MATHEMATICS 10C
TRIGONOMETRY
High School collaborative venture with
Jasper Place, Ross Sheppard and Victoria Schools
Jasper Place: Martin Fechner, Elisha Pinter, Nic Ryan, Suzan Saad
Ross Sheppard: Tim Gartke, Jeremy Klassen, Don Symes
Victoria: Kevin Bissoon
Facilitators: Greg McInulty (Consulting Services) and Gail Drouin (Alberta Education)
Editor: Rosalie Mazurok (Ross Sheppard High School)
Spring, 2009
Mathematics 10C
Trigonometry
Page 2 of 60
TABLE OF CONTENTS
STAGE 1 DESIRED RESULTS
PAGE
Big Idea
4
Enduring Understandings
4
Essential Questions
5
Knowledge
6
Skills
7
STAGE 2 ASSESSMENT EVIDENCE
Teacher Notes for Transfer Tasks
8
Transfer Tasks
Fast and Furious – Edmonton Drift
Teacher Notes for Fast and Furious and Rubric
Transfer Task
Rubric
Possible Solution
9
10 -12
13 - 14
15 - 20
The Extraordinary Race
Teacher Notes for The Extraordinary Race and Rubric
Transfer Task
Rubric
Possible Solution
21
22 - 29
30 - 31
32 - 39
STAGE 3 LEARNING PLANS
LL #4
Lesson #1
Pythagorean Theorem
40 - 44
Lesson #2
Developing The Tangent Ratio
45 - 48
Lesson #3
Developing Sine And Cosine Ratios
49 - 53
Lesson #4
Applications of Trigonometric Ratios in One and Two Triangle Questions
54 - 56
Lesson #5
Designing a Solution Using Trigonometry
57 - 59
Mathematics 10C
Trigonometry
Page 3 of 60
Mathematics 10C
Trigonometry
STAGE 1 DESIRED RESULTS
Big Idea:
It is important that students experience a wide variety of real-world applications
pertaining to trigonometry, so that they understand that trigonometry is a practical
branch of mathematics and a foundation on which more advanced concepts may be
added or built.
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer
to it often.
Enduring Understandings:
Students will understand that…

The Pythagorean Theorem utilizes the relationship between the sides in a right
triangle.

Trigonometry is based on a series of constant ratios.

Trigonometry utilizes the relationships between the sides and angles in a
triangle.

Connections exist between trigonometry and real-life situations.

Different strategies work in different situations.
Mathematics 10C
Trigonometry
Page 4 of 60
Essential Questions:

Where could trigonometry be used?

Where do sine, cosine and tangent come from?

Why use sine, cosine and tangent ratios?

Why are the sine ratio and the cosine ratio bounded by -1 and 1 inclusive and
the tangent ratio is not?

How do I decide which strategy to use?

What answer should be expected and how precise should it be?
Implementation note:
Ask students to consider one of the
essential questions every lesson or two.
Has their thinking changed or evolved?
Mathematics 10C
Trigonometry
Page 5 of 60
Knowledge:
Enduring
Understanding
Specific
Outcomes
Students will understand that…

the Pythagorean Theorem
utilizes the relationship between
the sides in a right triangle.
Knowledge that applies
to this Enduring
Understanding
Students will know …
*M4
Students will understand that…

Students will know …


trigonometry is based on a
series of constant ratios.
*M4


Students will understand that…

trigonometry utilizes the
relationships between the sides
and angles in a triangle.
connections exist between
trigonometry and real-life
situations.
*M4
different strategies work in
different situations.

primary trigonometric ratios.
Students will know …

*M4
Students will understand that…

that there exists a constant ratio
of the corresponding sides in
similar right triangles.
that the limits on values of sine
and cosine ratios are -1 and +1.
SOHCAHTOA is a mnemonic
device to help remember the
primary trigonometric ratios.
Students will know …
Students will understand that…

that the sum of the squares of the
sides in a right triangle equals the
square of the hypotenuse.
applications using angles of
elevation and angles of
depression.
Students will know …
*M4

that the primary trigonometric
ratios can be used for right
triangles only.
*M = Measurement
Mathematics 10C
Trigonometry
Page 6 of 60
Skills:
Enduring
Understanding
Specific
Outcomes
The student will understand that…
Skills that applies
to this enduring
understanding
Students will be able to…
*M4



the Pythagorean Theorem
utilizes the relationship between
the sides in a right triangle.
label triangles appropriately.
find the missing side in
a 2  b 2  c 2 , where c is the
hypotenuse in a right triangle.
The student will understand that…
Students will be able to…
*M4


trigonometry is based on a
series of constant ratios.

The student will understand that…

trigonometry utilizes the
relationships between the sides
and angles in a triangle.
Students will be able to…
*M4


The student will understand that…

connections exist between
trigonometry and real-life
situations.
different strategies work in
different situations.
use the primary trigonometric
ratios to find the missing parts of
right triangles.
use a scientific calculator in
solving trigonometric problems.
Students will be able to…
*M4
The student will understand that…

solve equations involving
proportions.
use the primary trigonometric
ratios.

use angle(s) of elevation and
angle(s) of depression in
applications.
Students will be able to…
*M4


design a strategy for decision
making.
solve problems using two
triangles.
*M = Measurement
Implementation note:
Teachers need to continually ask
themselves, if their students are
acquiring the knowledge and skills
needed for the unit.
Mathematics 10C
Trigonometry
Page 7 of 60
STAGE 2 Assessment Evidence
1
Desired Red Results
Fast and Furious – Edmonton Drift or The Extraordinary Race
Teacher Notes
There are two transfer tasks to evaluate student understanding of the concepts relating to
trigonometry. The teacher (or the student) will select one for completion. Photocopy-ready
versions of the two transfer tasks and rubric are included in this section.
Implementation note:
Students must be given the transfer task & rubric at
the beginning of the unit. They need to know how
they will be assessed and what they are working
toward.
Each student will:




demonstrate their understanding of the primary trigonometric ratios.
be able to solve right triangles.
demonstrate connections to “real life” situations.
develop strategies to be able to decide which method is appropriate for a
specific situation.
Mathematics 10C
Trigonometry
Page 8 of 60
Teacher Notes for Fast and Furious – Edmonton Drift Transfer Task
Students will need to request information regarding the amount of the fine from the “City of
Edmonton” (aka the teacher) regarding the fine.
Teacher Notes for Rubric

No score is awarded for the Insufficient/Blank column , because there is no evidence of
student performance.

Limited is considered a pass. The only failures come from Insufficient/Blank.

When work is judged to be Limited or Insufficient/Blank, the teacher makes decisions
about appropriate intervention to help the student improve.
Implementation note:
Teachers need to consider what performances and
products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Mathematics 10C
Trigonometry
Page 9 of 60
FAST AND FURIOUS: Edmonton Drift - Student Assessment Task
Situation
You have received the attached photo radar ticket in the mail.
Goal
To develop a case that identifies and proves mathematically the inaccuracies of your
speeding ticket.
Role
You are to analyze all aspects of the case and present evidence that would require the
judge to dismiss the case. You need to show either:

there is something wrong with the set up of the camera, or

that you were not speeding.
Audience
You are to present information, calculations and diagrams to a court that will prove
conclusively that the charge is not warranted.
Product / Performance
Your evidence should include:
 diagrams
 trigonometric calculations, and
 an explanation of your calculations: i.e.
o script
o prompt cards
o annotated calculations
Resources
You will require additional information to enable you to prove your case. Specific
information may be requested from "The City of Edmonton" (your teacher) such as:
 intersection blueprint
 aerial photograph
 specified measurements
Edmonton
Police Service
P
1897345SP
Name:
Photo Identification Officer
Const. Stefan O'Brien
ID # 789159321
Edmonton City Traffic
Enforcement
Address:
You are charged with the following Violation
Date and time
of offence:
Penalty
Assessment
□
23:13 21st
November
2008
Offence Charged:
Travelling at 62 km/h in a 50km/h
zone.
Birdseye View of the Intersection
Blueprint of Intersection
Assessment
Mathematics 10C
Trigonometry
Rubric
Level
Criteria
Excellent
4
Proficient
3
Adequate
2
Limited*
1
Insufficient /
Blank*
No score is
awarded
because there
is no evidence
of student
performance.
No data is
presented.
Performs
Calculations
Performs
precise and
explicit
calculations.
Performs
focused and
accurate
calculations.
Performs
appropriate
and generally
accurate
calculations.
Performs
superficial
and irrelevant
calculations.
Presents Data
Presentation of
data is
insightful and
astute.
Presentation
of data is
logical and
credible.
Presentation of
data is
simplistic and
plausible.
Presentation of
data is vague
and
inaccurate.
Explains
Choice
Shows a
solution for the
problem;
provides an
insightful
explanation.
Shows a
solution for
the problem;
provides a
logical
explanation.
Communicates
findings
Develops a
compelling and
precise
presentation
that fully
considers
purpose and
audience; uses
appropriate
mathematical
vocabulary,
notation and
symbolism.
Develops a
convincing
and logical
presentation
that mostly
considers
purpose and
audience;
uses
appropriate
mathematical
vocabulary,
notation and
symbolism.
Shows a
solution for the
problem;
provides
explanations
that are
complete but
vague.
Develops a
predictable
presentation
that partially
considers
purpose and
audience; uses
some
appropriate
mathematical
vocabulary,
notation and
symbolism.
Shows a
solution for the
problem;
provides
explanations
that are
incomplete or
confusing.
Develops an
unclear
presentation
with little
consideration
of purpose and
audience; uses
inappropriate
mathematical
vocabulary,
notation and
symbolism.
No explanation
is provided.
No findings are
communicated.
Glossary
accurate – free from errors
astute – shrewd and discerning
appropriate – suitable for the circumstances
compelling – convincing and persuasive
complete – including every necessary part
convincing – impressively clear or definite
credible – believable
explicit – expressing all details in a clear and obvious way
focused – concentrated on a particular thing
incomplete – partial
inaccurate – not correct
inappropriate – not suitable
insightful – a clear perception of something
irrelevant – not relevant or important
logical - based on facts, clear rational thought, and sensible reasoning
precise - detailed and specific
plausible – believable
predictable - happening or turning out in the way that might have been expected
simplistic – lacking detail
superficial - having little significance or substance
unclear – ambiguous or imprecise
vague - not clear in meaning or intention
Mathematics 10C
Trigonometry
Page 14 of 60
Fast and Furious: Edmonton Drift – Possible Solutions
Project Version 1
Diagram 1:
Given
3m
(city)
x
Given 3.01 m (city)
Streetlight
Camera
x  distance from the back of the car directly to the camera; hypotenuse.
x2  a 2  b2
x 2  (3) 2   3.01
2
x  (3)2   3.01
2
x  4.25 m to nearest hundredth
Mathematics 10C
Trigonometry
Page 15 of 60
Diagram 2: (3 dimensional) – side view
Streetlight
Camera
City given
3.2 m
(adjacent)
X
z
Distance calculated = 4.25 m
(opposite)
4.25
 1.328125
3.2
 4.25 
X  tan 1 

 3.2 
X  53° rounded to the nearest degree.
tan X 
 Camera should be set at 55° (given).
Further proof of math with the correct angle.
Mathematics 10C
Trigonometry
Page 16 of 60
Project Version 2
Diagram 1:
y
Given
3m
(city)
Given 3.71 m (city)
Streetlight
Camera
y  distance from the back of the car directly to the camera; hypotenuse.
y 2  a 2  b2
y 2  (3.71) 2   3
y
2
 3.71  3
2
2
y  4.77 m to the nearest hundredth.
Mathematics 10C
Trigonometry
Page 17 of 60
Diagram 2: (3 dimensional) – side view
Streetlight
Camera
Y
City given
3.6 m
(adjacent)
Distance calculated = 4.77 m
(opposite)
tan Y 
4.77
3.6
 4.77 
Y  tan 1 

 3.6 
Y  53° rounded to the nearest degree.
 Camera should be set at 55° (given).
Further proof of math with the correct angle.
Mathematics 10C
Trigonometry
Page 18 of 60
Project Version 3
Diagram 1:
z
Streetlight
Camera
Given
3m
(city)
Given 4.38 m (city)
z  distance from the back of the car directly to the camera; hypotenuse.
z 2  a 2  b2
z 2   3   4.38 
2
z
2
3   4.38
2
2
z  5.31 m to the nearest hundredth.
Mathematics 10C
Trigonometry
Page 19 of 60
Diagram 2: (3 dimensional) – side view
Streetlight
Camera
Z
City given
4.0 m
(adjacent)
Distance calculated = 5.31 m
tan Z 
5.31
4.0
 5.31 
Z  tan 1 

 4.0 
Z  53° rounded to the nearest degree.
 Camera should be set at 55° (given).
Further proof of math with the correct angle.
Mathematics 10C
Trigonometry
Page 20 of 60
Teacher Notes for The Extraordinary Race Transfer Task
This transfer task was inspired by the “Sapphire Heist” by Diane Stobbe and Renee Handfield. It
was revised by Martin Fechner, Don Symes, Tim Gartke and Jeremy Klassen.
Stage 4 is intended to be a multiple triangle problem in three dimensions. The pictures provided
are intended to give a visual representation of the horizontal triangle and the vertical triangle
separately. Students are expected to draw in the lines and use a protractor to find the angles.
Teacher Notes for Rubric

No score is awarded for the Insufficient/Blank column , because there is no evidence of
student performance.

Limited is considered a pass. The only failures come from Insufficient/Blank.

When work is judged to be Limited or Insufficient/Blank, the teacher makes decisions
about appropriate intervention to help the student improve.
Implementation note:
Teachers need to consider what performances and
products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Mathematics 10C
Trigonometry
Page 21 of 60
The Extraordinary Race - Student Assessment Task
Part A
You are the leader of a team participating in a reality TV show. On this leg of the
extraordinary race you need to journey from the starting point to the next checkpoint.
Your job is to choose a path for your team to follow. Choose a path, complete the
appropriate calculations, and put it in a convincing presentation.
Each section of this race to the ultimate prize requires you to determine an angle or a
distance before continuing on.
Checkpoint
Rope Bridge
Rope Bridge
I
J
H
Stage 5
Stage 5
F
1.0 km
G
I
1.8 km
1.7 km
H
G
Stage 4
1.9 km
F
1.6 km
52
46
Stage 3
Stage 3
E
Stage 2
57
D
2.1 km
B
Zip-line
B
D

A
Drop-off
30 m
point
C
8m
C
E
Stage 1
10 m
Zip-line

23 m
Information about the race:
 You are dropped onto a plateau by a helicopter.
 You must race to the edge of the cliff using the path provided, use a zip line to cross
the river to a point below the top of the cliff.
 From here you race to the next river crossing.
 At this point you need to use the limited resources that you have been provided to get
across the river AND up the cliff on the other side.
 Once you are at the top of the cliff it is a straight run to the checkpoint. Pace yourself,
but try to get there first.
For each stage you need to include an explanation of your strategy for solving the given
problems.
Stage 1: Choose a path (AB or AC) and determine the length of your chosen path.
Stage 2: At this point determine the angle of depression of the zip line. At point B the
height of the cliff is 10 m and the width of the river is 30 m.
At point C the height of the cliff is 8m and the width of the river is 23m.
Site B
Site C
Stage 3: For this leg of your journey you need to determine the distance to the river.
Use the information provided in the diagram to determine the distance, DF or EG.
Stage 4: At this point with your limited resources you need to be careful. You
need to determine how much rope you need to connect to the top of the cliff on the
far side.
For this stage two pictures of the river have been provided. One is from above and
the other is from the shore. Use the information provided about the specific
distances used and a protractor to determine the length straight from the shore to
the top of the cliff.
Site F
From the river.
B
F
From Satellite:
the line segment
drawn is 14.0 m.
F
14.0 m
Observers in boat
H
Site G
From the River
I
G
I
G
G
From the Satellite:
the lines segment
drawn is 39.0 m
39.0m
Stage 5: Here you will need to run as fast as possible to get to the checkpoint. You
need to determine the distance you will run so that you can pace yourself.
Use the information provided to determine the distance to the checkpoint (HJ or IJ).
Part B
From point J, you need to get to point Z. Sources have told us that the trail is 3.16
km, but you know that this does not get you to the end point. You will need to forge
your way across the desert for an undisclosed distance. To determine the supplies
you require, you must figure out how far this is.
Z
? km
3.16 km
θ α
J
α θ
1.0 km
1.7 km
1.8 km
Congratulations! If you solved this correctly, you win!
You’ve just won a ________________. (ask your teacher)
Assessment
Mathematics 10C
Trigonometry
Rubric
Level
Criteria
Excellent
4
Proficient
3
Adequate
2
Limited*
1
Insufficient /
Blank*
No score is
awarded
because there
is no evidence
of student
performance.
No data is
presented.
Performs
Calculations
Performs
precise and
explicit
calculations.
Performs
focused and
accurate
calculations.
Performs
appropriate
and generally
accurate
calculations.
Performs
superficial
and irrelevant
calculations.
Presents Data
Presentation of
data is
insightful and
astute.
Presentation
of data is
logical and
credible.
Presentation of
data is
simplistic and
plausible.
Presentation of
data is vague
and
inaccurate.
Explains
Choice
Shows a
solution for the
problem;
provides an
insightful
explanation.
Shows a
solution for
the problem;
provides a
logical
explanation.
Communicates
findings
Develops a
compelling and
precise
presentation
that fully
considers
purpose and
audience; uses
appropriate
mathematical
vocabulary,
notation and
symbolism.
Develops a
convincing
and logical
presentation
that mostly
considers
purpose and
audience;
uses
appropriate
mathematical
vocabulary,
notation and
symbolism.
Shows a
solution for the
problem;
provides
explanations
that are
complete but
vague.
Develops a
predictable
presentation
that partially
considers
purpose and
audience; uses
some
appropriate
mathematical
vocabulary,
notation and
symbolism.
Shows a
solution for the
problem;
provides
explanations
that are
incomplete or
confusing.
Develops an
unclear
presentation
with little
consideration
of purpose and
audience; uses
inappropriate
mathematical
vocabulary,
notation and
symbolism.
No explanation
is provided.
No findings are
communicated.
Glossary
accurate – free from errors
astute – shrewd and discerning
appropriate – suitable for the circumstances
compelling – convincing and persuasive
complete – including every necessary part
convincing – impressively clear or definite
credible – believable
explicit – expressing all details in a clear and obvious way
focused – concentrated on a particular thing
incomplete – partial
inaccurate – not correct
inappropriate – not suitable
insightful – a clear perception of something
irrelevant – not relevant or important
logical - based on facts, clear rational thought, and sensible reasoning
precise - detailed and specific
plausible – believable
predictable - happening or turning out in the way that might have been expected
simplistic – lacking detail
superficial - having little significance or substance
unclear – ambiguous or imprecise
vague - not clear in meaning or intention
The Extraordinary Race – Possible Solution
Part A
You are the leader of a team participating in a reality TV show. On this leg of the
extraordinary race you need to journey from the starting point to the next checkpoint.
Your job is to choose a path for your team to follow. Choose a path, complete the
appropriate calculations, and put it in a convincing presentation.
Each section of this race to the ultimate prize requires you to determine an angle or a
distance before continuing on.
Checkpoint
Rope Bridge
Rope Bridge
I
J
H
Stage 5
Stage 5
F
1.0 km
G
I
1.8 km
1.7 km
H
G
Stage 4
1.9 km
F
1.6 km
52
46
Stage 3
E
Stage 2
57
D
2.1 km
B
Zip-line
8m
C
E
Stage 1

23 m
Zip-line
B
10 m
D

A
Drop-off
point
30 m
Mathematics 10C
Trigonometry
C
Page 32 of 60
Information about the race:





You are dropped onto a plateau by a helicopter.
You must race to the edge of the cliff using the path provided, use a zip line to
cross the river to a point below the top of the cliff.
From here you race to the next river crossing.
At this point you need to use the limited resources that you have been provided to
get across the river AND up the cliff on the other side.
Once you are at the top of the cliff it is a straight run to the checkpoint. Pace
yourself, but try to get there first.
For each stage you need to include an explanation of your strategy for solving the
given problems.
Stage 1: Choose a path (AB or AC) and determine the length of your chosen path.
C
2.1km
57
B
b
c
A
Go from A to C.
OR from A to B.
c
2.1
2.1sin 57  c
c 1.8 km
b
2.1
2.1cos 57  b
b 1.1 km
sin 57 
cos 57 
Mathematics 10C
Trigonometry
Page 33 of 60
Stage 2: At this point determine the angle of depression of the zip line. At point B the
height of the cliff is 10 m and the width of the river is 30 m.
At point C the height of the cliff is 8m and the width of the river is 23m.
Site B

10 m
30 m
tan  
10
30
 10 

 30 
  tan 1 
 18
Site C

8m
23 m
23m
tan  
8
23
 8 

 23 
  tan 1 
 19
Mathematics 10C
Trigonometry
Page 34 of 60
Stage 3: For this leg of your journey you need to determine the distance to the river.
Use the information provided in the diagram to determine the distance, DF or EG.
Find the distance from E to G
G
x
1.9
1.9sin 52  x
1.5 km  x
sin 52 
1.9 m
52
x
 x 1.5 km
E
OR find the distance from F to D
1.6 km
46
F
x
1.6
1.6sin 46  x
sin 46 
1.2  x
x
 x 1.2 km
D
Mathematics 10C
Trigonometry
Page 35 of 60
Stage 4: At this point with your limited resources you need to be careful. You
need to determine how much rope you need to connect to the top of the cliff on the
far side.
For this stage two pictures of the river have been provided. One is from above and
the other is from the shore. Use the information provided about the specific
distances used and a protractor to determine the length straight from the shore to
the top of the cliff.
Site F
From the river.
20
F
15.0 m
x
From Satellite:
the line segment
drawn is 15.0 m.
x
15.0 m
25
Observers in boat
Angles are measured with a protractor.
Observer in boat
15.0
x
15.0
x
tan 25
x  32.1676
cos 20 
tan 25 
x
cos 20
32.2
r
cos 20
r  34.2 m
r
 x  32.2 m
Mathematics 10C
x
r
Trigonometry
Page 36 of 60
Site G
From the River
r
G
14
x
I
From the Satellite:
the lines segment
drawn is 39.0 m
x
39.0m
G
G
54
cos14 
x
39.0
39.0 tan 54  x
x  53.7 m
tan 54 
Mathematics 10C
x
r
x
cos14
53.7
r
cos14
r  55.3 m
r
Trigonometry
Page 37 of 60
Stage 5: Here you will need to run as fast as your team will go to get to the checkpoint.
You need to determine the distance you will run so that you can pace yourself.
Use the information provided to determine the distance to the checkpoint (HJ or IJ).
Find the distance from J to I
J
1.02  1.82  x 2
x
x  1.02  1.82
x  2.1 km
1.0 km
I
1.8 km
OR find the distance from H to J
J
j
x 2 1.7 2  1.02
y
1.0 km
x  1.7 2  1.02
x  2.0 km
H
1.7 km
Mathematics 10C
Trigonometry
Page 38 of 60
Part B
From point J, you need to get to point Z. Sources have told us that the trail is 3.16
km, but you know that this does not get you to the end point. You will need to forge
your way across the desert for an undisclosed distance. To determine the supplies
you require, you must figure out how far this is. .
Z
29
x
? km
3.16 km
θ α
J
α θ
tan  
1.8
1.0
1.0 km
 1.8 

 1.0 
  tan 1 
1.7 km
1.8 km
Z  29
  61
tan 29 
x
tan  
3.16
3.16 tan   x
x
?
x
tan 29
?  10.3 km
?
5.7  x
 x  5.7 km
Congratulations! If you solved this correctly, you win!
You’ve just won a ________________. (ask your teacher)
Mathematics 10C
Trigonometry
Page 39 of 60
STAGE 3 LEARNING PLANS
Lesson 1
Pythagorean Theorem
STAGE 1
BIG IDEA:
It is important that students experience a wide variety of real-world applications pertaining to
trigonometry, so that they understand that trigonometry is a practical branch of mathematics and a
foundation on which more advanced concepts may be added or built.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand that…



the Pythagorean Theorem utilizes the
relationship between the sides in a right
triangle.
trigonometry utilizes the relationships
between the sides and angles in a triangle.
connections exist between trigonometry
and real-life situations.



Where would trigonometry be used?
How do I decide which strategy to use?
What answer should be expected and how
precise should it be?
.
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to…

that the sum of the squares of the sides in
a right triangle equals the square of the
hypotenuse.

label triangles appropriately.

find the missing side in a  b  c ,
where c is the hypotenuse in a right
triangle.
2
2
2
Implementation note:
Each lesson is a conceptual unit and is not intended to
be taught on a one lesson per block basis. Each
represents a concept to be covered and can take
anywhere from part of a class to several classes to
complete.
Mathematics 10C
Trigonometry
Page 40 of 60
Lesson Summary
Students will be able to use the Pythagorean Theorem to determine the length of a
side that is missing from a right triangle. They should be able to complete this
problem regardless of whether the missing side is a leg or the hypotenuse.
Lesson Plan
Activating prior knowledge



What is the Pythagorean Theorem?
Can you use this theorem to solve any triangle?
DI: Do you know of any Pythagorean Triples? (Note to Teacher: These
were covered in Grade 8)
Teachers need to explain the generally accepted rules for labelling the sides and
the angles of a triangle. (Vertices are labelled with capital letters and the sides
opposite the vertices are labelled with the identical lower case letter.)
Use
and
to demonstrate
C
c
a
b
H
A
Q
p
r
R
P
q
the labelling of triangles.
Mathematics 10C
Trigonometry
Page 41 of 60
Review and Preview
Solve for the variable in the following two questions.
6
x
3
10
y
4
You may choose to use the Pythagorean Theorem applets to illustrate the relationship in
the theorem.
Activities for Pythagorean Theorem
A Classroom Investigation
Materials needed
a rough floor plan of the classroom including walls, doors. (Other features you may
want to include to help students orient themselves could be windows, whiteboards,
teacher desks, etc.) (Note: Extra copies of the floor plans may be used for lesson 6).
tape measures or meter sticks
Objective
Students will use the Pythagorean Theorem to find the shortest distance to some object
of their choosing in the classroom.
Mathematics 10C
Trigonometry
Page 42 of 60
Method
Step 1. Pass out a floor plan to each student
2. Have students locate their desk on the floor plan and label it as Point X (need
not be exact).
3. Have each student individually choose an object on or near a wall in the
classroom (i.e. Clock, pencil sharpener, door, friend?) and place this on the floor
plan as Point Y.
4. Have students measure the shortest distance from their own location to the wall
the object is on and show this point on the floor plan as Point Z and state the
measured length on the floor plan. Note: this length should be Perpendicular to
the wall the object is on.
5. Measure the length along the wall from Point Z to Y and state this length on the
floor plan.
6. Determine the shortest distance from your seat to the object.
Extension (Optional)

Is this actually the shortest distance? (No, it does not take a vertical distance into
account)

Have students determine the distance to an object high on the wall (i.e. Clock,
poster, corner of ceiling) using floor distance and height. In this case, the
perpendicular lengths will be the horizontal (floor) and vertical (wall) lengths. The
students will not be able to rely on walls to orient the right angles.
Challenge (Optional)
Prepare students for two triangle problems by solving distances using 3 Dimensions.
Going Beyond
Give a question with a radical length. Give a question involving 13 , for example,
find the length of a diagonal in a box.
Mathematics 10C
Trigonometry
Page 43 of 60
Resources
McGraw Hill Math 10
Foundations and Pre-calculus Mathematics 10 (Pearson)
Pythagorean Theorem Applet
http://www.ies.co.jp/math/java/samples/pytha2.html
another applet
http://www.usna.edu/MathDept/mdm/pyth.html
Supporting
An applet given in the resources could also be used for extra support for a student.
Glossary
hypotenuse – the side opposite the right angle (also the longest side)
Pythagorean Theorem – the square of the length of the hypotenuse equals the sum of
the squares of the lengths of the legs of the right triangle
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in
more than one division. Some terms have animations to illustrate meanings.
Mathematics 10C
Trigonometry
Page 44 of 60
Lesson 2
Developing the Tangent Ratio
STAGE 1
BIG IDEA:
It is important that students experience a wide variety of real-world applications pertaining to
trigonometry, so that they understand that trigonometry is a practical branch of mathematics and a
foundation on which more advanced concepts may be added or built.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand that…

trigonometry is based on a series of
constant ratios.
trigonometry utilizes the relationships
between the sides and angles in a triangle.
connections exist between trigonometry
and real-life situations.




Where would trigonometry be used?
What answer should be expected and how
precise should it be?
.
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to…


the tangent ratio
that there exists a constant ratio of the
corresponding sides in similar right
triangles.



use the tangent ratio.
solve equations involving proportions.
use a scientific calculator in solving
trigonometric problems.
Lesson Summary
The tangent ratio can be used to find missing sides or missing angles in a right triangle.
Lesson Plan
Review Prior Knowledge


What identifies a triangle as a right triangle?
Draw a right-angled triangle and identify the opposite side, adjacent side and
hypotenuse.
Mathematics 10C
Trigonometry
Page 45 of 60
Today’s Challenge
Do we have enough information here to solve for the unknown angle? If so, is your
solution the most efficient method?
5

7
What observations can you make about the relationship between the lengths of the
vertical and horizontal sides? Can this be used to find the measure of Angle A? Note:
You may use applets referenced in Resources.
What observations can you make about the relationship between the lengths of the
vertical and horizontal sides? Can this be used to find the measure of Angle A? Note:
You may use applets referenced in Resources.
Conclusion
The ratio of opposite side to adjacent side of similar right triangles is constant and is
called the Tangent Ratio.
Mathematics 10C
Trigonometry
Page 46 of 60
Discussion Points

Are there limits on the values of the tangents?

Can the tangent ratio be negative? Can it be 2/3? Can it be 2000? Why or why not?

How can we use the tangent ratio to solve for unknown angles in a right triangle?

What information can be found on your calculator?
Teacher-Led Classroom Example
Cupid’s Bow (A tale of Woe of Juliette and her Romeo)
Cupid shoots his arrows straight and true. Juliette is on a balcony. The balcony is 4m
above the ground. Cupid is hovering behind a tree, 8.5 m above the ground. His
ground distance to the balcony is 15.2m. Draw a diagram and label all appropriate
information. (Note: Teacher should ensure students also include the horizontal from the
balcony to the tree). Use the diagram to determine the angle from the horizontal that
Cupid must aim his bow in order to hit Juliette. Is this an angle of Elevation or
Depression? Why? Romeo is standing directly below Juliette and he sees Cupid. At
what angle to the horizontal is Romeo’s line of vision? Is this an angle of Elevation or
Depression? Why?
How can we use the tangent ratio to solve for unknown sides in a right triangle?
Going Beyond
Use the tangent box applet to explore the relationship between the angle and the
tangent value in conjunction with the tangent function in your calculator.
Mathematics 10C
Trigonometry
Page 47 of 60
Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 2.7)
Math 10 (McGraw Hill: sec 3.3)
similar triangles applet
http://ronblond.com/M10/SimilarTriangles.APPLET/index.html
tangent box applet
http://www.ies.co.jp/math/products/trig/applets/tanbox/tanbox.html
Glossary
adjacent side – the leg in a right triangle that connects the angle to the right angle
opposite side – the leg in a right triangle across from the angle
similar triangles – two triangles with the same shape
tangent ratio – the ratio of the opposite side to the adjacent side in a right triangle
Mathematics 10C
Trigonometry
Page 48 of 60
Lesson 3
Developing Sine and Cosine Ratios
STAGE 1
BIG IDEA:
It is important that students experience a wide variety of real-world applications pertaining to
trigonometry, so that they understand that trigonometry is a practical branch of mathematics and a
foundation on which more advanced concepts may be added or built.
.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:

Students will understand that…




trigonometry is based on a series of
constant ratios.
trigonometry utilizes the relationships
between the sides and angles in a triangle.
connections exist between trigonometry
and real-life situations.
different strategies work in different
situations.



Where do sine, cosine and tangent come
from?
Why do we use sine, cosine and tangent
ratios?
Why are the sine ratio and the cosine ratio
bounded by -1 and 1 inclusive and the
tangent ratio is not?
What answer should be expected and how
precise should it be?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to…



the primary trigonometric ratios.
that there exists a constant ratio of the
corresponding sides in similar right
triangles.
SOHCAHTOA is a mnemonic device to
help remember the primary trigonometric
ratios.



solve equations involving proportions.
use primary trigonometric ratios to find the
missing parts of right triangles.
use a scientific calculator in solving
trigonometric problems.
Lesson Summary
Students will develop an understanding of the sine and cosine ratios and how they relate
to triangles.
Mathematics 10C
Trigonometry
Page 49 of 60
Lesson Plan
Challenge
Do we have enough information here to solve for the unknown angle? If so, is
your solution the most efficient method?
12

7
Investigation
Complete either “Investigate Trigonometric Ratios” in Section 3.2 of McGraw Hill’s Math
10 or “Construct Understanding” in Section 2.4 of Pearson’s Foundations and Precalculus Mathematics 10.
Classroom Activity
Materials
construction paper
class set of scissors
rulers
protractors
Mathematics 10C
Trigonometry
Page 50 of 60
Method
Students may work individually or in pairs.
Step 1.
Have each student (group) use a ruler to draw a straight line from one edge of
the construction paper to an adjacent edge and then cut along that line. Note
to teacher: the result should be that each student has now cut out a right
triangle (sizes and angles will differ).
2. Have students choose one of the acute angles and determine the sine and
cosine ratios.
3. Create a table similar to that of the text’s investigation on the board.
4. Have students enter the information for their angle on the board. NOTE: have
students enter their angles from smallest to largest measure.
Classroom Discussion

Are there limits on the values of sine and cosine?

What are the limits?

Why are there limits?

How precise a value is appropriate for sine and cosine?
Extension

What pattern do you notice in the values of sine and cosine as the angles increase?
You may wish to assign questions in which the student must use sine or
cosine to solve for side lengths of triangles.
The interactive trig explore applet demonstrates the relationship between the sine
measure and the sides of the triangle. This applet allows you to use all of the trig
ratios.
The sine box applet and the cosine box applet can also be used to introduce or
show the relationship between the sine and cosine of an angle and the ratio of the
sides.
Mathematics 10C
Trigonometry
Page 51 of 60
Students can find a missing angle in a triangle if the other two are known by
simply subtracting them from 180 .
Students will be able to apply their skills isolating a variable in a proportion to find
either the leg or hypotenuse of a triangle.
Show students how to calculate an angle measure given two sides.
Students need to be told what it means to solve a triangle.
Resources
Math 10 (McGraw Hill: sec 3.2)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 2.4, 2.5)
interactive trig explore applet
http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2
=AB.MATH.JR.SHAP.TRIG&lesson=html/object_interactives/trigonometry/use_it.html
sine box applet
http://www.ies.co.jp/math/java/trig/sinBox/sinBox.html
cosine box applet
http://www.ies.co.jp/math/products/trig/applets/cosbox/cosbox.html
Supporting
Further, and possibly individual, practice with the applets.
Mathematics 10C
Trigonometry
Page 52 of 60
Assessment
Solve the following triangle in as many ways as possible. Choose one technique
that you did not use to solve for a specific unknown, and use it to verify your
answer. This will require you to solve for the same unknown in more than one
way.
14
55
Glossary
 – Greek letter “theta” often used to represent an unknown angle
cosine – the ratio of the adjacent side to the hypotenuse in a right triangle
inverse operations – a mathematical operation that reverses another operation
sine – the ratio of the opposite side to the hypotenuse in a right triangle
supplementary angles – two angles whose sum is 180o
solve – give a solution to a problem (i.e. find all the missing sides and angles in a
triangle)
Mathematics 10C
Trigonometry
Page 53 of 60
Lesson 4
Applications of Trigonometric Ratios in One and Two Triangle Questions
STAGE 1
BIG IDEA:
It is important that students experience a wide variety of real-world applications pertaining to
trigonometry, so that they understand that trigonometry is a practical branch of mathematics and a
foundation on which more advanced concepts may be added or built.
.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand that…





different strategies work in different
situations
the Pythagorean Theorem utilizes the
relationship between the sides in a right
triangle
connections exist between trigonometry
and real-life situations.
trigonometry exploits the relationships
between the sides and angles in a triangle
trigonometry is based on a series of
constant ratios

How do I decide which strategy to use?

What answer should be expected and
how precise should It be?

Why do we use sine and cosine ratios?

Where would trigonometry be used?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to…




the primary trigonometric ratios.
that the sum of the squares of the sides in
a right triangle equals the square of the
hypotenuse.
applications using angles of elevation and
angles of depression.
that the primary trigonometric ratios can
be used for right triangles only.

label triangles appropriately.

find the missing side in a  b  c ,
where c is the hypotenuse in a right
triangle.
solve equations involving proportions.
use primary trigonometric ratios to find the
missing parts of right triangles.
design a strategy for decision making.
solve problems using two triangles.




2
2
2
Lesson Summary
Students will be able to recognize situations which require trigonometry. Students
will be able to model scenarios where trigonometry would be useful.
Students will be able to interpret word problems pictorially and solve.
Mathematics 10C
Trigonometry
Page 54 of 60
Lesson Plan
Class Example
The angle of elevation from the top of the Pi Hotel to the Sigma Office Building is
17  . The angle of depression to the base of the Sigma Office Building is 60  . The
height of the Pi Hotel is 150 metres. Determine the height of the Sigma Office
Building to the nearest tenth of a metre.
Challenge:
Develop a real-life situation that requires a trigonometric solution.
You may wish to use some of the students’ responses to the above challenge as
classroom practice or as questions for evaluative purposes.
Students could be encouraged to try to determine the height of a cliff across a river if
given a tape measure and a tool for measuring angles (This could also be the
challenge).
One could measure the sides of a desk and predict the length of the diagonal. Students
could go outside and determine the height of the school; students could determine
angles given various measurements and then use clinometers to verify them.
Students should also practice pictorially representing and solving word problems. Work
may be assigned from textbook. Practice both one and two triangle questions.
Review the relationships between angles formed by a transversal cutting two parallel
lines.
Going Beyond
Students could develop their own trigonometric problems.
Mathematics 10C
Trigonometry
Page 55 of 60
Resources
Foundations and Pre-calculus Mathematics 10(Pearson: sec 2.6, 2.7)
Math 10 (McGraw Hill: sec 3.3)
Supporting
If needed, students could go back to the ratio applets. Students could be encouraged to
come up with their own problems to solve (i.e. given a diagram, develop their own
scenario).
Glossary
angle of elevation – the angle between a horizontal line of sight and the line of sight up
to an object
angle of depression – the angle between a horizontal line of sight and the line of sight
down to an object
Mathematics 10C
Trigonometry
Page 56 of 60
Lesson 5
Designing a Solution Using Trigonometry
STAGE 1
BIG IDEA:
It is important that students experience a wide variety of real-world applications pertaining to
trigonometry, so that they understand that trigonometry is a practical branch of mathematics and a
foundation on which more advanced concepts may be added.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand that…



the Pythagorean Theorem utilizes the
relationship between the sides in a right
triangle.
trigonometry is based on a series of
constant ratios.
trigonometry utilizes the relationships
between the sides and angles in a triangle.

Where would I use trigonometry?

Why do we use sine, cosine and tangent
ratios?

How do I decide which strategy to use?

What answer should be expected and how
precise should it be?
.

connections exist between trigonometry
and real-life situations.

different strategies work in different
situations.
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to…



the primary trigonometric ratios.
applications using angles of elevation and
angles of depression.
that the primary trigonometric ratios can
be used for right triangles only.

label triangles appropriately.

find the missing side in a  b  c ,
where c is the hypotenuse in a right
triangle.
solve equations involving proportions.
use primary trigonometric ratios to find the
missing parts of right triangles.
use a scientific calculator in solving
trigonometric problems.
use angle(s) of elevation and angle(s) of
depression in applications.
design a strategy for decision making.
solve problems using two triangles.






Mathematics 10C
Trigonometry
2
2
2
Page 57 of 60
Lesson Summary
This lesson incorporates all of the trigonometry from this unit. Students should be able
to interpret word problems that require multiple steps to solve.
Lesson Plan
Materials
floor plans (such as those provided in Lesson 1)
tape measures or metre sticks
Objective
Students will use trigonometry to solve a problem involving more than one triangle.
Student Task
Determine the angle between your line of sight or vision from your desk and each end of
the white board.
Students should draw a diagram that includes one line from their desk that is
perpendicular to the board, and two more lines (one to each end of the board). The
student can then solve for 2 triangles, one on either side of the perpendicular line.
Follow-up Discussion

Can/Did you use the following to find the angles? Why or Why not?
o
o
o
o

Pythagorean Theorem
tangent ratio
sine ratio
cosine ratio
Is one of the above better than the others?
Mathematics 10C
Trigonometry
Page 58 of 60
Extension

Experiment with different desk locations to determine ideal angles of vision for
students in your classroom.

Experiment with different desk locations to determine ideal distances from the board
for students in your classroom.

Choose a format that will use this information to create your ideal classroom.
Present this to your class. (Note to teacher: the format could be a scale model, an
architectural drawing using CAD (Computer Assisted Design Software), a proposal
letter to the school board, etc.)
Resources
Math 10 (McGraw Hill: sec 3.3)
Foundations and Pre-calculus Mathematics 10(Pearson: sec 2.6, 2.7)
Mathematics 10C
Trigonometry
Page 59 of 60
ACKNOWLEDGEMENTS
Pictures or Digital Images
Pages 11, 12
http://flickr.com
Pages 22, 24, 25, 30, 32, 33
Photographs provided by Jeremy Klassen
Mathematics 10C
Trigonometry
Page 60 of 60