUCC | Secondary II Math 108
Unit H.13
Rectangular and Polar Complex Number
A Solidify Understanding Task 13
Name_____________________________________
Hour___________
You have graphed complex number in rectangular form and you have graph in polar coordinates.
When working with complex numbers it is often advantageous to write the number in polar form
(also called trigonometric form). This form is particularly useful when finding powers and roots
of complex numbers.
A complex number in rectangular form is z  a  bi
A complex number in polar form is z  r  cos  i sin  , therefore a  r cos , b  r sin , based
b
on this relationship r  a2  b2 and tan 
a
Convert the following complex numbers into polar form.
3  3i
2 2 3 i


Convert the following complex numbers in polar form to rectangular (standard form).


 3 
 3  
 
  
z  4  cos 
z  2 6  cos    i sin  
  i sin

 4 
 4 
6
 6 


Like all functions in math we should be able to multiple and divide them.
How do you think you would do that to polar forms of complex numbers?
NUCC | Secondary II Math 109
Unit H.13
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Find the polar form of the complex number.
1.
2.
z1  2  2i 3
3.
z3  4 3  4i
4.
z2  1  i 3
z4  2  2i
Set
Represent the complex number graphically, and find the rectangular (standard) form of
the number.
3
5.
6.
2  cos120  i sin120 
 cos330  i sin330 
2
7.



6  cos  i sin 
3
3

8.
3
3 

4  cos
 i sin 
2
2 

Perform the operation and leave the result in trigonometric form use the complex number
reference above if necessary.
 

   

 
9.
3  cos 3  i sin 3    4  cos 6  i sin 6  
  

 
10.
2
2 

2  cos
 i sin 
3
3 

2
2 

4  cos
 i sin 
9
9 

11.
z1 z2
12.
z3
z4
Go!
Find the polar form of the complex number. Approximate your values to three decimal
places. (Remember to figure out the correct quadrant of the complex number when you
find the value for θ)
13. 6  8i
14. 12  5i
NUCC | Secondary II Math 110
Unit H.13
Solutions:
NUCC | Secondary II Math 111
Unit H.14
H.14 POWERS (MULTIPLY/DIVIDE COMPLEX
NUMBERS)
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Students will learn how do multiply and divide complex numbers in polar form.
Core Standards Focus H.N.CN.4 Represent complex numbers on the complex plane in rectangular and
polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a
given complex number represent the same number.
H.N.CN.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically
on the complex plane; use properties of this representation for computation.
H.N.CN.6 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the
midpoint of a segment as the average of the numbers at its endpoints.
Launch (Whole Class): Give students the task sheet and tell them we are going to start by trying
to show them the polar number multiplication form yesterday came from. Have student multiply
z1 z2 , tell them to try foil if they are stuck. Once they have done the multiplication have them
compare with their partner.
Explore (Individual, Small Group or Pairs): Lead a discussion where you have a student come
up and show their complex number multiplication. Then have students give the angle addition or
subtraction formula from unit 3 to help them simplify. The derivation of the quotient formula is
left for students as a homework problem.
NUCC | Secondary II Math 112
Unit H.14
One you have shown them the product rule of a polar number; have them continue with the task
sheet. They will find DeMoivre’s formula for powers my applying the product rule of complex
numbers.
Discuss (Whole Class or Group): Make sure that they have the connections. They need in this
section before you start on the homework.
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 113
Unit H.14
Mathematics Content
Cluster Title: Represent complex numbers and their operations on the complex plane.
Standard H.N.CN.4: Represent complex numbers on the complex plane in rectangular and
polar form (including real and imaginary numbers), and explain why the rectangular and polar
forms of a given complex number represent the same number.
Concepts and Skills to Master



Convert between the rectangular form, z  x  yi , and polar form, z  r (cos  i sin  ) , of a
complex number.
Graph complex numbers on a complex plane in both rectangular and polar form.
Justify rectangular and polar forms of a complex number as representing the same number.
Critical Background Knowledge




Complex numbers
Graphing polar coordinates
Trigonometric identities on the unit circle
Modulus
Academic Vocabulary
complex plane, rectangular form, polar form, modulus, argument
Suggested Instructional Strategies


Plot a complex number represented in rectangular form on the complex plane.
Lead students to see the relationship between (x,y) and (r,θ).
Skills Based Task:
Express the complex number z  3  i in
polar form. Plot this number on the complex
plane.
Problem Task:
Given the complex number in polar form
z  r (cos  i sin  ) , what is the polar form of
 z ? Justify your answer with both verbal and
algebraic arguments.
Some Useful Websites:
NUCC | Secondary II Math 114
Unit H.14
Mathematics Content
Cluster Title: Represent complex numbers and their operations on the complex plane.
Standard H.N.CN.5: Represent addition, subtraction, multiplication, and conjugation of
complex numbers geometrically on the complex plane; use properties of this representation for
computation. For example, ( 1  3i ) 3  8 because ( 1  3i ) has modulus 2 and argument
120°.
Concepts and Skills to Master



Represent geometrically the sum, difference, product, and conjugation of complex numbers
on the complex plane.
Show that the conjugate of a complex number in the complex plane is the reflection across
the x-axis.
Evaluate the power of a complex number, in rectangular form, using the polar form of the
complex number.
Critical Background Knowledge


Complex numbers
Complex plane
Academic Vocabulary
complex plane
Suggested Instructional Strategies


Use properties of parallelograms for addition and subtraction of complex numbers, and use
properties of similar triangles for multiplication of complex numbers.
Approach addition, subtraction, and multiplication of complex numbers as vectors by
showing that when multiplying two vectors, you add the arguments to find the resulting
argument, and multiply the moduli to find the resulting modulus.
Skills Based Task:
Problem Task:
Find the sum and product of 2 + 3i and 4 + 2i
graphically and algebraically.
Find two sets of complex numbers whose
differences are equal. Justify graphically.
Some Useful Websites:
NUCC | Secondary II Math 115
Unit H.14
Mathematics Content
Cluster Title: Represent complex numbers and their operations on the complex plane.
Standard H.N.CN.6: Calculate the distance between numbers in the complex plane as the
modulus of the difference, and the midpoint of a segment as the average of the numbers at its
endpoints.
Concepts and Skills to Master


Show that the distance between two complex numbers is equivalent to the modulus of the
difference by applying the distance formula.
Find the midpoint of a segment between two complex numbers by taking the average of the
numbers at its endpoints using the midpoint formula.
Critical Background Knowledge




Distance formula
Midpoint formula
Modulus
Complex plane
Academic Vocabulary
complex plane, modulus
Suggested Instructional Strategies

Use graphical representations to show relationships between distance formula and the
modulus of the difference, and the relationship between a segments midpoint and the average
of its endpoints.
Skills Based Task:
Find the distance and the midpoint between
2  3i and 1  5i .
Problem Task:
A treasure is hidden in the complex plane.
Follow the sequence of events: From the origin,
travel to 1  3i, then travel to Point A located at
2  5i, noting the distance and direction
traveled. Now return to the origin. Travel the
same distance and direction to find Point B. The
treasure will be halfway between Point A and
Point B. Give the coordinate location of the
treasure.
Some Useful Websites:
NUCC | Secondary II Math 116
Unit H.14
Powers
A Develop Understanding Task 14
Name_____________________________________
Hour___________
Is there a shortcut for  a  bi  ?
n
While raising real numbers to a whole number power is a relatively easy task using exponents
raising a complex number to a power that has two components, presents a more challenging task.
We can use the tools that we have learned, but this becomes a daunting task when expanding a
8
number like  2  2i  . We will use DeMoivre’s Theorem to make this task simpler.
Yesterday we looked at multiplication of complex numbers. You have already multiplied
complex numbers in rectangular form and polar form we want to formalize the method we used
yesterday, by showing where it comes from.
z1  r1  cos a  i sin a  and z2  r2  cos b  i sin b  . Find the product of z1 z2
Now use the sum and difference formulas for sine and cosine to simplify the expression you
found.
2
 

 
Use the formula you simplified to find  6  cos  i sin  
6
6 
 
11
11 
2
2 


Given z1  2  cos
 i sin
 i sin 
 and z2  3  cos
6
6 
3
3 


Find z1 z2 and
z1
z2
Given z  r  cos  i sin 
Show that z2  r 2  cos  2   i sin 2  
Use z2 results to find z 3 and z 4
Make a prediction for the general term z n
This general prediction is called DeMoivre’s Theorem



Find z 6 if z  2  cos  i sin 
6
6

NUCC | Secondary II Math 117
Unit H.14
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Change to polar form.
1.
2  2i
3.
4 3  4i
2. 6  8i
4.  3i
Set
Simplify
5.
 2  2i  4i 
6.
2
1  3i
Redo problems 5 and 6 by converting them into polar form and simplify and leave answer
polar form.
7.
8.
Convert problems 7 and 8 back to rectangular and see if they match the answer to 5 and 6.
9.
10.
11.
Show that
r1  cos a  i sin a  r1
  cos  a  b   i sin  a  b   Hint: Multiply the conjugate of
r2  cos b  i sin b  r2
the denominator.
NUCC | Secondary II Math 118
Unit H.14
Go!
Simplify
12.
 2  2i 
4

13.  3  i

6
NUCC | Secondary II Math 119
Unit H.14
Solutions:
NUCC | Secondary II Math 120