MATHEMATICS 20-1
Trigonometry
High School collaborative venture with
Edmonton Christian, Harry Ainlay, J. Percy Page, Jasper
Place, Millwoods Christian, Ross Sheppard and W. P.
Wagner
Edm Christian High: Aaron Trimble
Harry Ainlay: Ben Luchkow
Harry Ainlay: Darwin Holt
Harry Ainlay: Lareina Rezewski
Harry Ainlay: Mike Shrimpton
J. Percy Page: Debbie Younger
Jasper Place: Matt Kates
Jasper Place: Sue Dvorack
Millwoods Christian: Patrick Ypma
Ross Sheppard: Patricia Elder
W. P. Wagner: Amber Steinhauer
Facilitator: John Scammell (Consulting Services)
Editor: Jim Reed (Contracted)
2010 - 2011
Mathematics 20-1
Trigonometry
Page 2 of 48
TABLE OF CONTENTS
STAGE 1
DESIRED RESULTS
PAGE
Big Idea
4
Enduring Understandings
4
Essential Questions
4
Knowledge
5
Skills
6
STAGE 2
ASSESSMENT EVIDENCE
Transfer Task (on a separate page which could be photocopied & handed out to
students)
“Tri” Alberta
Teacher Notes for Transfer Task and Rubric
Transfer Task and Rubric
Rubric
Possible Solution
8
9
12
14
STAGE 3 LEARNING PLANS
Lesson #1
Angles in Standard Position
18
Lesson #2
Reference Triangles & Trigonometry Ratios for Angles 0˚ - 360˚
24
Lesson #3
Applying the CAST Rule
29
Lesson #4
Special Angles 0-30-45-60-90
33
Lesson #5 The Sine Law
38
Lesson #6
The Cosine Law
42
Lesson #7
The Ambiguous Case
45
Mathematics 20-1
Trigonometry
Page 3 of 48
Mathematics 20-1
Trigonometry
STAGE 1
Desired Results
Big Idea:
The sine law and cosine law allows us to work with non right triangles. Applications of
trigonometry include surveying, navigation, construction, and calculus.
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to
it often.
Enduring Understandings:
Students will understand …




Angles in a circle can be expressed in a variety of ways.
Primary trigonometric ratios and Pythagorean Theorem only work for right
triangles, while the sine and cosine law will work for all triangles.
Each primary trigonometric ratio is positive in two quadrants and negative in
two quadrants between 0o and 360o (CAST).
Special triangles are useful for determining exact value of trigonometric ratios
with reference angles 0, 30o, 45o and 60o.
Essential Questions:




What is triangulation?
How is a negative angle possible?
How many different ways can you estimate the height of a mountain?
Why is it easier to find the exact value of some trigonometric ratios?
Implementation note:
Ask students to consider one of the
essential questions every lesson or two.
Has their thinking changed or evolved?
Mathematics 20-1
Trigonometry
Page 4 of 48
Knowledge:
Enduring
Understanding
List enduring
understandings (the
fewer the better)
Students will understand…
 Angles in a circle can
be expressed in a
variety of ways.
Students will understand…
 Primary trigonometric
ratios and
Pythagorean Theorem
only work for right
triangles, while the
sine and cosine law
will work for all
triangles.
Students will understand…
 Each primary
trigonometry ratio is
positive in two
quadrants and
negative in two
quadrants between 0o
and 360o (CAST).
Students will understand…
 Special triangles are
useful for determining
exact value of
trigonometric ratios
with reference angles
0, 30o, 45o and 60o.
8888
I*T =
Specific
Outcomes
List the reference
# from the
Alberta Program
of Studies
*T1, T2, T3
T2, T3
T1, T2
T1, T2
Description of
Knowledge
The paraphrased outcome that the group is
targeting
Students will know …
 an angle in standard position, given the
measure of the angle
 without the use of technology, the value of sin
θ, cos θ or tan θ, given any point P (x, y) on the
terminal arm of angle θ, where θ = 0º, 90º,
180º, 270º or 360º
 the sign of a given trigonometric ratio for a
given angle, without the use of technology, and
explain
 the patterns in and among the values of the
sine, cosine and tangent ratios for angles from
0° to 360°
Students will know …
 the Pythagorean theorem or the distance
formula, can be used to calculate the distance
from the origin to a point P (x, y) on the
terminal arm of an angle
 some contextual problems can be solved using
trigonometric ratios
Students will know …
 illustrate, using examples, that the points P ( x,
y), P (−x, y), P (−x,− y) and P (x,− y) are points
on the terminal sides of angles in standard
position that have the same reference angle
Students will know …
 determine the exact value of the sine, cosine or
tangent of a given angle with a reference angle
of 30º, 45º or 60º
 describe patterns in and among the values of
the sine, cosine and tangent ratios for angles
from 0° to 360°
Trigonometry
Mathematics 20-1
Trigonometry
Page 5 of 48
Skills:
Enduring
Understanding
List enduring
understandings (the
fewer the better)
Students will understand…

Students will be able to…
T1
 sketch an angle in standard position, given the
measure of the angle
 determine the reference angle for an angle in
standard position
 explain, using examples, how to determine the
angles from 0° to 360° that have the same
reference angle as a given angle
 illustrate, using examples, that any angle from
90° to 360° is the reflection in the x-axis and/or
the y-axis of its reference angle
 determine the quadrant in which a given angle
in standard position terminates
 draw an angle in standard position given any
point P (x, y) on the terminal arm of the angle
Students will be able to…
T2,3
 determine, using the Pythagorean theorem or
the distance formula, the distance from the
origin to a point P (x, y) on the terminal arm of
an angle
 solve a contextual problem, using trigonometric
ratios
 sketch a diagram to represent a problem that
involves a triangle without a right angle
 solve, using primary trigonometric ratios, a
triangle that is not a right triangle
 explain the steps in a given proof of the sine
law or cosine law
 sketch a diagram and solve a problem, using
the cosine law
 sketch a diagram and solve a problem, using
the sine law
Primary trigonometric
ratios and
Pythagorean Theorem
only work for right
triangles, while the
Sine and cosine law
will work for all
triangles.
Students will understand…

Description of
Skills
The paraphrased outcome that the group is
targeting
Angles in a circle can
be expressed in a
variety of ways.
Students will understand…

Specific
Outcomes
List the reference
# from the
Alberta Program
of Studies
Students will be able to…
T1, T2
 illustrate, using examples, that the points P (x,
y), P (−x, y), P (−x,− y) and P (x,− y) are points
on the terminal sides of angles in standard
position that have the same reference angle
Each ratio is positive in
two quadrants and
negative in two
quadrants between 0o
Mathematics 20-1
Trigonometry
Page 6 of 48
 determine the value of sin θ, cos θ or tan θ,
given any point P (x, y) on the terminal arm of
angle θ
 determine the sign of a given trigonometric
ratio for a given angle, without the use of
technology, and explain
 solve, for all values of θ, an equation of the
form sin θ = a or cos θ = a, where −1 ≤ a ≤ 1,
and an equation of the form tan θ = a, where a
is a real number
 determine the exact value of the sine, cosine or
tangent of a given angle with a reference angle
of 30º, 45º or 60º
 describe patterns in and among the values of
the sine, cosine and tangent ratios for angles
from 0° to 360°
and 360o (CAST).
Students will understand…

Special triangles are
useful for determining
exact value of
trigonometric ratios
with reference angles
0, 30o, 40o and 60o
*T = Trigonometry
Mathematics 20-1
Students will be able to…
T2


determine, without the use of technology, the
value of sin θ, cos θ or tan θ, given any point P
(x, y) on the terminal arm of angle θ, where θ =
0º, 90º, 180º, 270º or 360º
determine the exact value of the sine, cosine or
tangent of a given angle with a reference angle
of 30º, 45º or 60º
Implementation note:
Teachers need to continually ask
themselves, if their students are
acquiring the knowledge and skills
needed for the unit.
Trigonometry
Page 7 of 48
STAGE 2
Assessment Evidence
1 Desired Results Desired Results
Desired Results Desired Results
“Tri” Alberta
Teacher Notes
There is one transfer task to evaluate student understanding of the concepts in relation
to Trigonometry. A photocopy-ready version of the transfer task is 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:


Estimate the distances and angles between four chosen locations, where
Edmonton is the origin. The estimations will be verified using the sine law and
cosine law.
Solve for reference, principle and, co-terminal angles, and approximate
coordinates of each location.
Materials:




ruler
protractor
map – provided in transfer task
Internet

You may wish to assign Part A as a take home assignment. This will
eliminate the need for Internet access.
Mathematics 20-1
Trigonometry
Page 8 of 48
“Tri” Alberta - Student Assessment Task
You have access to your own helicopter to visit four locations over two
days. You can only visit two locations per day and for each excursion, you
must visit locations in adjacent quadrants. You are travelling back to
Edmonton at the end of each day. Your two-day trip must include all 4
quadrants.
Part A:



Choose four locations in Alberta, each in a different quadrant.
You must research two important facts to explain why you are
choosing these locations. (for example: Vegreville has a giant
Ukrainian egg (pysanka).
Draw two triangles of your excursions, starting and ending in
Edmonton. (Each triangle must contain two locations in adjacent
quadrants and Edmonton.)
Part B:







Excursion One
Measure all three sides of one triangle using a ruler. Use the map
scale to determine the actual distances. (Round the distances to
the nearest km.)
Determine the measure of all three angles using the cosine law.
(Round your answers to the nearest degree.)
Verify each angle using the sine law.
Part C:

Background Information
Excursion Two
Measure the distances from Edmonton to the other two locations in
the second triangle. Using a ruler convert the measured distances
to the actual distances using the map scale.
Measure the angle at Edmonton, in between those two distances,
using a protractor to the nearest degree.
Determine the distance between the two non-Edmonton locations
using the cosine law.
Determine the other two angles using the sine law.
Verify algebraically the measured angle and distances of the
previous bullets. Verify the angle using the sine law, and sides
using the cosine law.
“Tri” Alberta - Student Assessment Task
Part D:
Reference Angles
 Name and measure the reference angles for each one of your
locations (Use a protractor to the nearest degree.)
 Name and determine both the principle, and negative co-terminal
angles for each location.
 Provide an approximate coordinate for each location. (Use the scale
on the map and round to the nearest km.)
Glossary
adjacent quadrants – Two quadrants beside each other
ambiguous case – From the given information the solution for the triangle is not
clear: there might be one triangle, two triangles, or no triangle [Math 20-1 (McGrawHill Ryerson: page 104)]
angle in standard position - The location of an angle in the plane in which the vertex
is at the origin, the initial arm lies along the positive x-axis, and the terminal arm is free
to rotate
cosine law - the relationship between the cosine of an angle and the lengths of the
three sides of any triangle
c2 = a2 + b2 = 2ab cos C
coterminal angle – An angle in standard position with the same initial arm and
terminal arm as the principal angle. Adding or subtracting the principal angle by a
multiple of 360° finds coterminal angles.
exact value – Answers involving radicals or fractions are exact, unlike approximated
decimal values [Math 20-1 (McGraw-Hill Ryerson: page 587)]
oblique triangle – A triangle that is not a right triangle
principle angle - The smallest positive angle
quadrantal angle – an angle in standard position where the terminal arm is on the xor y-axis. Examples are 0°, 90°, 180°, 270° and 360°.
reference angle – The acute angle formed by the terminal arm of an angle in
standard position and the x-axis
rotation angles:
 positive angle - An angle in standard position swept out by a counterclockwise
rotation of its terminal arm
 negative angle - An angle in standard position swept out by a clockwise rotation
of its terminal arm
reference triangle – A right triangle with a reference angle as one of its vertices
sine law - The lengths of the sides are proportional to the sines of the opposite angles
a
b
c
=
=
sin A sin B sin C
Assessment
Mathematics 20-1
Trigonometry - Rubric
Level
Criteria
Excellent
4
Proficient
3
Adequate
2
Limited*
1
Insufficient /
Blank*
Student is able to
determine and
verify three out of
six angles and
sides in both or
either oblique
triangles
Student is able to
determine and
verify three out of
six angles and
sides in both or
either oblique
triangles
Student is able to
determine and/or
verify one or two
angles or sides in
either or both
oblique triangles
Student is unable
to determine or
verify any angles
or sides in either
oblique triangle
Student is able to
determine and/or
verify one or two
angles or sides in
either or both
oblique triangles
Student is unable
to determine or
verify any angles
or sides in either
oblique triangle
Performs
Algebraic
Operations and
Verification using
Sine Law (Parts
B and C)
Student is able to
determine and
verify all angles
and sides in both
oblique triangles
Student is able to
determine and
verify four or five
out of six angles
and sides in both
oblique triangles
Performs
Algebraic
Operations and
Verification using
Cosine Law
(Parts B and C)
Student is able to
determine and
verify all angles
and sides in both
oblique triangles
Student is able to
determine and
verify four or five
out of six angles
and sides in both
oblique triangles
Solving for
Reference,
Principle,
Negative Coterminal Angles
and Coordinates
(Part D)
Student is able to
solve all angles
and coordinates
for each location
Student is able to
solve twelve out
of sixteen angles
and coordinates
for each location
Student is able to
solve eight out of
sixteen angles
and coordinates
for each location
Student is able to
solve four out of
sixteen angles
and coordinates
for each location
Student is unable
to solve any
angles or
coordinates for
each location
Presentation
(Parts A – D)
Student has
presented all clear
and accurate
diagrams
solutions, and
Provides relevant
reasons for
locations chosen
Student has
presented most
clear and accurate
diagrams,
solutions, and
provides relevant
reasons for
locations chosen.
Student has
presented some
clear and accurate
diagrams,
solutions, and
provides relevant
reasons for
locations chosen.
Student has
presented no clear
and accurate
diagrams,
solutions, and fails
to provide relevant
reasons for
locations chosen
Student has not
presented.
Possible Solution to “Tri” Alberta
Mathematics 20-1
Trigonometry
Page 14 of 48


Answers will vary depending on the locations students chose.
The following solutions are from Edmonton to Fort Chipewyan to Spirit River for
excursion one and Edmonton to Crowsnest Pass to Oyen for excursion two.
Part A:
B.
Fort Chipewyan
- oldest community in Alberta
- population 1012
C.
Spirit River - "Chepi Sepe" - Cree for Ghost or Spirit River
-
D.
Crowsnest Pass – Frank Slide covered the city
- "Burmis Tree" – 700 year old tree died and fell but was re-built
E.
Oyen - 1908 Andrew Oyen walked from Spokane, Washington to Oyen
- Canada’s National Women’s Hockey team coach (2 olympic gold medals) is
from Oyen
Part B:
Measured Distances:
Edmonton (A) to Fort Chipewyan (B) = 9.8 cm x 60 = 588 km
Edmonton ( A) to Spirit River (C) = 6.7 x 60 = 402 km
Fort Chipewyan (B) to Spirit River (C) = 9.3 x 60 = 558 km
Solved Angles:
4022 + 5882 - 5582
cos A =
2 ´ 402 ´ 588
Ð A = 65.5°
cos B =
5582 + 5882 - 4022
2 ´ 558 ´ 588
Ð B = 41.0°
cos C =
5582 + 4022 - 5882
2 ´ 558 ´ 402
Ð C = 73.5°
Mathematics 20-1
Trigonometry
Page 15 of 48
Verify:
sin A sin 41.0
=
558
402
Ð A = 65.5°
sin 65.5
sin B
=
558
402
Ð B = 41.0°
sin 65
sin C
=
558
588
Ð C = 73.5°
Part C:
Measured Distances:
Ð DAE = 53.0°
Edmonton (A) to Crowsnest Pass (D) = 7.2 cm x 60 = 432 km
Edmonton (A) to Owen (E) = 5.3 x 60 = 318 km
Calculations
(DE)2 = 3182 + 4322 – 2 x 318 x 432 x cos 53.0o
DE = 349.9 km
sin D sin 53.0
=
318
349.9
Ð D = 46.5°
sin E sin 53.0
=
432
349.9
Ð E = 80.4°
Mathematics 20-1
Trigonometry
Page 16 of 48
Verify:
(AD)2 = 3182 + 349.92 – 2 x 318 x 349.9 x cos 80.4o
DE = 431.8 km
(AE)2 = 4322 + 349.92 – 2 x 432 x 349.9 x cos 46.5o
AE = 317.7 km
Part D:
<BAQ
reference angle = 77o
principle angle = 77 o
coterminal angle = -283o
B = (126, 576)
<PAC
reference angle = 38o
principle angle = 142o
coterminal angle = -218o
C = (-324, 246)
<PAD
reference angle = 81o
principle angle = 261 o
coterminal angle = -99 o
D = (-66, -426)
<QAE
reference angle = 46o
principle angle = 314 o
coterminal angle = -46o
E = (222, -228)
Mathematics 20-1
Trigonometry
Page 17 of 48
STAGE 3
Learning Plans
Lesson 1
Angles in Standard Position
STAGE 1
BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of
trigonometry include surveying, navigation, construction, and calculus.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …





Angles in a circle can be expressed in a
variety of ways.
Special triangles are useful for determining
exact value of trigonometric ratios with
reference angles 0o, 30o, 40o and 60o.

What is triangulation?
How is a negative angle possible?
How many different ways can you estimate
the height of a mountain?
Why is it easier to find the exact value of
some trigonometric ratios?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …




an angle in standard position, given the
measure of the angle
the reference angle for an angle in standard
position
the exact value of the sine, cosine or tangent
of a given angle with a reference angle of 30º,
45º or 60º






sketch an angle in standard position, given
the measure of the angle
determine the reference angle for an angle in
standard position
explain, using examples, how to determine
the angles from 0° to 360° that have the same
reference angle as a given angle
illustrate, using examples, that any angle from
90° to 360° is the reflection in the x-axis
and/or the y-axis of its reference angle
determine the quadrant in which a given angle
in standard position terminates
draw an angle in standard position given any
point P (x, y) on the terminal arm of the angle
illustrate, using examples, that the points P (
x, y), P (−x, y), P (−x,− y) and P (x,− y) are
points on the terminal sides of angles in
standard position that have the same
reference angle
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 20-1
Trigonometry
Page 18 of 48
Lesson Summary

Demonstrate/explore angles in standard position.
Lesson Plan
Hook
Show a video clip of someone doing a three-sixty (360˚ turn).
Video 1: http://www.youtube.com/watch?v=wLnx3Utj9ik&NR=1
Video 2: http://www.youtube.com/watch?v=bwWfSgUYggc&feature=related
Video 1
Video 2
 files were added to the EPSB Understanding by Design share site
Ask them to estimate how many degrees the person spun. Have students try to do a 90˚, a
180˚, a 270˚ turn (others if you deem it important to do so. Ask which way they spun
(clockwise/counter-clockwise). Did the people who went in opposite directions create the
same angle? Discuss.
Lesson Goal
Students will demonstrate an understanding of angles in standard position (0˚ - 360˚).
Activate Prior Knowledge
 Discuss the idea of 360˚ as a circle (skateboarding, snowboarding etc.)
 Remind students about acute/right/obtuse/straight/reflex angles
 Review the concept of Cartesian plane, numbering of quadrants and how to locate
points on the plane.
Mathematics 20-1
Trigonometry
Page 19 of 48
Lesson
1. Define the parts of an angle (initial arm, terminal arm, vertex, rotation angle, standard
position).
2. Have students draw a coordinate plane, and draw lines estimating multiples of 30˚ and
45˚. Check with a protractor if required.
3. Define reference angle, find a reference angle in all of the examples students have
already done.
Use this applet to show the reflections of the angles in the other 3 quadrants. Talk about
the colours of the initial arm vs. the terminal arm (blue vs. red).
source: http://www.ronblond.com/MathGlossary/Division04/TrigCircle/
Mathematics 20-1
Trigonometry
Page 20 of 48
Option 1: Give students a reference angle and a quadrant and have them tell you the
rotation angle.
Option 2: Give a rotation angle and ask for the reference angle and quadrant number.
Example:
Reference 
30˚
Quadrant
4
Rotation 
210˚
67˚
2
130˚
4. Use examples to illustrate that the points P (x, y), P (−x, y), P (−x, −y) and P (x, −y) are
points on the terminal sides of angles in standard position that have the same reference
angle.
5. Give the students the point (6, 8) for P (x, y), and asks them to write down the reflected
points in quadrants II, III and IV:



P (−x, y)
P (−x, −y)
P (x, −y).
6. Briefly discuss the connection between the original point given and a triangle. From the
original point on the terminal arm (and the reflected points in other 3 quadrants) draw a
vertical line to the x-axis.
Question: What do you notice about the 4 triangles?
Answer: They are congruent and are right-angled.
Refer back to the applet.
source: http://www.ronblond.com/MathGlossary/Division04/TrigCircle/
Mathematics 20-1
Trigonometry
Page 21 of 48
The next lesson will focus on the right-angled triangles that we have created. The angles
that determine the height of these triangles are called reference angles.
Going Beyond
Discuss or have students research the concepts of negative angles versus positive angles,
co-terminal angles, principal angles etc.
http://staff.argyll.epsb.ca/jreed/math30p/trigonometry/angles.htm
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 2.1, pages 74-87)
Ron Blond’s Trig Applet http://www.ronblond.com/MathGlossary/Division04/TrigCircle/
Supporting
Assessment
Mathematics 20-1
Trigonometry
Page 22 of 48
Option 1: Exit slip showing rotation angle, quadrant, reference angle. Have students fill in the
blanks.
Option 2: Exit slip “It is 3:15 pm, if you rewind your clock and the minute hand rotates 120˚,
what time is it?”
Answer: 11:55 am
Glossary
angle in standard position - The location of an angle in the plane in which the vertex is at
the origin, the initial arm lies along the positive x-axis, and the terminal arm is free to rotate
exact value – Answers involving radicals or fractions are exact, unlike approximated decimal
values [Math 20-1 (McGraw-Hill Ryerson: page 587)]
initial arm - For an angle in standard position, the arm along the positive x-axis
reference angle – The acute angle formed by the terminal arm of an angle in standard
position and the x-axis
rotation angles:
 positive angle - An angle in standard position swept out by a counterclockwise rotation
of its terminal arm
 negative angle - An angle in standard position swept out by a clockwise rotation of its
terminal arm
terminal arm – For an angle in standard position, the arm that is free to rotate
vertex - Common endpoint of two rays that form the angle.
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.
Other
Mathematics 20-1
Trigonometry
Page 23 of 48
Lesson 2
Reference Triangles & Trigonometry Ratios for Angles 0˚ - 360˚
STAGE 1
BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of
trigonometry include surveying, navigation, construction, and calculus.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …





Angles in a circle can be expressed in a
variety of ways.
Each primary trigonometric ratio is positive in
two quadrants and negative in two quadrants
between 0o and 360o (CAST).
Special triangles are useful for determining
exact value of trigonometric ratios with
reference angles 0o, 30o, 40o and 60o.
KNOWLEDGE:
Students will know …







SKILLS:
Students will be able to …
an angle in standard position, given the
measure of the angle
the reference angle for an angle in standard
position
the Pythagorean theorem or the distance
formula, can be used to calculate the distance
from the origin to a point P (x, y) on the
terminal arm of an angle
the sign of a given trigonometric ratio for a
given angle, without the use of technology,
and explain
the exact value of the sine, cosine or tangent
of a given angle with a reference angle of 30º,
45º or 60º
the patterns in and among the values of the
sine, cosine and tangent ratios for angles from
0° to 360°
some contextual problems can be solved
using trigonometric ratios
Mathematics 20-1

What is triangulation?
How many different ways can you estimate
the height of a mountain?
Why is it easier to find the exact value of
some trigonometric ratios?
Trigonometry
 determine, using the Pythagorean theorem or
the distance formula, the distance from the
origin to a point P (x, y) on the terminal arm of
an angle
 determine the value of sin θ, cos θ or tan θ,
given any point P (x, y) on the terminal arm of
angle θ.
 determine, without the use of technology, the
value of sin θ, cos θ or tan θ, given any point
P (x, y) on the terminal arm of angle θ, where
θ = 0º, 90º, 180º, 270º or 360º
 determine the sign of a given trigonometric
ratio for a given angle, without the use of
technology, and explain
 solve, for all values of θ, an equation of the
form sin θ = a or cos θ = a, where −1 ≤ a ≤ 1,
and an equation of the form tan θ = a, where a
is a real number
 determine the exact value of the sine, cosine
or tangent of a given angle with a reference
angle of 30º, 45º or 60º
 describe patterns in and among the values of
the sine, cosine and tangent ratios for angles
from 0° to 360°
 solve a contextual problem, using
trigonometric ratios
Page 24 of 48
Lesson Summary



Relate the 3 primary trigonometric ratios to angles in standard position.
Determine the sign of a given trig ratio for a given angle, without the use of technology
and explain.
Describe patterns in and among the values of sine, cosine, and tangent ratios for
angles from 0˚ - 360˚.
Lesson Plan
Hook – Look at the school flagpole. How can we measure to the top? Go out to the pole with
a clinometer and a tape measure.
Lesson Goal
Relate SOH CAH TOA and Pythagorean theorem to reference angles.
Activate Prior Knowledge


Review SOH CAH TOA and Pythagorean Theorem using a right triangle.
Define the primary trigonometric ratios and the sides of a triangle.
Lesson
Revisit the triangle and its congruent reflections from the previous day where
P (x, y) = P (6, 8)
Define: Reference Triangle.
Take each triangle separately and use the Pythagorean theorem to find the distance from
point P to the origin. Does the distance vary in the other 3 quadrants?
Show the primary trigonometric ratios in conjunction with the terminal arm in each quadrant.
Q: What do we know about the reference triangles?
A: They are all congruent.
Have students calculate the reference angle and the rotation angles.
Mathematics 20-1
Trigonometry
Page 25 of 48
Continue using the 4 quadrants and ask “Where is cosine positive? Where is sine positive?
Where is tan positive?” Define the CAST rule.
You may want to return to Ron Blond’s applet at this point and look at it again.
source: http://www.ronblond.com/MathGlossary/Division04/TrigCircle/
Ask students if the given trig ratios will be positive or negative.
tan 217˚
cos 122˚
sin 300˚
cos 50˚
Mathematics 20-1
Trigonometry
Page 26 of 48
In which quadrant is each of the following located?
4
cos q = 5
2
sin q =
7
tan q = -2
1
2
sin q = -0.759
tan q = 1.5
cos q =
Revisit the CAST rule by looking at the results of the above activity.
As a demonstration, draw a Cartesian plane, choose a point in quadrant I, and draw a
terminal arm through the point. Determine:
 distance from the origin to the point
 the exact value of the sin θ, cos θ, and tan θ
 the reference angle
Student Pairs Activity: Students pick any point in quadrant I and draw a terminal arm.
Students should then swap papers and determine:
 distance from the origin to the point
 the exact value of the sin θ, cos θ, and tan θ
 the reference angle
Going Beyond
Continue the above lesson using negative angles and angles beyond 360 
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 2.2, pages 88-99)
source: http://www.ronblond.com/MathGlossary/Division04/TrigCircle/
Supporting
Mathematics 20-1
Trigonometry
Page 27 of 48
Assessment
Glossary
angle in standard position - The location of an angle in the plane in which the vertex is at
the origin, the initial arm lies along the positive x-axis, and the terminal arm is free to rotate
initial arm - For an angle in standard position, the arm along the positive x-axis
quadrantal angle – an angle in standard position where the terminal arm is on the x- or yaxis. Examples are 0°, 90°, 180°, 270° and 360°.
reference angle – The acute angle formed by the terminal arm of an angle in standard
position and the x-axis
rotation angles:
 positive angle - An angle in standard position swept out by a counterclockwise rotation
of its terminal arm
 negative angle - An angle in standard position swept out by a clockwise rotation of its
terminal arm
reference triangle – A right triangle with a reference angle as one of its vertices
terminal arm – For an angle in standard position, the arm that is free to rotate
vertex - Common endpoint of two rays that form the angle.
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.
Other
Mathematics 20-1
Trigonometry
Page 28 of 48
Lesson 3
Applying the CAST Rule
STAGE 1
BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of
trigonometry include surveying, navigation, construction, and calculus.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …




Angles in a circle can be expressed in a
variety of ways.
Each primary trigonometric ratio is positive in
two quadrants and negative in two quadrants
between 0o and 360o (CAST).

What is triangulation?
How many different ways can you estimate
the height of a mountain?
Why is it easier to find the exact value of
some trigonometric ratios?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 the sign of a given trigonometric ratio for a
given angle, without the use of technology,
and explain
 the exact value of the sine, cosine or tangent
of a given angle with a reference angle of 30º,
45º or 60º
 the patterns in and among the values of the
sine, cosine and tangent ratios for angles from
0° to 360°
 some contextual problems can be solved
using trigonometric ratios
Mathematics 20-1
Trigonometry
 determine the value of sin θ, cos θ or tan θ,
given any point P (x, y) on the terminal arm of
angle θ
 determine, without the use of technology, the
value of sin θ, cos θ or tan θ, given any point
P (x, y) on the terminal arm of angle θ, where
θ = 0º, 90º, 180º, 270º or 360º
 determine the sign of a given trigonometric
ratio for a given angle, without the use of
technology, and explain
 solve, for all values of θ, an equation of the
form sin θ = a or cos θ = a, where −1 ≤ a ≤ 1,
and an equation of the form tan θ = a, where a
is a real number
 determine the exact value of the sine, cosine
or tangent of a given angle with a reference
angle of 30º, 45º or 60º
 describe patterns in and among the values of
the sine, cosine and tangent ratios for angles
from 0° to 360°
 solve a contextual problem, using
trigonometric ratios
Page 29 of 48
Lesson Summary


Review the CAST rule
Solve an equation of the form sin θ = a or cos θ = a, for all values of θ, where a  1 ,
and tan θ = a, where a is real.
Lesson Plan - Applying the CAST Rule
Hook
Ask students for “real-life” uses of trigonometry (navigation, construction industry, astronomy,
space exploration, design etc.). Make a list. Then show this video for fun, which shows 5
ways that trigonometry can be used in your everyday life.
Video 3: http://www.youtube.com/watch?v=T_19ZxaCP3g&feature=related
(apologies to non-Gaga fans)
Video 3
 file was added to the EPSB Understanding by Design share site
Lesson Goal
Use the CAST rule and reference triangles to solve trig equations where the angle is between
0˚ and 360˚.
Activate Prior Knowledge
Review the CAST rule and reference triangles. May want to use Ron Blond’s applet again
here.
source: http://www.ronblond.com/MathGlossary/Division04/TrigCircle/
Lesson
Optional: Make a large coordinate plane using masking tape on the floor. Have students
stand on the outside of the grid.
Teacher says: “Tan Negative”, or “Sin Positive”, etc. and students must run to a quadrant
where that is correct.
Draw the coordinate plane on the board, ask the students to put CAST in the correct
quadrants. Ask them the meaning of it.
Mathematics 20-1
Trigonometry
Page 30 of 48
Review reference angles and triangles by giving them this question or one like it:
Examples:
1. Ask if θ = 140o, what is the reference angle? Draw the reference triangle and find all other
angles that have the same reference angle. Use your calculator to find sin θ, cos θ, and
tan θ of every one of those angles in all four quadrants.
1
,
2
a) Find cos θ and tan θ as exact values.
b) Find all possible values of θ, where θ is between 0˚ and 360˚.
1
3. If cos q = ,
2
a) Find cos θ and tan θ as exact values.
b) Find all possible values of θ, where θ is between 0˚ and 360˚.
1
4. If tan q = ,
2
a) Find cos θ and tan θ as exact values.
b) Find all possible values of θ, where θ is between 0 and 360˚.
2. If sin q =
5. Point P (-1, -8) is on the terminal arm of an angle. Find the angle.
6. If cos q = -0.327 , Find all possible values of θ, where θ is between 0˚ and 360˚. Round
to the nearest degree.
Going Beyond
9
and a sin q + b = 0 etc. (You may want to teach the
16
quadratics unit first, so students are familiar with how to solve.)
Solve equations like sin2 q =
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 2.2)
http://www.ronblond.com/MathGlossary/Division04/TrigCircle/
5 Ways to use Trigonometry in Everyday Life
Video 3: http://www.youtube.com/watch?v=T_19ZxaCP3g&feature=related
Video 3
 file was added to the EPSB Understanding by Design share site
Mathematics 20-1
Trigonometry
Page 31 of 48
Supporting
Assessment
4
Exit Slip: If sin q = , find all possible values of θ, where θ is between 0 and 360 degrees.
9
Round to the nearest degree.
Glossary
angle in standard position - The location of an angle in the plane in which the vertex is at
the origin, the initial arm lies along the positive x-axis, and the terminal arm is free to rotate
initial arm - For an angle in standard position, the arm along the positive x-axis
quadrantal angle – an angle in standard position where the terminal arm is on the x- or yaxis. Examples are 0°, 90°, 180°, 270° and 360°.
reference angle – The acute angle formed by the terminal arm of an angle in standard
position and the x-axis
rotation angles:
 positive angle - An angle in standard position swept out by a counterclockwise rotation
of its terminal arm
 negative angle - An angle in standard position swept out by a clockwise rotation of its
terminal arm
reference triangle – A right triangle with a reference angle as one of its vertices
terminal arm – For an angle in standard position, the arm that is free to rotate
vertex - Common endpoint of two rays that form the angle.
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.
Other
Mathematics 20-1
Trigonometry
Page 32 of 48
Lesson 4
Special Angles 0-30-45-60-90
STAGE 1
BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of
trigonometry include surveying, navigation, construction, and calculus.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …






Angles in a circle can be expressed in a
variety of ways.
Each primary trigonometric ratio is positive in
two quadrants and negative in two quadrants
between 0o and 360o (CAST).
Special triangles are useful for determining
exact value of trigonometric ratios with
reference angles 0o, 30o, 40o and 60o.

What is triangulation?
How is a negative angle possible?
How many different ways can you estimate
the height of a mountain?
Why is it easier to find the exact value of
some trigonometric ratios?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 the reference angle for an angle in standard
position
 the Pythagorean theorem or the distance
formula, can be used to calculate the distance
from the origin to a point P (x, y) on the
terminal arm of an angle
 the sign of a given trigonometric ratio for a
given angle, without the use of technology,
and explain
 the exact value of the sine, cosine or tangent
of a given angle with a reference angle of 30º,
45º or 60º
 the patterns in and among the values of the
sine, cosine and tangent ratios for angles from
0° to 360°
 some contextual problems can be solved
using trigonometric ratios
Mathematics 20-1
Trigonometry
 determine, using the Pythagorean theorem or
the distance formula, the distance from the
origin to a point P (x, y) on the terminal arm of
an angle
 determine the value of sin θ, cos θ or tan θ,
given any point P (x, y) on the terminal arm of
angle θ
 determine, without the use of technology, the
value of sin θ, cos θ or tan θ, given any point
P (x, y) on the terminal arm of angle θ, where
θ = 0º, 90º, 180º, 270º or 360º
 determine the sign of a given trigonometric
ratio for a given angle, without the use of
technology, and explain
 solve, for all values of θ, an equation of the
form sin θ = a or cos θ = a, where −1 ≤ a ≤ 1,
and an equation of the form tan θ = a, where a
is a real number
 determine the exact value of the sine, cosine
or tangent of a given angle with a reference
angle of 30º, 45º or 60º
 describe patterns in and among the values of
the sine, cosine and tangent ratios for angles
from 0° to 360°
 solve a contextual problem, using
trigonometric ratios
Page 33 of 48
Lesson Summary


Determine the exact value of sine, cosine, or tangent with a given angle, with a
reference angle of 0º, 30º, 45º, 60º or 90º.
Determine all possible angles between 0º and 360º, without the use of technology,
given an exact trigonometric ratio.
Lesson Plan
Hook
See www.nga.gov/.../sculpturegarden/sculpture/sculpture12.shtm link to
structure/sculpture built entirely of equilateral and isosceles triangles. It’s famous.
Lesson Goal
To determine the exact value of sine, cosine, or tangent with a given angle, with a reference
angle of 30º, 45º or 60º and determine all possible angles between 0º and 360º, without the
use of technology, given an exact trigonometric ratio.
Activate Prior Knowledge
Review previous day’s homework.
Lesson
Draw a random square on the page (big or small). Tell students to decide how long the side
is (1 unit, 2 units, 3 units etc.) Draw a diagonal and ask how many degrees the other 2 angles
are. Given your chosen lengths, find the length of the diagonal. Find the 3 primary
trigonometric ratios for the angle. Record the answers from 3 students and draw their
diagrams on the board and students will see that similar triangles were created. The answers
are the same no matter what size of square they started with. (Note: Hopefully you are
teaching this AFTER you have taught Rational Expressions, since they will have to
rationalise the denominator in order to see that the trig ratios are equal.)
2
1
1
Mathematics 20-1
3 2
2
2
Trigonometry
3
Page 34 of 48
3
Although any square will work, since all of the other triangles drawn are similar triangles,
discuss why it is easiest to work with the unit square. Confirm the answers for the
trigonometry ratios using the calculator. Discuss what an exact value of a trigonometry ratio
is vs a rounded value.
Next, start with an equilateral triangle with sides of 2 units, cut it in half to make 2 right
triangles (in order to use SOH CAH TOA). One of the angles becomes 30º. Find the sine,
cosine and tangent ratios of the 30º and 60º angles.
Ask the students if it would make a difference if our original isosceles triangle only had oneunit lengths.
Those steps should lead to these 2 special triangles. Memorise these triangles.
Examples:
1. Using the CAST rule and these special triangles, find:
a) sin 240°
Suggested steps:
 Draw the reference triangle on the coordinate plane and find the reference angle.
Mathematics 20-1
Trigonometry
Page 35 of 48


Have the students decide whether the sine ratio is positive or negative in that
quadrant.
In this case, since 60º is the reference angle, students should refer to the 30-6090 triangle and use the sine ratio for 60º.

Therefore the sin 240º is equal to
- 3
.
2
b) cos 150º
c) tan 315º
2. Given a trigonometric ratio, without the use of technology, find all possible values of θ
from 0º to 360º.
1
a) sin q = 2
Suggested steps:
 Using the cast rule, draw all possible reference triangles in the coordinate plane.
 Fill in the ratio (in this case the opposite and adjacent sides) to determine the
special triangle & reference angle that will be used to solve the problem.
 Using the diagram, fill in the reference angle and calculate both of the rotation
angles.
 Check your answer using your calculator.
b) cos q =
3
2
c) tan q = 3
3. Draw a right triangle that includes a 30º angle in
x
standard position. Draw dotted lines to indicate 20º, 10º,
0
and 5º. Discuss what is happening to the lengths of the
30 
initial and terminal arm and the triangle height as the
x
angle in standard position approaches 0º. Also discuss
what is happening to the second acute angle.
 The initial and terminal arms are approaching the same length; the height is
approaching 0.
 As the first approaches 0º, the second acute angle approaches 90º. The discussion
should include that 0º and 90º are complementary angles. Can both exist in the same
right triangle?
 Determine the primary trigonometry ratios for right triangles with a 0º angle in standard
position.
Use the same starting triangle. Draw dotted lines to indicate 45º, 60º, 70º, 80º, and 85º
angles in standard position. Discuss what is happening to the lengths of the initial and
terminal arm and the triangle height.
 The terminal arm and height are approaching the same length; the initial arm is
approaching 0º.
 Determine the primary trigonometry ratios from the right triangle with a 90º angle in
standard position.
Mathematics 20-1
Trigonometry
Page 36 of 48
If you would like to emphasize the limit as the 2 acute angles approach 0º and 90º,
consider: http://www.learnalberta.ca/content/memg/index.html
or http://staff.argyll.epsb.ca/jreed/math9/strand3/trigonometry.htm . Select [Trig Ratios] [Functions].
Going Beyond
Use the unit circle instead of or in addition to triangles. Go beyond 360º and/or deal with
negative angles.
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 2.2)
Supporting
Assessment
Glossary
angle in standard position - The location of an angle in the plane in which the vertex is at
the origin, the initial arm lies along the positive x-axis, and the terminal arm is free to rotate
quadrantal angle – an angle in standard position where the terminal arm is on the x- or yaxis. Examples are 0°, 90°, 180°, 270° and 360°.
reference angle – The acute angle formed by the terminal arm of an angle in standard
position and the x-axis
rotation angles:
 positive angle - An angle in standard position swept out by a counterclockwise rotation
of its terminal arm
 negative angle - An angle in standard position swept out by a clockwise rotation of its
terminal arm
reference triangle – A right triangle with a reference angle as one of its vertices
Other
Mathematics 20-1
Trigonometry
Page 37 of 48
Lesson 5
The Sine Law
STAGE 1
BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of
trigonometry include surveying, navigation, construction, and calculus.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …





Angles in a circle can be expressed in a
variety of ways.
Primary trigonometric ratios and Pythagorean
Theorem only work for right triangles, while
sine and cosine law will work for all triangles.

What is triangulation?
How is a negative angle possible?
How many different ways can you estimate
the height of a mountain?
Why is it easier to find the exact value of
some trigonometric ratios?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …






an angle in standard position, given the
measure of the angle
the reference angle for an angle in standard
position
the Pythagorean theorem or the distance
formula, can be used to calculate the distance
from the origin to a point P (x, y) on the
terminal arm of an angle
the sign of a given trigonometric ratio for a
given angle, without the use of technology,
and explain
the exact value of the sine, cosine or tangent
of a given angle with a reference angle of 30º,
45º or 60º
some contextual problems can be solved
using trigonometric ratios
 sketch a diagram to represent a problem that
involves a triangle without a right angle
 solve, using primary trigonometric ratios, a
triangle that is not a right triangle
 explain the steps in a given proof of the sine
law or cosine law
 sketch a diagram and solve a problem, using
the sine law
Lesson Summary


To teach sine law by showing relationships between the angles in a triangle and their
opposite sides.
Sketch a diagram and solve a problem, using the sine law
Mathematics 20-1
Trigonometry
Page 38 of 48
Lesson Plan
Hook
Consider allowing students to explore a resource that restricts examples to right angle
triangles. At the end of the exploration illicit that there are other kinds of triangles (oblique)
that we need to be able to calculate angle and side values for.
source
Lesson Goal
Students can see the relationship between angles and the length of opposite sides and use
the sine law to find unknown sides and angle measures. Students will realize that sine law
can be used when a triangle does not have a 90º angle, but an angle, its opposite side, and
at least one other angle or side is known.
Activate Prior Knowledge
Students will be using a protractor to determine angle size(s) and a ruler to determine side
length(s). They will need to set up a ratio using appropriate sides and angles.
Mathematics 20-1
Trigonometry
Page 39 of 48
Lesson
Discuss the relationship between the sine of the angle and the opposite side.
Teach the sine law to show how this relationship can be used to find unknown sides and
angles. Stress to students that the sine law can only be used if you are given (or can find) the
angle, its opposite side, and at least one other angle or side.
a
b
c
=
=
sin A sin B sin C
Go through examples finding:
 unknown sides
 unknown angles.
Provide examples to be given where sine law must be used to solve triangles. Problems
should include examples where students must sketch a diagram and solve a problem using
the sine law.
Going Beyond
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 2.3, pages 100-113)
Supporting
The Sine Law applet can be used for a
visual. As you change the length of one
side, the other sides and angles change
accordingly. The ratios are shown and
change accordingly as well.
http://staff.argyll.epsb.ca/jreed/math9/strand
3/sine_law.htm
Mathematics 20-1
Trigonometry
Page 40 of 48
Assessment
An exit slip can be used to test their knowledge using two triangles…one finding an unknown
side and one finding the unknown angle.
Glossary
ambiguous case – From the given information the solution for the triangle is not clear: there
might be one triangle, two triangles, or no triangle [Math 20-1 (McGraw-Hill Ryerson: page
104)]
oblique triangle – A triangle that is not a right triangle
opposite side - The side opposite the reference angle
opposite angle – The angle opposite a particular side
ratio – A comparison of numbers or quantities
sine law – The lengths of the sides are proportional to the
sines of the opposite angles
a
b
c
=
=
sin A sin B sin C
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.
Other
Mathematics 20-1
Trigonometry
Page 41 of 48
Lesson 6
The Cosine Law
STAGE 1
BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of
trigonometry include surveying, navigation, construction, and calculus.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …





Angles in a circle can be expressed in a
variety of ways.
Primary trigonometric ratios and Pythagorean
Theorem only work for right triangles, while
sine and cosine law will work for all triangles.

What is triangulation?
How is a negative angle possible?
How many different ways can you estimate
the height of a mountain?
Why is it easier to find the exact value of
some trigonometric ratios?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …






an angle in standard position, given the
measure of the angle
the reference angle for an angle in standard
position
the Pythagorean theorem or the distance
formula, can be used to calculate the distance
from the origin to a point P (x, y) on the
terminal arm of an angle
the sign of a given trigonometric ratio for a
given angle, without the use of technology,
and explain
the exact value of the sine, cosine or tangent
of a given angle with a reference angle of 30º,
45º or 60º
some contextual problems can be solved
using trigonometric ratios
 sketch a diagram to represent a problem that
involves a triangle without a right angle
 solve, using primary trigonometric ratios, a
triangle that is not a right triangle
 explain the steps in a given proof of the sine
law or cosine law
 sketch a diagram and solve a problem, using
the cosine law
Lesson Summary
Using cosine law when given a triangle with
 2 side lengths and included angle size or
 all 3 side lengths
Sketch a diagram and solve a problem using the cosine law
Explain the steps in a given proof of the cosine law.
Mathematics 20-1
Trigonometry
Page 42 of 48
Lesson Plan
Hook
Solve oblique triangles that can be solved with the sine law and a few that require the cosine
law. Have students recognize that the sine law will not work for all oblique triangles.
Lesson Goal
Students will recognize the scenario in which cosine law would be used, and able to use it to
solve a triangle.
Activate Prior Knowledge
Teachers may want to review some algebraic manipulations.
Lesson
Explain the steps in a given proof of the cosine law.
Draw a triangle that has 2 sides and the enclosed angle. Ask students if the sine law could
be used to solve the triangle. Students will realize that because there is not a “pair” (angle
with its side), the sine law cannot be used.
Cosine law is introduced as the only method to solving a non-right angle triangle with this
information.
Finding sides:
Use c2 = a2 + b2 = 2ab cos C, to find the unknown side length. Once this has been found,
the sine law or the cosine law may be used to find other unknown values.
Examples should be done to practice finding the unknown side, using the cosine law.
Finding angles:
Use c2 = a2 + b2 = 2ab cos C, to find the unknown angle. Once this has been found, the
sine law can be used to continue, or cosine law may be used again.
Examples should be done to practice finding the unknown angles, using the cosine law.
Provide examples where cosine law must be used to solve triangles. Problems should
include examples where students must sketch a diagram and solve a problem using the
cosine law.
Mathematics 20-1
Trigonometry
Page 43 of 48
Going Beyond
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 2.4, pages 114-125)
Supporting
The following applet may be used to show
the cosine law while changing lengths of
sides or angle measurements
http://staff.argyll.epsb.ca/jreed/math9/s
trand3/a_law.htm
Assessment
Glossary
cosine law – the relationship between the cosine of an angle and the lengths of the three
sides of any triangle:
c2 = a2 + b2 = 2ab cos C
oblique triangle – A triangle that is not a right triangle
Other
Mathematics 20-1
Trigonometry
Page 44 of 48
Lesson 7
The Ambiguous Case
STAGE 1
BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of
trigonometry include surveying, navigation, construction, and calculus.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …





Angles in a circle can be expressed in a
variety of ways.
Primary trigonometric ratios and Pythagorean
Theorem only work for right triangles, while
sine and cosine law will work for all triangles.

What is triangulation?
How is a negative angle possible?
How many different ways can you estimate
the height of a mountain?
Why is it easier to find the exact value of
some trigonometric ratios?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …





the reference angle for an angle in standard
position
the Pythagorean theorem or the distance
formula, can be used to calculate the distance
from the origin to a point P (x, y) on the
terminal arm of an angle
the exact value of the sine, cosine or tangent
of a given angle with a reference angle of 30º,
45º or 60º
some contextual problems can be solved
using trigonometric ratios




sketch a diagram to represent a problem that
involves a triangle without a right angle
solve, using primary trigonometric ratios, a
triangle that is not a right triangle
explain the steps in a given proof of the sine
law or cosine law
sketch a diagram and solve a problem, using
the sine law
describe and explain situations in which a
problem may have no solution, one solution or
two solutions
Lesson Summary


Describe and explain situations in which a problem may have no solution, one solution,
or two solutions.
Sketch a diagram and solve a problem using the sine law.
Mathematics 20-1
Trigonometry
Page 45 of 48
Lesson Plan - The Ambiguous Case
Hook
If you have a computer, use an applet that shows the ambiguous case. John Scammel
created two applets using Geogebra.
AmbiguousCase.ggb
AmbiguousCase2.ggb
 files were added to the EPSB Understanding by Design share site
Simulate what the applet does, using:
1. straws (uncut, 7 cm, 5 cm, 4 cm, 3 cm, 2 cm) and a protractor
2. a geometry set, pencil & paper.
Sample straws and protractor activity:
 Make an approximately 37o degree
angle with the uncut and 7 cm straws
as shown in the diagram.
 Check how many triangles can be
made with the 2 cm, 3 cm and 4 cm
straws.
The students should notice that the:
 2 cm straw gives no solution
 3 cm straw gives one solution
 4 cm straw gives two possible solutions.
Lesson Goal
Describe and explain situations in which a problem may have no solution, one solution, or two
solutions.
Sketch a diagram and solve a problem using the sine law.
Activate Prior Knowledge
Review the sine law.
Mathematics 20-1
Trigonometry
Page 46 of 48
Lesson
After going through the opening activity, students should understand that there are conditions
where there is no solution, one solution, or two solutions, given a specific angle and one fixed
side.
Define Ambiguous Case.
You may want to go over the general case in your text.
Discuss that when
 a = h, there is one solution
 h < a < b there are 2 solutions
 a < h , there is no solution.
b
a
h
a
Example:
In △ABC, <A = 120o, a = 20 cm, b = 15 cm determine all possible values for <B to the nearest
degree.
(Teacher note: there are 2 answers, 41o and 139o)
Do a few more examples, one where the entire triangle is solved.
Exit Slip:
In △ABC, Ð A = 50o, a = 9.5 cm, b = 7.5 cm. Determine all possible values for Ð C.
Recommendation: Do another lesson (Lesson 8) with mixed problem-solving, using sine
law, cosine law, ambiguous case and primary trigonometry ratios.
Going Beyond
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 2.3)
Ambiguous Case Video:
http://www.youtube.com/watch?v=ksBaHrVqhyo&playnext=1&list=PL57A3218714EA7A89
Video 4
 file was added to the EPSB Understanding by Design share site
Mathematics 20-1
Trigonometry
Page 47 of 48
Supporting
Assessment
Questions from text.
Glossary
ambiguous case – From the given information the
solution for the triangle is not clear: there might be one
triangle, two triangles, or no triangle [Math 20-1
(McGraw-Hill Ryerson: page 104)]
As shown, the given conditions (side a, side b, and A)
can produce more than one triangle: an obtuse-angled
triangle and an acute-angled triangle.”
Mathematics Discovery Dictionary
oblique triangle – A triangle that is not a right triangle
opposite side - the side opposite the reference angle
opposite angle – the angle opposite a particular side
ratio - a comparison of numbers or quantities.
sine law – The sides are proportional to the sines of the opposite angles
a
b
c
=
=
sin A sin B sin C
Other
Mathematics 20-1
Trigonometry
Page 48 of 48