Mathematics 20-1 - Activating Students in High School Math Web viewMathematics 20-1 Trigonometry...

MATHEMATICS 20-1

Trigonometry

High School collaborative venture withEdmonton Christian, Harry Ainlay, J. Percy Page, Jasper Place,

Millwoods Christian, Ross Sheppard and W. P. Wagner

Edm Christian High: Aaron TrimbleHarry Ainlay: Ben LuchkowHarry Ainlay: Darwin HoltHarry Ainlay: Lareina RezewskiHarry Ainlay: Mike ShrimptonJ. Percy Page: Debbie YoungerJasper Place: Matt KatesJasper Place: Sue DvorackMillwoods Christian: Patrick YpmaRoss Sheppard: Patricia ElderW. P. Wagner: Amber Steinhauer

Facilitator: John Scammell (Consulting Services)Editor: Jim Reed (Contracted)

2010 - 2011

Mathematics 20-1 Trigonometry Page 2 of 52

TABLE OF CONTENTS

STAGE 1 DESIRED RESULTS PAGE

Big Idea

Enduring Understandings

Essential Questions

4

4

4

Knowledge

Skills

5

6

STAGE 2 ASSESSMENT EVIDENCE

Transfer Task (on a separate page which could be photocopied & handed out to students)

“Tri” AlbertaTeacher Notes for Transfer Task and RubricTransfer Task and RubricRubricPossible Solution

89

1214

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


Implementation note:Post the BIG IDEA in a prominentplace in your classroom and refer to it often.

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

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.

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?

Knowledge:


EnduringUnderstandingList enduring understandings (the fewer the better)

SpecificOutcomesList the reference # from the Alberta Program of Studies

Description ofKnowledgeThe paraphrased outcome that the group is targeting

Students will understand… Angles in a circle can

be expressed in a variety of ways.

*T1, T2, T3Students 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 understand… Primary trigonometric

ratios and Pythagorean Theorem only work for right triangles, while the sine and cosine law will work for all triangles.

T2, T3Students 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 understand… Each primary

trigonometry ratio is positive in two quadrants and negative in two quadrants between 0o and 360o (CAST).

T1, T2Students 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 understand… Special triangles are

useful for determining exact value of trigonometric ratios with reference angles 0, 30o, 45o and 60o.

T1, T2Students 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°

8888I*T = Trigonometry

Skills:


EnduringUnderstandingList enduring understandings (the fewer the better)

SpecificOutcomesList the reference # from the Alberta Program of Studies

Description ofSkillsThe paraphrased outcome that the group is targeting

Students will understand…

Angles in a circle can be expressed in a variety of ways.

T1Students will be able to…

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


Primary trigonometric ratios and Pythagorean Theorem only work for right triangles, while the Sine and cosine law will work for all triangles.

T2,3Students will be able to…

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


Each ratio is positive in two quadrants and negative in two quadrants between 0o and 360o (CAST).

T1, T2Students will be able to…

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

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


Implementation note:Teachers need to continually askthemselves, if their students are acquiring the knowledge and skills needed for the unit.

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



Special triangles are useful for determining exact value of trigonometric ratios with reference angles 0, 30o, 40o and 60o

T2Students will be able to…

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


*T = Trigonometry


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.

STAGE 2 Assessment Evidence

1 Desired Results Desired ResultsDesired 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.

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.


“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: Background Information

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: 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 (McGraw-Hill 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 x- or 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-axisrotation 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 anglesa

sin A= b

sin B= c

sin C

Assessment

Mathematics 20-1

Trigonometry - Rubric

LevelCriteria

Excellent4

Proficient3

Adequate2

Limited*1

Insufficient / Blank*

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

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

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

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

Solving for Reference, Principle, Negative Co-terminal 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, andProvides 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


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:

cos A = 4022+5882−5582

2×402×588

∠A = 65.5°

cos B =5582+5882−4022

2×558×588

∠B = 41.0°

cos C =5582+4022−5882

2×558×402

∠C = 73.5°


Verify:sin A558

=sin 41 .0∘

402

∠A = 65.5°

sin 65 .5∘

558=sin B

402

∠B = 41.0°

sin 65∘

558=sin C

588

∠C = 73.5°

Part C:

Measured Distances:

∠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 kmsin D318

=sin 53. 0∘

349. 9

∠D = 46.5°sin E432

=sin 53. 0∘

349 .9

∠E = 80.4°


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:

<BAQreference angle = 77o principle angle = 77 o coterminal angle = -283o

B = (126, 576)

<PACreference angle = 38o principle angle = 142o coterminal angle = -218o

C = (-324, 246)

<PADreference angle = 81o principle angle = 261 o coterminal angle = -99 o

D = (-66, -426)

<QAEreference angle = 46o principle angle = 314 o coterminal angle = -46o

E = (222, -228)


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.

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:



Special triangles are useful for determining exact value of trigonometric ratios with reference angles 0o, 30o, 40o 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?

KNOWLEDGE:

Students will know …

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

SKILLS:

Students will be able to …

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


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=1Video 2: http://www.youtube.com/watch?v=bwWfSgUYggc&feature=related

Video 1Video 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.


https://sites.google.com/a/share.epsb.ca/trio-workshop-files/

http://www.youtube.com/watch?v=bwWfSgUYggc&feature=related

http://www.youtube.com/watch?v=wLnx3Utj9ik&NR=1

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/


http://www.ronblond.com/MathGlossary/Division04/TrigCircle/

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 Quadrant Rotation 30˚ 4

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/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

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



http://staff.argyll.epsb.ca/jreed/math30p/trigonometry/angles.htm

Glossary


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




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.

Other


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.

http://www.learnalberta.ca/content/memg/index.html

http://www.ronblond.com/MathGlossary/Division04/Vertex%20(of%20an%20angle)/index.html

Lesson 2

Reference Triangles & Trigonometry Ratios for Angles 0˚ - 360˚

STAGE 1





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



KNOWLEDGE: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





SKILLS:



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





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.

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.


Ask students if the given trig ratios will be positive or negative.tan 217˚cos 122˚sin 300˚cos 50˚



In which quadrant is each of the following located?

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



Supporting



Assessment

Glossary










Other





Lesson 3

Applying the CAST Rule

STAGE 1







What is triangulation? How many different ways can you estimate



KNOWLEDGE:






SKILLS:










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 , 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˚.


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.




http://www.youtube.com/watch?v=T_19ZxaCP3g&feature=related

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.

2. If sin θ=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˚.

3. If cosθ=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˚.

4. If tan θ=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˚.

5. Point P (-1, -8) is on the terminal arm of an angle. Find the angle.

6. If cosθ=−0 .327 , Find all possible values of θ, where θ is between 0˚ and 360˚. Round to the nearest degree.

Going Beyond

Solve equations like sin2 θ= 9

16 and a sin θ+b=0 etc. (You may want to teach the quadratics unit first, so students are familiar with how to solve.)

Resources

Math 20-1 (McGraw-Hill Ryerson: sec 2.2)


5 Ways to use Trigonometry in Everyday Life Video 3: http://www.youtube.com/watch?v=T_19ZxaCP3g&feature=related




http://www.youtube.com/watch?v=T_19ZxaCP3g&feature=related


Supporting

Assessment

Exit Slip: If sin θ=4

9 , find all possible values of θ, where θ is between 0 and 360 degrees. Round to the nearest degree.

Glossary










Other





Lesson 4

Special Angles 0-30-45-60-90

STAGE 1











KNOWLEDGE:








SKILLS:











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/sculpture 12 .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.


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.)


http://www.google.ca/search?q=Moondog%20sculpture&rls=com.microsoft:en-us&oe=UTF-8&startIndex=&startPage=1&rlz=&redir_esc=&um=1&ie=UTF-8&tbm=isch&source=og&sa=N&hl=en&tab=wi&biw=1259&bih=622

1

1 2

2

3

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 one-unit lengths.

Those steps should lead to these 2 special triangles. Memorise these triangles.

Examples:1. Using the CAST rule and these special triangles, find:


http://www.learnalberta.ca/content/memg/Division03/Equilateral%20Triangle/index.html

0

x

x

a) sin 240°Suggested steps:

Draw the reference triangle on the coordinate plane and find the reference angle. 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 30-60-

90 triangle and use the sine ratio for 60º.

Therefore the sin 240º is equal to .

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.

a) 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)

c)

3. Draw a right triangle that includes a 30º angle in standard position. Draw dotted lines to indicate 20º, 10º, and 5º. Discuss what is happening to the lengths of the initial and terminal arm and the triangle height as the 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.

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.htmlor 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


Supporting

Assessment

Glossary







http://staff.argyll.epsb.ca/jreed/math9/strand3/trigonometry.htm



Other


Lesson 5

The Sine Law

STAGE 1





Primary trigonometric ratios and Pythagorean Theorem only work for right triangles, while sine and cosine law will work for all triangles.





KNOWLEDGE:








SKILLS:







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


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 GoalStudents 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 KnowledgeStudents 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.


http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRI&lesson=html/object_interactives/trigonometry/explore_it.html

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.

asin A

= bsin B

= csin 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


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/strand3/sine_law.htm




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



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

asin A

= bsin B

= csin C

Other




http://www.learnalberta.ca/content/memg/Division03/Ratio/index.html

http://www.learnalberta.ca/content/memg/Division03/Opposite%20Angles/index.html

Lesson 6

The Cosine Law

STAGE 1










KNOWLEDGE:








SKILLS:





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 lawExplain the steps in a given proof of the cosine law.



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.


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.


Going Beyond

Resources


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/strand3/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


Other

Lesson 7

The Ambiguous Case


http://staff.argyll.epsb.ca/jreed/math9/strand3/cosine_law.htm

http://staff.argyll.epsb.ca/jreed/math9/strand3/cosine_law.htm

STAGE 1










KNOWLEDGE:






SKILLS:






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.


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.ggbAmbiguousCase2.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 protractor2. 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.


Review the sine law.



http://www.geogebra.org/cms/

aab h

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.

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, ∠A = 50o, a = 9.5 cm, b = 7.5 cm. Determine all possible values for ∠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


Ambiguous Case Video: http://www.youtube.com/watch?v=ksBaHrVqhyo&playnext=1&list=PL57A3218714EA7A89




http://www.youtube.com/watch?v=ksBaHrVqhyo&playnext=1&list=PL57A3218714EA7A89

http://www.youtube.com/watch?v=ksBaHrVqhyo&playnext=1&list=PL57A3218714EA7A89

Supporting

Assessment

Questions from text.

Glossary


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


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 anglesa

sin A= b

sin B= c

sin C

Other


http://www.learnalberta.ca/content/memg/Division03/Ratio/index.html

http://www.learnalberta.ca/content/memg/Division03/Opposite%20Angles/index.html

http://www.mathematicsdictionary.com/english/vmd/full/a/ambiguouscase.htm

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