Download chapter 8: acute triangle trigonometry

CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY
Specific Expectations Addressed in the Chapter
• Explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that the
ratio of the side lengths equals the ratio of the sines of the opposite angles; follow the algebraic development of the sine law
and identify the application of solving systems of equations [student reproduction of the development of the formula is not
required]). [8.1, 8.2]
• Explore the development of the cosine law within acute triangles (e.g., use dynamic geometry software to verify the cosine
law; follow the algebraic development of the cosine law and identify its relationship to the Pythagorean theorem and the
cosine ratio [student reproduction of the development of the formula is not required]). [8.3, 8.4]
• Determine the measures of sides and angles in acute triangles, using the sine law and the cosine law. [8.2, 8.4, 8.5,
Chapter Task]
• Solve problems involving the measures of sides and angles in acute triangles. [8.2, 8.4, 8.5, Chapter Task]
Prerequisite Skills Needed for the Chapter
• Solve problems involving proportions.
• Apply the primary trigonometric ratios to determine side lengths and angle measures.
• Solve problems involving the properties of interior angles of a triangle and angles formed by parallel lines.
• Apply the Pythagorean theorem to determine side lengths.
Copyright © 2011 by Nelson Education Ltd.
What “big ideas” should students develop in this chapter?
Students who have successfully completed the work of this chapter and who understand the
essential concepts and procedures will know the following:
length of opposite side
• The ratio
is the same for all three angle–side pairs
sin (angle)
in an acute triangle.
a
b
c
• The sine law states that in any acute triangle,+ABC,
.
=
=
sin A sin B sin C
• The sine law can be used to solve a problem modelled by an acute triangle if you can
determine two sides and the angle opposite one of these sides, or two angles and any side.
• The cosine law is an extension of the Pythagorean theorem to triangles that do not have a
right angle.
• The cosine law states that in any acute triangle,+ABC,
a2 = b2 + c2 – 2 bc cos A
b2 = a2 + c2 – 2 ac cos B
c2 = a2 + b2 – 2 ab cos C
• The cosine law can be used to solve a problem modelled by an acute triangle if you can
determine two sides and the angle between them, or all three sides.
• If a real-world problem can be modelled using an acute triangle, unknown
measurements can be determined using the sine law or the cosine law,
sometimes along with the primary trigonometric ratios.
Chapter 8 Introduction
| 289
Chapter 8: Planning Chart
Pacing
10 days
Lesson Title
Lesson Goal
Getting Started, pp. 422–425
Use concepts and skills developed
prior to this chapter.
2 days
ruler;
protractor;
Diagnostic Test
Lesson 8.1: Exploring the Sine
Law, pp. 426–427
Explore the relationship between
each side in an acute triangle and
the sine of its opposite angle.
1 day
dynamic geometry software, or ruler
and protractor
Lesson 8.2: Applying the Sine
Law, pp. 428–434
Use the sine law to calculate
unknown side lengths and angle
measures in acute triangles.
1 day
ruler;
Lesson 8.2 Extra Practice
Lesson 8.3: Exploring the Cosine
Law, pp. 437–439
Explore the relationship between
side lengths and angle measures in
a triangle using the cosines of
angles.
1 day
dynamic geometry software
Lesson 8.4: Applying the Cosine
Law, pp. 440–445
Use the cosine law to calculate
unknown measures of sides and
angles in acute triangles.
1 day
ruler;
Lesson 8.4 Extra Practice
Lesson 8.5: Solving Acute Triangle
Problems, pp. 446–451
Solve problems using the primary
trigonometric ratios and the sine
and cosine laws.
1 day
ruler;
Lesson 8.5 Extra Practice
3 days
Mid-Chapter Review Extra Practice;
Chapter Review Extra Practice;
Chapter Test;
Chapters 7–8 Cumulative Review
Copyright © 2011 by Nelson Education Ltd.
Mid-Chapter Review, pp. 435–436
Chapter Review, pp. 452–453
Chapter Self-Test, p. 454
Curious Math, p. 439
Chapter Task, p. 455
Materials/Masters Needed
290 |
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
CHAPTER OPENER
Using the Chapter Opener
Copyright © 2011 by Nelson Education Ltd.
Introduce the chapter by discussing the photograph and map on pages 420
and 421 of the Student Book.
The photograph shows a view of the Toronto waterfront, and the map shows
Toronto, Hamilton, and St. Catharines. A triangle has been drawn on the
map to connect the three cities. The distance between Toronto and Hamilton
and the distance between Hamilton and St. Catharines have been marked on
the map, as well as the angle formed by the lines between these cities.
Invite students to explain what they remember about the three primary
trigonometric ratios: sine, cosine, and tangent. Ask: What information is
shown on the map? Encourage discussion about the following question:
Why can you not use a primary trigonometric ratio to directly calculate the
distance by air from St. Catharines to Toronto? Ask students to estimate the
distance and to justify their estimates.
Tell students that, in this chapter, they will develop two methods for solving
problems that can be modelled using acute triangles. After completing the
chapter, return to the question on page 421. Have students calculate the
distance in pairs or groups and then share their solutions. Alternatively, you
could calculate the distance with the class after completing Lesson 8.5.
Chapter 8 Opener
|
291
GETTING STARTED
Using the Words You Need to Know
Students might read each sentence and select the best term to complete it,
read each term and search for the sentence in which it fits, or use a
combination of these strategies.
After students have recorded all the terms to complete the sentences, ask
them to share their strategies. Some students may know all the terms. Others
may have used a process of elimination. Then read each term and ask
questions such as these: How would you define this term? What other words
could you use to explain it? How could you illustrate it? Emphasize that
each term can be defined and illustrated in different ways.
Using the Skills and Concepts You Need
When discussing angle relationships, ensure that students relate each
relationship to the angles shown. Work through the example in the Student
Book (or similar examples, if you would like students to have experience
with more examples), and encourage students to ask questions about the
reasons given in the table. Ask students to look over the Practice questions
to see if there are any questions they do not know how to solve. Have
students work on the Practice questions in class, and assign any unfinished
questions for homework.
Student Book Pages 422–425
Preparation and Planning
Pacing
5−10 min
Words You Need to
Know
40−45 min Skills and Concepts You
Need
45−55 min Applying What You
Know
Materials
ƒ ruler
ƒ protractor
Nelson Website
http://www.nelson.com/math
Introduce the activity by modelling the situation. Ask for a volunteer to
mark the position of the soccer net on the board, and have students estimate
Marco’s position. Then have students work in pairs. After students finish,
have some of the pairs describe their strategies for determining the width of
the net. Ask if there is more than one way to solve the problem. Also ask if
there is more than one primary trigonometric ratio that can be used to
calculate the width of the soccer net using the two right triangles.
Answers to Applying What You Know
A. The angle formed by the posts and Marco’s position is 75°. Since the
sum of the angle measures in a triangle is 180°, the total number of
degrees for the other two angles is 180° – 75°, or 105°. If one of the
angles formed by Marco’s position and the goalposts were 90°, the other
angle would be 15°. However, the sides of the triangle that are between
Marco and the goalposts are 5.5 m and 6.5 m. These lengths are close to
the same length, so the angles formed by the goalposts and Marco’s
position are close to the same measure. None of the angles is a right
angle, so Marco’s position does not form a right triangle.
B. A primary trigonometric ratio cannot be used to calculate the width of
the net directly because the triangle formed by Marco’s position and the
goalposts is not a right triangle.
292 |
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
Copyright © 2011 by Nelson Education Ltd.
Using the Applying What You Know
C. Draw h from a goalpost perpendicular to the opposite side.
h
6 .5
6.5 sin 75º = h
6.28 = h
The height of the triangle is about 6.28 m.
E.–F. Begin with the right triangle that contains x and the 75º angle.
Calculate the third side, x, of the triangle using the cosine ratio.
x
cos 75º =
6 .5
6.5 cos 75º = x
1.68 = x
Then calculate y.
y = 5.5 − x
y = 5.5 – 1.68
y = 3.82
Let the width of the soccer net, in metres, be n. Use the Pythagorean
theorem to calculate n.
n2 = y2 + h2
n2 = 3.822 + 6.282
n2 = 54.03
n = 54.03
n = 7.35
The width of the soccer net is about 7.4 m.
Copyright © 2011 by Nelson Education Ltd.
D.
sin 75º =
Initial Assessment
What You Will See Students Doing…
When students understand…
If students misunderstand…
Students articulately explain why a primary trigonometric ratio
cannot be used to calculate the width of the soccer net
directly.
Students may not realize that the triangle is not a right
triangle. They may think that a primary trigonometric ratio can
be used to solve a triangle that is not a right triangle.
Students draw the height of the triangle correctly.
Students may not know how to show the height of a triangle
from a vertex at the goalposts so that either the 5.5 m length
or the 6.5 m length can be used in a right triangle with a 75°
angle.
Students apply the sine ratio, the cosine ratio, and then the
Pythagorean theorem to determine the width of the net.
Students may not understand how to use the sine ratio,
cosine ratio, or Pythagorean theorem or how to connect the
information for the two triangles they formed.
Students write a concluding statement that includes the
correct answer to the question asked.
Students may not connect the results of their calculations with
the situation. They may not use the correct units or answer
the question.
Chapter 8 Getting Started
|
293
8.1 EXPLORING THE SINE LAW
GOAL
Explore the relationship between each
side in an acute triangle and the sine
of its opposite angle.
Lesson at a Glance
Student Book Pages 426–427
Prerequisite Skills/Concepts
• Solve problems involving the properties of interior angles of a triangle.
• Solve problems involving proportions.
Specific Expectation
• Explore the development of the sine law within acute triangles (e.g., use
dynamic geometry software to determine that the ratio of the side lengths
equals the ratio of the sines of the opposite angles[; follow the algebraic
development of the sine law and identify the application of solving
systems of equations [student reproduction of the development of the
formula is not required])].
Preparation and Planning
Pacing
5–10 min Introduction
35−45 min Teaching and Learning
10–15 min Consolidation
Materials
ƒ dynamic geometry software, or ruler
and protractor
Recommended Practice
Questions 1, 2, 3, 4
Nelson Website
http://www.nelson.com/math
Mathematical Process Focus
Copyright © 2011 by Nelson Education Ltd.
• Reasoning and Proving
• Connecting
MATH BACKGROUND | LESSON OVERVIEW
• In this lesson, students explore the sine law relationship between sides and angles in an acute triangle.
• The exploration leads to an understanding that in any acute triangle,+ABC, the ratio
equivalent for all three angle–side pairs and
294 |
a
sin A
=
b
sin B
=
c
sin C
.
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
length of opposite side
sin (angle)
is
1
Introducing the Lesson
(5 to 10 min)
Have students work in small groups. Assign a primary trigonometric ratio to
each group, and ask students to discuss the ratio for a few minutes. To help
students get started, suggest topics such as:
• type of triangle for which a primary trigonometric ratio is used
• relationship represented by each primary trigonometric ratio
• use of a primary trigonometric ratio to determine a side length
• use of a primary trigonometric ratio to determine an angle measure
• connection between tangent and slope
• diagram to model a primary trigonometric ratio
• problem that involves a primary trigonometric ratio in the solution
Then ask a member of each group to describe the ratio in a few words, with
or without a diagram.
Pose these questions: How is an acute triangle different from a right
triangle? Can you use primary trigonometric ratios to solve acute triangles?
Why not?
2
Teaching and Learning
(35 to 45 min)
Explore the Math
Have students work through the exploration in pairs. Ensure that they
construct only acute triangles, except in part F.
Copyright © 2011 by Nelson Education Ltd.
If students are using dynamic geometry software, the following ideas may
be helpful:
• Ensure that students can use the software to measure angles in a triangle
and lengths of line segments, as noted in the Tech Support in the margin.
• Place more experienced users of the software with students who are less
proficient. Encourage students to seek assistance from their neighbours
when necessary.
• Remind students to choose the precision for lengths and angle measures,
as well as the units.
• Ensure that students know how to use the calculator in the dynamic
length of opposite side
.
geometry software to determine the ratio
sin (angle)
• For part F, students could drag vertices to obtain an angle of almost
exactly 90º, create a segment and rotate it, or create a right triangle by
constructing a perpendicular line. When they drag the right triangle, the
angle must remain a right angle.
8.1: Exploring the Sine Law
| 295
If students are using rulers and protractors, have them measure the side
lengths to tenths of a centimetre and the angles to the nearest degree. The
accuracy will be less exact than the accuracy with dynamic geometry
software.
Discuss why results can vary because of precision in measurements and
because of rounding, when using the calculator in the dynamic geometry
software or when using a scientific calculator.
Answers to Explore the Math
A.–B. Answers may vary, e.g.,
length of opposite side
Angle
Side
Sine
∠A = 52.3º
a = 2.6
sin A = 0.79
a
= 3.3
sin A
∠B = 55.5º
b = 2.8
sin B = 0.82
b
= 3.3
sin B
∠C = 72.2º
c = 3.2
sin C = 0.95
c
= 3.3
sin C
C. The ratio is the same for all three angles in my triangle.
D.
296 |
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
Copyright © 2011 by Nelson Education Ltd.
sin (angle)
length of opposite side
Angle
Side
Sine
∠A = 26.3º
a = 2.6
sin A = 0.44
a
= 6.0
sin A
∠B = 65.2º
b = 5.4
sin B = 0.91
b
= 6.0
sin B
∠C = 88.5º
c = 6.0
sin C = 1.00
c
= 6.0
sin C
sin (angle)
length of opposite side
is the same for all three angles in this
sin (angle)
triangle, but it is different from the ratio for the angles in the first
triangle.
length of opposite side
E. The ratio
is always the same for all three angles
sin (angle)
in a triangle.
F.
Copyright © 2011 by Nelson Education Ltd.
The ratio
length of opposite side
Angle
Side
Sine
∠A = 48.2º
a = 3.9
sin A = 0.75
a
= 5.2
sin A
∠B = 41.8º
b = 3.5
sin B = 0.67
b
= 5.2
sin B
∠C = 90.0º
c = 5.2
sin C = 1.00
c
= 5.2
sin C
sin (angle)
In a right triangle, the ratio is the same for all three angles.
For some of the tables, if I calculated the answers using the
measurements in the other columns, some results would be slightly
different than my results using the software calculator. The differences
are because of the precision in the measurements and because of
rounding.
8.1: Exploring the Sine Law
| 297
G. Replacing the sine ratio with the cosine or tangent ratio does not give the
same results. The cosine ratios are not the same for all three angles in a
triangle. Similarly, the tangent ratios are not the same.
H. i) In an acute triangle, the ratio of the length of a side to the sine of the
opposite angle is equivalent for all three angle–side pairs.
a
b
c
ii) In an acute triangle ABC,
.
=
=
sin A sin B sin C
Answers to Reflecting
I. Since the relationship involves the sine ratio, an appropriate name could
be the sine law.
J. Yes. Write the ratio of the length of the known side to the sine of the
angle opposite the side. Create an equivalent ratio for the unknown side
length and the sine of the opposite known angle, or for the length of the
known side and the sine of the unknown opposite angle. Equate the two
ratios, and solve for the unknown. If you know two of the angles, you
can calculate the third angle and then use ratios to solve for the unknown
side length.
3
Consolidation
(10 to 15 min)
Students should understand that the ratio of the length of a side to the sine of
the opposite angle is equivalent for all three angle–side pairs in an acute
triangle. Students should also understand that this relationship can be used
to determine unknown side lengths and angle measures in an acute triangle
if enough information is known.
Copyright © 2011 by Nelson Education Ltd.
Students should be able to answer the Further Your Understanding questions
independently.
298 |
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
8.2 APPLYING THE SINE LAW
GOAL
Use the sine law to calculate unknown
side lengths and angle measures in
acute triangles.
Lesson at a Glance
Prerequisite Skills/Concepts
Student Book Pages 428–434
length of opposite side
is the same for all three
sin (angle)
angle–side pairs in an acute triangle.
• Apply the primary trigonometric ratios to determine side lengths and
angle measures.
• Solve problems involving the properties of interior angles of a triangle
and angles formed by parallel lines.
• Solve problems involving proportions.
• Understand that the ratio
Specific Expectations
• [Explore the development of the sine law within acute triangles (e.g., use
dynamic geometry software to determine that the ratio of the side lengths
equals the ratio of the sines of the opposite angles;] follow the algebraic
development of the sine law and identify the application of solving
systems of equations [student reproduction of the development of the
formula is not required]).
• Determine the measures of sides and angles in acute triangles, using the
sine law [and the cosine law].
• Solve problems involving the measures of sides and angles in acute
triangles.
Preparation and Planning
Pacing
5−10 min Introduction
30−40 min Teaching and Learning
15−20 min Consolidation
Materials
ƒ ruler
Recommended Practice
Questions 3, 4, 6, 7, 8, 9, 11, 14
Key Assessment Question
Question 7
Extra Practice
Lesson 8.2 Extra Practice
New Vocabulary/Symbols
sine law
Nelson Website
http://www.nelson.com/math
Copyright © 2011 by Nelson Education Ltd.
Mathematical Process Focus
• Problem Solving
• Selecting Tools and Computational Strategies
• Representing
MATH BACKGROUND | LESSON OVERVIEW
• In Lesson 8.1, students explored the relationship between each side in an acute triangle and the sine of its opposite
angle. They discovered that the ratio
length of opposite side
sin (angle)
is equivalent for all three angle–side pairs.
• In Lesson 8.2, students learn the name of this relationship: the sine law.
• Students use the sine law to determine unknown side lengths and angle measures in acute triangles. They also use
the sine law to solve problems that involve acute triangles.
8.2: Applying the Sine Law
| 299
1
Introducing the Lesson
(5 to 10 min)
Initiate a discussion about the exploration for Lesson 8.1 by inviting
students to explain their discoveries in their own words and to illustrate their
discoveries on the board, as appropriate. Alternatively, students could spend
a few minutes discussing the exploration in groups. Ask them to decide
what they think was their most important discovery in the exploration. Have
a student from each group present their decision to the class.
2
Teaching and Learning
(30 to 40 min)
Learn About the Math
Example 1 shows that the sine law is true for all acute triangles. It allows
students to discuss a proof of the sine law. Help students understand how
this proof extends their exploration for Lesson 8.1, in which they obtained
equivalent ratios. Using the prompts, lead students through the steps in the
proof.
A. Ben needed to create two right triangles. He used one right triangle to
determine an expression for sin B and the other right triangle to
determine an expression for sin C. Since AD was the opposite side in
+ABD and in+ACD, he was able to relate the two triangles.
B. If Ben drew a perpendicular line segment from vertex C to side AB, he
a
b
and
are equal.
would be able to show that the ratios
sin A
sin B
C. The expressions c sin B and b sin C both describe AD and can be set
equal to each other. Instead of dividing both sides of the equation by
sin C and then by sin B, both sides can be divided by b and then by c.
This will still result in an equation that describes only one side of the
sin B sin C
triangle on each side:
. Similarly, dividing the second
=
b
c
sin A sin C
. Therefore,
equation by a and then by c will result in
=
a
c
it makes sense that the sine law can also be written in the form
sin A sin B sin C
.
=
=
a
b
c
300 |
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
Copyright © 2011 by Nelson Education Ltd.
Answers to Reflecting
3
Consolidation
(15 to 20 min)
Apply the Math
Using the Solved Examples
In Example 2, three angle measures and the length of one side of an acute
triangle are given, so one angle–side pair and the measure of another angle
can be used to determine a side length. Students could talk about the
solution in pairs, before a class discussion. Encourage students to take an
active role in explaining the steps in the example in their own words. Pose
questions such as these:
• How do you think Elizabeth decided which angle measures to use?
• If she had known only the measures of ∠A and ∠C, how could she have
solved the problem?
• Why does it make sense that side b is shorter than side a?
In Example 3, two side lengths and one angle measure are given, so one
angle–side pair and the length of another side can be used to determine an
angle measure. Discuss how using the form of the sine law with sine in the
numerator makes it easier to calculate the solution. Since the final step in the
solution uses the inverse of the sine ratio, a brief review of the inverse may
be helpful.
Example 4 presents a problem that can be modelled with an acute triangle.
Two angles and one side are given, but not an angle–side pair. Students
should understand how the measure of the third angle is determined by
using the sum of the angles in a triangle.
Copyright © 2011 by Nelson Education Ltd.
Answer to the Key Assessment Question
Students should draw a diagram to help them visualize the parallelogram in
the multi-step problem for question 7. Students could use the information
about parallelograms from Lesson 2.4, In Summary, as a reference. After
students complete the solution, ask them to explain the steps.
7. The long sides of the parallelogram measure 15.4 cm, to the nearest
tenth of a centimetre.
Closing
Question 14 gives students an opportunity to summarize a situation in which
the sine law can be used to solve a problem. Students might complete the
question on their own or with a partner. After students have completed the
question, have them share their solutions:
• Invite a student to explain her or his solution to the class or a group. Then
ask for a few different solutions. Emphasize that measurements can be
determined in different ways. Ask: How are all the solutions the same?
How are some solutions different from other solutions?
8.2: Applying the Sine Law
| 301
• Alternatively, have a student draw the triangle on the board. Then ask the
class: What could you calculate? How? What could you calculate next?
How? When finished, provide time for students to compare their own
solution with the class solution.
Ensure that students realize how the sum of the interior angles of a triangle
can be used to determine ∠R.
Assessment and Differentiating Instruction
What You Will See Students Doing…
When students understand…
If students misunderstand…
Students write the sine law correctly, with the value to be
determined in the numerator.
Students may be unable to identify opposite angles and sides.
They may substitute incorrectly because of errors when
interpreting measurements or drawing diagrams. They may
not write the sine law with the value to be determined in the
numerator.
Students solve an equation correctly and state the results in a
concluding sentence.
Students cannot solve proportions by isolating the value to be
determined and then calculating and rounding. They may not
remember how to use the inverse sine, or they may not state
the solution in a concluding sentence to answer the question,
using the correct units.
Students create and label a parallelogram correctly to model
the situation.
Students cannot create or label a parallelogram correctly to
model the situation. They may label the given sides
incorrectly.
Students realize that they need to determine the angle
opposite a short side of the parallelogram. Then they use the
sum of the interior angles of a triangle to determine the angle
opposite a long side of the parallelogram.
Students may have difficulty with multi-step problems. They
may not realize that they need to determine the angle
opposite a short side of the parallelogram before they can
determine the angle opposite a long side.
Students use the sine law to determine the length of the long
sides of the parallelogram.
Students may correctly calculate the angle opposite a short
side but use this angle in the sine law to determine the length
of a long side.
Differentiating Instruction | How You Can Respond
EXTRA SUPPORT
1. To help students visualize a triangle, guide them as they draw a diagram with both the given and unknown information
marked. Suggest circling the information for each ratio in a different colour. Help them note whether they are determining a
side length or angle measure, and remind them to place this value in the numerator of the sine law.
2. Suggest strategies for checking accuracy. Students might divide the sine of each angle by the length of the opposite side to
check that they are equal, divide the length of each side by the sine of the opposite angle to check that they are equal, or
add the angle measures to check that the sum is 180°.
EXTRA CHALLENGE
1. Students could create problems that can be solved using the sine law and then solve their problems.
2. Have students investigate real-world situations in which the sine law would be useful. The contexts used for the Practising
questions in Lesson 8.2 might help students think of situations to investigate.
302 |
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
Copyright © 2011 by Nelson Education Ltd.
Key Assessment Question 7
MID-CHAPTER REVIEW
Big Ideas Covered So Far
• The ratio
length of opposite side
sin (angle)
is the same for all three angle–side pairs in an acute triangle.
• The sine law states that in any acute triangle,+ABC,
a
sin A
=
b
sin B
=
c
sin C
.
• The sine law can be used to solve a problem modelled by an acute triangle if you can determine two sides and the angle
opposite one of these sides, or two angles and any side.
Using the Frequently Asked Questions
Have students keep their Student Books closed. Display the Frequently Asked Questions on a board. Have students discuss the
questions and use the discussion to draw out what the class thinks are good answers. Then have students compare the class
answers with the answers on Student Book page 435. Students can refer to the answers to the Frequently Asked Questions as
they work through the Practice Questions.
Using the Mid-Chapter Review
Ask students if they have any questions about any of the topics covered so far in the chapter. Review any topics that students
would benefit from considering again. Assign Practice Questions for class work and for homework.
To gain greater insight into students’ understanding of the material covered so far in the chapter, you may want to ask questions
such as the following:
• How do you decide which sides and angles to use in the ratios for the sine law?
• How are solutions that use the sine law to determine a side length the same as solutions that use the sine law to determine
an angle measure? How are these solutions different?
• The sine law can only be used if certain information about an acute triangle is given. Explain why.
• How would you explain the inverse sine ratio? What other way could you explain it?
Copyright © 2011 by Nelson Education Ltd.
• What have you learned about the sine law?
Chapter 8 Mid-Chapter Review |
303
Lesson at a Glance
GOAL
Explore the relationship between side
lengths and angle measures in a
triangle using the cosines of angles.
Prerequisite Skills/Concepts
• Solve problems involving proportions.
• Apply the Pythagorean theorem to determine side lengths.
Specific Expectation
• Explore the development of the cosine law within acute triangles (e.g.,
use dynamic geometry software to verify the cosine law[; follow the
algebraic development of the cosine law and identify its relationship to
the Pythagorean theorem and the cosine ratio [student reproduction of the
development of the formula is not required]).]
Mathematical Process Focus
• Reasoning and Proving
• Connecting
Student Book Pages 437–439
Preparation and Planning
Pacing
5−10 min Introduction
35−45 min Teaching and Learning
10−15 min Consolidation
Materials
ƒ dynamic geometry software
Recommended Practice
Questions 1, 2, 3, 4, 5
New Vocabulary/Symbols
cosine law
Nelson Website
http://www.nelson.com/math
MATH BACKGROUND | LESSON OVERVIEW
• Students should understand the Pythagorean theorem.
• In this lesson, students explore the cosine law for an acute triangle,+ABC:
a2 = b2 + c2 – 2 bc cos A
b2 = a2 + c2 – 2 ac cos B
c2 = a2 + b2 – 2 ab cos C
• The cosine law is an extension of the Pythagorean theorem to triangles that do not have a right angle.
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Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
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8.3 EXPLORING THE COSINE LAW
1
Introducing the Lesson
(5 to 10 min)
Begin by reviewing the Pythagorean theorem:
• Draw and label an acute triangle,+ABC.
• Ask students why the Pythagorean theorem cannot be used to determine
the measure of c. (The triangle does not have a right angle.)
• Then ask how students would show that the triangle does not have a right
angle. (Measure the sides, and show that the side lengths do not satisfy
the Pythagorean theorem.)
Tell students that they will be exploring how the Pythagorean theorem can
be modified to work with triangles that are not right triangles, such as
+ABC.
2
Teaching and Learning
(35 to 45 min)
Explore the Math
Dynamic geometry software allows students to experiment with a variety of
different triangles, without having to draw each triangle on paper, and to
measure with a degree of accuracy that is not possible using a protractor and
ruler.
Copyright © 2011 by Nelson Education Ltd.
Have students work through the exploration in pairs or on their own. They
could complete the exploration on their own, but consult a partner when
necessary or helpful. Ensure that they construct only acute triangles, except
in part C.
Previous experience with dynamic geometry software should make it easier
for students to work through the exploration. Ensure that students
understand how to use the software to measure angles and sides in a
triangle. Also ensure that they understand how to choose the precision for
side lengths and angle measures, as well as the units. Students can use the
calculator in the software for some of the calculations. Discuss how
variations in results can occur because of precision in measurements and
because of rounding, when using dynamic geometry software or when using
a ruler and protractor.
Part H provides a basis for discussing how the results and conclusions are
related.
If necessary, this exploration could be completed as a demonstration.
8.3: Exploring the Cosine Law
| 305
Answers to Explore the Math
A.–B.
C. c2 = a2 + b2, or 2.82 = 2.72 + 0.82
c2
a2 + b2
a2 + b2 – c2
2ab cos C
90º
7.9
7.9
0.0
0.0
2.8
54.2º
7.9
19.0
11.1
11.1
2.3
2.8
54.3º
7.9
17.2
9.2
9.2
1.6
3.0
2.8
67.3º
7.9
11.7
3.8
3.8
2.6
1.4
2.8
83.4º
7.9
8.7
0.9
0.8
Triangle
a
b
c
1
2.7
0.8
2.8
2
3.2
3.0
3
3.5
4
5
∠C
If I calculated the answers for the four columns on the right using
the measurements in the other columns, the results would be
slightly different than my results using the software calculator.
The differences are because of the precision in the measurements
and because of rounding.
F. The values of a2 + b2 – c2 and 2 ab cos C are equivalent.
H. I can conclude that 2 ab cos C = a2 + b2 – c2 in an acute triangle.
306 |
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
Copyright © 2011 by Nelson Education Ltd.
D.–E., G.
Answers to Reflecting
I. When ∠C = 90º, 2 ab cos C = 0. This happens because cos 90º = 0.
J. The closer ∠C is to 90º, the closer the value of 2 ab cos C is to 0.
K. i) The cosine law can be used to relate a2 to b2 + c2 by
a2 = b2 + c2 – 2 bc cos A.
ii) The cosine law can be used to relate b2 to a2 + c2 by
b2 = a2 + c2 – 2 ac cos C.
3
Consolidation
(10 to 15 min)
Students should understand that the cosine law relates each side in a triangle
to the other two sides and the cosine of the angle opposite the first side:
a2 = b2 + c2 – 2 bc cos A
b2 = a2 + c2 – 2 ac cos B
c2 = a2 + b2 – 2 ab cos C
Students may notice that the values of a2 + b2 – c2 and 2 ab cos C are very
close but not exactly equivalent. Ask students to suggest reasons for this.
Copyright © 2011 by Nelson Education Ltd.
Students should be able to answer the Further Your Understanding questions
independently.
8.3: Exploring the Cosine Law
| 307
Curious Math
This Curious Math feature provides students with an opportunity to determine whether there is a relationship that relates the
tangents of the angles in a triangle to the sides, like the sine law and cosine law do. Students explore, using dynamic geometry
software, ratios that involve two sides and their tangents. Students should work through the questions individually.
Answers to Curious Math
Answers will vary, e.g.,
1.–4. If I calculated using the measurements for side lengths and angle measures, the results would be slightly
different than my results using the software calculator. The differences are because of the precision in the
measurements and because of rounding.
⎛1
⎞
tan ⎜ ( A − B ) ⎟
a−b
⎝2
⎠ are equal.
The ratios
and
⎛1
⎞
a+b
tan ⎜ ( A + B ) ⎟
⎝2
⎠
b−c
6. a)
=
b+c
308 |
⎞
⎛1
tan ⎜ (B − C ) ⎟
⎠
⎝2
⎛1
⎞
tan ⎜ (B + C ) ⎟
2
⎝
⎠
⎞
⎛1
tan ⎜ ( A − B ) ⎟
2
⎠.
⎝
⎛1
⎞
tan ⎜ ( A + B ) ⎟
⎝2
⎠
a−c
b)
=
a+c
⎞
⎛1
tan ⎜ ( A − C ) ⎟
⎠
⎝2
⎛1
⎞
tan ⎜ ( A + C ) ⎟
2
⎝
⎠
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
Copyright © 2011 by Nelson Education Ltd.
a−b
5. The tangent law is
=
a+b
8.4 APPLYING THE COSINE LAW
Lesson at a Glance
GOAL
Use the cosine law to calculate
unknown measures of sides and
angles in acute triangles.
Prerequisite Skills/Concepts
• Apply the Pythagorean theorem to determine side lengths.
• Understand the cosine law for acute triangles.
• Apply the primary trigonometric ratios to determine side lengths and
angle measures.
• Solve problems involving the properties of interior angles of a triangle
and angles formed by parallel lines.
Specific Expectations
• [Explore the development of the cosine law within acute triangles (e.g.,
use dynamic geometry software to verify the cosine law;] follow the
algebraic development of the cosine law and identify its relationship to
the Pythagorean theorem and the cosine ratio [student reproduction of the
development of the formula is not required]).
• Determine the measures of sides and angles in acute triangles, using [the
sine law and] the cosine law.
• Solve problems involving the measures of sides and angles in acute
triangles.
Student Book Pages 440–445
Preparation and Planning
Pacing
10 min
Introduction
20−25 min Teaching and Learning
30−35 min Consolidation
Materials
ƒ ruler
Recommended Practice
Questions 3, 5, 7, 8, 9, 11, 15
Key Assessment Question
Question 7
Extra Practice
Lesson 8.4 Extra Practice
Nelson Website
http://www.nelson.com/math
Mathematical Process Focus
Copyright © 2011 by Nelson Education Ltd.
• Problem Solving
• Selecting Tools and Computational Strategies
• Representing
MATH BACKGROUND | LESSON OVERVIEW
• In Lesson 8.3, students explored the cosine law and discovered how it extends the Pythagorean theorem to triangles
that do not have a right angle.
• In this lesson, students use the cosine law to calculate unknown side lengths and angle measures in acute triangles.
• Students solve problems that can be modelled by an acute triangle when two sides and the angle between them are
given or all three sides are given.
8.4: Applying the Cosine Law
| 309
1
Introducing the Lesson
(10 min)
Have students discuss, in pairs, what they discovered in Lesson 8.3. Ask
them to jot a few notes or draw a few diagrams about their discoveries to
prepare for a class discussion.
Initiate a class discussion about the exploration in Lesson 8.3. Invite pairs to
take turns describing their discoveries, displaying or sketching diagrams as
appropriate. If necessary, prompt students to include the following ideas:
• The cosine law extends the Pythagorean theorem to triangles that are not
right triangles.
• The cosine law is true for all acute triangles.
• For any acute triangle ABC,
a2 = b2 + c2 – 2 bc cos A
b2 = a2 + c2 – 2 ac cos B
c2 = a2 + b2 – 2 ab cos C
Summarize and discuss the cosine law: the square of one side equals the
sum of the squares of the other two sides less twice the product of the other
two sides and the cosine of the angle between these two sides. Then sketch
acute triangles with other vertices, such as X, Y, and Z, and ask students to
write the cosine law for these triangles.
2
Teaching and Learning
(20 to 25 min)
Example 1 proves that the cosine law is true for all acute triangles. It gives
students another opportunity to follow the algebraic development of the
cosine law and to identify the relationship between the cosine law and the
Pythagorean theorem. Ask students to explain how Heather’s solution is the
same as their exploration of the cosine law in Lesson 8.3. Ask why Heather
divided the triangle into two right triangles, as in the proof of the sine law in
Lesson 8.2.
Students could work on the Reflecting questions in pairs, prior to the class
discussion. Encourage discussion of the answers, and ask different students
to express the answers in their own words.
Answers to Reflecting
A. By dividing the acute triangle ABC into two right triangles, Heather was
able to use the Pythagorean theorem to express h as the height of+ADB
and+ADC. This allowed Heather to equate the two expressions for h.
Thus, dividing+ABC into two right triangles allowed her to relate the
sides and angles in+ABC.
310 |
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
Copyright © 2011 by Nelson Education Ltd.
Learn About the Math
B. If Heather had substituted a – x for y instead of a – y for x, her result
would have been the same. Her solution would have been different,
however, with the cosine law written using the cosine of ∠B instead of
∠C . After substituting and simplifying, the equation would have been
c2 – x2 = b2 – (a – x)2
c2 – x2 + (a – x)2 = b2
2
2
c – x + a2 – 2 ax + x2 = b2
c2 + a2 – 2 ax = b2
(substituting c cos B for x)
2
2
2
c + a – 2 ac cos B = b or
b2 = a2 + c2 – 2 ac cos B
3
Consolidation
(30 to 35 min)
Apply the Math
Using the Solved Examples
Copyright © 2011 by Nelson Education Ltd.
In Example 2, the cosine law is used to calculate the length of a side when
the other two sides and the angle between these sides are known. Have
students use the sine law to write an equation they could use to determine
the length of CB. Ask: Why could Justin not use the sine law to determine
the length of CB? Elicit from students that Justin would need to know the
measure of an angle opposite AC or AB. Then ask: How did Justin know
which version of the cosine law to use? Ensure that students realize the need
to determine the square root of a2, as in the Pythagorean theorem.
In Example 3, the cosine law is used to calculate the measure of an angle in
an acute triangle when the three side lengths are known. Discuss why the
measure of at least one angle is needed to apply the sine law, while the
cosine law can be applied without knowing any angle measure if all three
side lengths are known. Ask students how Darcy knew which version of the
cosine law to use.
Answer to the Key Assessment Question
For question 7, students would benefit from drawing their own diagram and
labelling the unknown with a variable that could be used in the cosine law.
They might find it easier if they labelled each vertex in the triangle. Ensure
that students realize why the answer should be expressed to the nearest
centimetre (because the given dimensions are to the nearest centimetre) and
the word “about” should be used (because the answer is approximate).
7. The diameter of the top of the cone is about 11 cm.
Closing
Question 15 gives students an opportunity to develop their own real-life
problem involving the cosine law. Ask students for examples of real-life
situations that could be modelled by the given triangle. Examples might
include a map, a sports field, sports equipment, or musical instruments.
8.4: Applying the Cosine Law
| 311
Assessment and Differentiating Instruction
What You Will See Students Doing…
When students understand…
If students misunderstand…
Students solve problems by applying the cosine law.
Students may have difficulty choosing the relevant version of
the cosine law. They may have difficulty drawing or
interpreting diagrams.
Students correctly solve equations for an unknown side or
angle measure.
Students may choose the correct version of the cosine law but
not be able to solve the equation accurately. They may make
errors when calculating or when using the inverse of the
cosine to determine the measure of an unknown angle.
Students understand that the cosine law can be used to solve
problems when two sides and the angle between them are
known or when all three sides are known.
Students may be unable to distinguish the conditions when
the cosine law should be used, or they may incorrectly attempt
to use the sine law.
Key Assessment Question 7
Students write a correct equation for the diameter of the top of
the cone in terms of the sides of the cone and the angle at the
bottom of the cone, using the cosine law.
Students may write the cosine law incorrectly. They may
forget to square one or all of the sides.
Students solve the equation correctly to determine the
diameter of the top of the cone.
Students may write the cosine law correctly but make errors
when solving the equation.
Students write an appropriate concluding sentence, with the
length expressed to the nearest centimetre.
Students may not express the answer to the given precision,
or they may make rounding errors. They may not write a
concluding sentence that answers the question.
Differentiating Instruction | How You Can Respond
1. If students are having difficulty understanding when the cosine law should be used, have them try to use the sine law to
write two ratios of sides and the angles that are opposite these sides. When students equate the two ratios, they will realize
that the proportion involves two unknowns and cannot be used.
2. If students are having difficulty solving the equations correctly, review the Pythagorean theorem. Include situations in which
the unknown is the hypotenuse and the unknown is one of the other two sides.
3. To help students understand and remember the cosine law, provide a variety of strategies:
• Guide students to write the cosine law in their own words, using a diagram for reference. Ask students to share their
descriptions and discuss how their descriptions are the same. For example, the square of one side equals the sum of
the squares of the other two sides, less twice the product of these two sides and the cosine of the angle opposite the
first side.
• To help students remember the cosine law, ask them how thinking about the Pythagorean theorem could help.
• Have students draw a triangle and write the cosine law for the triangle, using the Student Book for reference. They could
label their diagram to show the connections to the cosine law. Encourage creativity with the choice of methods, such as
arrows, colours, diagrams, or pictures.
Ask students to devise their own strategies, as well, and share their strategies with others.
EXTRA CHALLENGE
1. Challenge students to write true and false statements about the cosine law. Have them share their statements with
classmates and discuss which are true and which are false.
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Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
Copyright © 2011 by Nelson Education Ltd.
EXTRA SUPPORT
8.5 SOLVING ACUTE TRIANGLE
PROBLEMS
Lesson at a Glance
GOAL
Solve problems using the primary
trigonometric ratios and the sine and
cosine laws.
Prerequisite Skills/Concepts
Student Book Pages 446–451
• Apply the primary trigonometric ratios to determine side lengths and
angle measures.
• Solve problems involving the properties of interior angles of a triangle
and angles formed by parallel lines.
• Apply the Pythagorean theorem to determine side lengths.
• Use the sine law or cosine law to determine unknown side lengths and
angle measures.
Specific Expectations
• Determine the measures of sides and angles in acute triangles, using the
sine law and the cosine law.
• Solve problems involving the measures of sides and angles in acute
triangles.
Mathematical Process Focus
Copyright © 2011 by Nelson Education Ltd.
•
•
•
•
Problem Solving
Reasoning and Proving
Selecting Tools and Computational Strategies
Representing
Preparation and Planning
Pacing
5−10 min Introduction
25−30 min Teaching and Learning
25−30 min Consolidation
Materials
ƒ ruler
Recommended Practice
Questions 3, 4, 5, 6, 7, 8, 9, 14, 16
Key Assessment Question
Question 9
Extra Practice
Lesson 8.5 Extra Practice
Nelson Website
http://www.nelson.com/math
MATH BACKGROUND | LESSON OVERVIEW
• This lesson focuses on strategies that can be used to solve problems modelled by one or more triangles. Students
use the primary trigonometric ratios, the Pythagorean theorem, the sum of the angles in a triangle, the sine law, and
the cosine law.
• Students determine whether the sine law or the cosine law is required to solve a problem and then devise a strategy
for problem solving.
8.5: Solving Acute Triangle Problems
| 313
1
Introducing the Lesson
(5 to 10 min)
Present students with examples of triangles, including three right triangles
and four acute triangles with measurements for the following given:
two sides and the angle opposite one of these sides, two angles and one side,
two sides and the contained angle, and three sides.
Ask students to identify the measurements they would need to determine to
solve each triangle and the strategy they would use. Ask them to explain
how they would decide between the sine law and the cosine law.
2
Teaching and Learning
(25 to 30 min)
Learn About the Math
Example 1 illustrates a multi-step problem involving an acute triangle
divided into two right triangles. As you discuss the problem with the class,
ask these questions:
Answers to Reflecting
A. I think Vlad started with the right triangle that contained x because he
knew it contained a 75º angle. He used this angle for the sine ratio.
B. Vlad could have used the tangent ratio to determine the base of the right
triangle that contained x. Then he could have subtracted this value from
2000 m to determine the base of the triangle that contained y. He could
have used the inverse tangent ratio to determine θ and the sine ratio to
determine y.
C. After Vlad knew the values of x and y, he could have used the sine law
sin θ sin 75°
to write
. Then he could have solved for the value of θ.
=
x
y
314 |
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
Copyright © 2011 by Nelson Education Ltd.
• How did Vlad use the information at the top of the page to draw his
diagram?
• How do you know which angles in the acute triangle are equal to the
given angle and the angle of depression?
• How would you explain Vlad’s use of the cosine law?
To help students develop their own strategies for solving problems, have
them explain, in their own words, the steps that Vlad used to solve the
problem.
3
Consolidation
(25 to 30 min)
Apply the Math
Using the Solved Examples
Example 2 presents a problem involving directions. Discuss the notation that
is used to describe the directions. You could have students relate the
directions to objects in the classroom or places in the community. Have
students work in pairs to draw a clearly labelled diagram. Ask them how
they know that the cosine law is the correct law to use.
Example 3 presents a problem that is solved using the sine law and the sine
ratio. Ask students if Marnie could have used the cosine law to determine x.
Draw attention to the fact that the solution begins with a situation in which
two angles and one side are given and the length of another side needs to be
determined.
Answer to the Key Assessment Question
Students will likely find it helpful to draw a diagram for question 9, such as
the one to the right. If students need help dividing the acute triangle into two
right triangles, direct their attention to the examples. Discuss how to use the
sum of the angles in a triangle to determine the measure of ∠FBA . Students
can use the sine law to determine x and then the sine ratio to determine the
height, h.
9. The altitude of the airplane is about 276 m.
Copyright © 2011 by Nelson Education Ltd.
Closing
For question 16, remind students that their problem needs to involve a reallife situation that can be modelled by an acute triangle. Also remind students
that they are asked to explain what must be done to solve the problem; they
are not asked to write the solution. Students might decide to research data to
use in their problem. Have students work in pairs to create a problem,
diagram, and explanation. Alternatively, they could work on their own and
then trade problems with a partner to check. Students’ problems can be
shared with the class, to provide additional problems to solve.
8.5: Solving Acute Triangle Problems
| 315
Assessment and Differentiating Instruction
What You Will See Students Doing…
When students understand…
If students misunderstand…
Students apply appropriate strategies, including angle
relationships, primary trigonometric ratios, the sine law, and
the cosine law to solve problems.
Students may not be able to devise strategies to solve
problems. Students may not remember strategies for
determining side lengths and angle measures or how to apply
these strategies. They may not be able to distinguish between
the measurements needed for the sine law and the
measurements needed for the cosine law.
Students solve multi-step problems.
Students may have difficulty working backwards to decide
what measurement they need to determine so they can solve
a problem. They may not understand the connections
between steps.
Students accurately solve problems that involve more than
one triangle.
Students may not be able to visualize connections between
triangles in a diagram. They may have difficulty drawing a
diagram that connects triangles. They may not be able to
predict how a value can be determined for one triangle and
then used for another triangle in the diagram.
Key Assessment Question 9
Students draw a clearly labelled diagram.
Students may label the diagram incorrectly. They may be
unsure about where to label the given angles of elevation or
where to draw the altitude.
Students determine the angle opposite the given distance and
use the sine law to determine the distance from Fred to the
airplane.
Students may be unable to devise a strategy to solve the
problem.
Students use the sine ratio to calculate the height of the
airplane.
Students may set up the proportion correctly but make errors
when solving it.
EXTRA SUPPORT
1. If students are not sure whether to use the sine law or the cosine law to solve a problem, they can refer to In Summary.
They can match the given information with one of the triangles in the table. Students may find it helpful to create their own
table so they can mark the sides and angles in different colours or record measurements. They could also include the
primary trigonometric ratios, the sum of the angles in a triangle, and angle relationships. Display students’ tables for class
reference.
2. Students may find it helpful to explain their solutions in writing, like the student thinking in the examples. They could also
explain their solutions to a partner.
EXTRA CHALLENGE
1. Pose the following question: Is it possible to alter the cosine law when it is applied to an isosceles triangle? Explain.
2. Have students change their solution to a problem to show an error that they think might be made. Also have them record the
reason they think this is a possible error. Students can trade solutions and find the errors.
3. Have students create a trouble-shooting guide or a list of hints for solving acute triangle problems. This could be as brief as
one item, or several pages. Students could include notes, problems and solutions, examples, diagrams, and explanations.
316 |
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
Copyright © 2011 by Nelson Education Ltd.
Differentiating Instruction | How You Can Respond
CHAPTER REVIEW
Big Ideas Covered So Far
• The ratio
length of opposite side
sin (angle)
is the same for all three angle–side pairs in an acute triangle.
• The sine law states that in any acute triangle,+ABC,
a
sin A
=
b
sin B
=
c
sin C
.
• The sine law can be used to solve a problem modelled by an acute triangle if you can determine two sides and the angle
opposite one of these sides, or two angles and any side.
• The cosine law is an extension of the Pythagorean theorem to triangles that do not have a right angle.
• The cosine law states that in any acute triangle,+ABC,
a2 = b2 + c2 – 2 bc cos A
b2 = a2 + c2 – 2 ac cos B
c2 = a2 + b2 – 2 ab cos C
• The cosine law can be used to solve a problem modelled by an acute triangle if you can determine two sides and the angle
between them, or all three sides.
• If a real-world problem can be modelled using an acute triangle, unknown measurements can be determined using the sine
law or the cosine law, sometimes along with the primary trigonometric ratios.
Using the Frequently Asked Questions
Have students keep their Student Books closed. Display the Frequently Asked Questions on a board. Have students discuss the
questions and use the discussion to draw out what the class thinks are good answers. Then have students compare the class
answers with the answers on Student Book page 452. Students can refer to the answers to the Frequently Asked Questions as
they work through the Practice Questions.
Using the Chapter Review
Copyright © 2011 by Nelson Education Ltd.
Ask students if they have any questions about any of the topics covered so far in the chapter. Review any topics that students
would benefit from considering again. Assign Practice Questions for class work and for homework.
To gain greater insight into students’ understanding of the material covered so far in the chapter, you may want to ask questions
such as the following:
• How do you decide whether to use the sine law or the cosine law?
• How does drawing a diagram help you apply the sine law or cosine law?
• What is the connection between the inverse cosine ratio and the cosine ratio? How is the inverse cosine ratio the same as
the inverse sine ratio?
• The cosine law can only be used to solve some acute triangles. Why?
• If you knew the measures of two angles and the length of one side in a triangle, what property of triangles would allow you to
use the sine law, even if the given side was not opposite one of the known angles? Explain how this property would allow you
to use the sine law.
• What question can you ask about the cosine law? How might someone answer your question? What other question can you
ask about the cosine law?
Chapter 8 Review
|
317
CHAPTER 8 TEST
For further assessment items, please use Nelson's Computerized Assessment Bank.
1. Determine the indicated side length or angle measure in each triangle.
a)
b)
2. Solve each triangle, using the given information.
a) In+ABC, ∠A = 88°, a = 15 cm, and c = 8 cm.
b) In+DEF, ∠F = 72°, d = 8.0 cm, and e = 6.0 cm.
3. The lengths of the sides in+RST are 6.0 cm, 12.0 cm, and 13.0 cm. Determine the
measure of the least angle in the triangle.
4. A canoeist paddles 3.8 km in a N63ºE direction and then 4.2 km in a N35ºW direction.
Determine the length of the direct path to the canoeist’s final position.
6. Ahmed used a graphic design program to create a logo for his new business. The logo
contains a triangle with sides that are 30 cm, 35 cm, and 24 cm long. Determine the angle
measures and the positions of the angles in relation to the sides of the triangle.
7. The three sides of a triangular sail measure 9.0 m, 11.0 m, and 14.0 m. Calculate the total
area of the sail.
318 |
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
Copyright © 2011 by Nelson Education Ltd.
5. From Jennifer’s position on the finish line of a rally course, she can see the flags that mark
the final two check-in points. One flag is 720 m directly in front of her, and the other flag
is at an angle of 70°. The distance between the flags is 800 m. How far is the second flag
from the finish line?
CHAPTER 8 TEST ANSWERS
1. a) a = 73.9 cm, b = 40.3 cm
b) θ = 51º
2. a) ∠B = 60º, ∠C = 32º, b = 13 cm
b) ∠D = 65º, ∠E = 43º, f = 8.4 cm
3. about 27º
4. about 5.3 km
5. 671 m or 670 m, depending on rounding
6. about 58°: angle opposite the 30 cm side; about 80°: angle opposite the 35 cm side;
about 42°: angle opposite the 24 cm side
Copyright © 2011 by Nelson Education Ltd.
7. 49.5 m2
Chapter 8 Test Answers
| 319
CHAPTER TASK
Dangerous Triangles
Specific Expectations
• Determine the measures of sides and angles in acute triangles, using the
sine law and the cosine law.
• Solve problems involving the measures of sides and angles in acute
triangles.
Introducing the Chapter Task (Whole Class)
Introduce the task on Student Book page 455 by asking students what they
know about the Marysburgh Vortex or the Bermuda Triangle. Direct their
attention to the maps, relating the locations on the maps to places where
students have lived, visited, or studied. Ask students to suggest strategies for
estimating distances on the map of the Marysburgh Vortex. Point out that
the map of the Bermuda Triangle is not drawn to the same scale as the map
of the Marysburgh Vortex.
Student Book Page 455
Preparation and Planning
Pacing
5−10 min
Introducing the Chapter
Task
50−55 min Using the Chapter Task
Materials
ƒ ruler
Nelson Website
http://www.nelson.com/math
Using the Chapter Task
Have students work individually on the task. Students should clearly explain
how they determined the area of the Marysburgh Vortex and the area of the
Bermuda Triangle.
Remind students to use the Task Checklist to help them produce an
excellent solution. As students work through the task, observe them
individually to see how they are interpreting and carrying out the task.
Use the Assessment of Learning chart as a guide for assessing students’
work.
Adapting the Task
You can adapt the task in the Student Book to suit the needs of your
students. For example:
• Have students work in pairs so that stronger students can help students
who are experiencing difficulty.
• Have weaker students draw the heights of the two triangles before
beginning the task. The heights will help them form strategies they can
use to determine the areas of the triangles.
• Have stronger students research information about the Marysburgh
Vortex and the Bermuda Triangle or about similar areas that interest
them.
320 |
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
Copyright © 2011 by Nelson Education Ltd.
Assessing Students’ Work
Assessment of Learning—What to Look for in Student Work…
Assessment Strategy: Interview/Observation and Product Marking
Level of Performance
1
2
3
4
Knowledge and
Understanding
demonstrates
limited knowledge of
content
demonstrates some
knowledge of
content
demonstrates
considerable
knowledge of
content
demonstrates
thorough
knowledge of
content
Understanding of
mathematical concepts
demonstrates
limited
understanding of
concepts (e.g.,
incorrectly estimates
the lengths of the
sides that form the
triangle of the
Marysburgh Vortex;
is unable to
determine the area)
demonstrates some
understanding of
concepts (e.g.,
estimates the
lengths of the sides
that form the triangle
of the Marysburgh
Vortex; determines
the area, with some
errors)
demonstrates
considerable
understanding of
concepts (e.g.,
reasonably
estimates the
lengths of the sides
that form the triangle
of the Marysburgh
Vortex; determines
the area, with only
minor errors)
demonstrates
thorough
understanding of
concepts (e.g.,
closely estimates the
lengths of the sides
that form the triangle
of the Marysburgh
Vortex; correctly
determines the area)
Thinking
uses planning skills
with limited
effectiveness
uses planning skills
with some
effectiveness
uses planning skills
with considerable
effectiveness
uses planning skills
with a high degree
of effectiveness
Use of processing skills
• carrying out a plan
• looking back at the
solution
uses processing
skills with limited
effectiveness
uses processing
skills with some
effectiveness
uses processing
skills with
considerable
effectiveness
uses processing
skills with a high
degree of
effectiveness
Use of critical/creative
thinking processes
uses critical/creative
thinking processes
with limited
effectiveness (e.g., is
unable to justify the
decision about which
region is more
dangerous for
sailing)
uses critical/creative
thinking processes
with some
effectiveness (e.g.,
incompletely justifies
the decision about
which area is more
dangerous for
sailing)
uses critical/creative
thinking processes
with considerable
effectiveness (e.g.,
justifies the decision
by comparing the
areas and the
number of wrecks)
uses critical/creative
thinking processes
with a high degree
of effectiveness
(e.g., justifies the
decision in detail by
comparing the areas
and the number of
wrecks)
Knowledge of content
Copyright © 2011 by Nelson Education Ltd.
Use of planning skills
• understanding the
problem
• making a plan for solving
the problem
Chapter 8 Assessment of Learning
|
321
Assessment of Learning—What to Look for in Student Work…
Level of Performance
1
2
3
4
Communication
expresses and
organizes
mathematical
thinking with limited
effectiveness
expresses and
organizes
mathematical
thinking with some
effectiveness
expresses and
organizes
mathematical
thinking with
considerable
effectiveness
expresses and
organizes
mathematical
thinking with a high
degree of
effectiveness
Communication for different
audiences and purposes in
oral, visual, and written
forms
communicates for
different audiences
and purposes with
limited effectiveness
(e.g., is unable to
communicate the
justification for the
decision)
communicates for
different audiences
and purposes with
some effectiveness
(e.g., partially
communicates the
justification for the
decision)
communicates for
different audiences
and purposes with
considerable
effectiveness (e.g.,
communicates the
justification for the
decision in a clear
and organized way)
communicates for
different audiences
and purposes with a
high degree of
effectiveness (e.g.,
communicates the
justification for the
decision in an
exceptionally clear
and organized way)
Use of conventions,
vocabulary, and
terminology of the discipline
in oral, visual, and written
forms
uses conventions,
vocabulary, and
terminology of the
discipline with
limited effectiveness
uses conventions,
vocabulary, and
terminology of the
discipline with some
effectiveness
uses conventions,
vocabulary, and
terminology of the
discipline with
considerable
effectiveness
uses conventions,
vocabulary, and
terminology of the
discipline with a high
degree of
effectiveness
Application
applies knowledge
and skills in familiar
contexts with limited
effectiveness (e.g., is
unable to use the
trigonometry learned
in the chapter to
determine the areas
of the triangles)
applies knowledge
and skills in familiar
contexts with some
effectiveness (e.g.,
applies the
trigonometry learned
in the chapter to
determine the areas,
with several errors)
applies knowledge
and skills in familiar
contexts with
considerable
effectiveness (e.g.,
applies the
trigonometry learned
in the chapter to
determine the areas,
with only minor
errors)
applies knowledge
and skills in familiar
contexts with a
high degree of
effectiveness (e.g.,
applies the
trigonometry learned
in the chapter to
determine the areas
accurately)
Transfer of knowledge and
skills to new contexts
transfers knowledge
and skills to new
contexts with limited
effectiveness
transfers knowledge
and skills to new
contexts with some
effectiveness
transfers knowledge
and skills to new
contexts with
considerable
effectiveness
transfers knowledge
and skills to new
contexts with a
high degree of
effectiveness
Making connections within
and between various
contexts
makes connections
within and between
various contexts with
limited effectiveness
makes connections
within and between
various contexts with
some effectiveness
makes connections
within and between
various contexts with
considerable
effectiveness
makes connections
within and between
various contexts with
a high degree of
effectiveness
Expression and
organization of ideas and
mathematical thinking,
using oral, visual, and
written forms
Application of knowledge
and skills in familiar
contexts
322 |
Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
Copyright © 2011 by Nelson Education Ltd.
Assessment Strategy: Interview/Observation and Product Marking
Copyright © 2011 by Nelson Education Ltd.
CHAPTERS 7–8 CUMULATIVE REVIEW
1. B; students who answered incorrectly may need to review similar triangles in Lesson 7.1.
2. B; students who answered incorrectly may need to review solving problems using similar
triangles in Lesson 7.2.
3. D; students who answered incorrectly may need to review the primary trigonometric ratios in
Lesson 7.4.
4. C; students who answered incorrectly may need to review using the inverses of the primary
trigonometric ratios to determine an angle measurement in Lesson 7.4.
5. D; students who answered incorrectly may need to review solving for a side length using
a primary trigonometric ratio in Lesson 7.5.
6. A; students who answered incorrectly may need to review solving for a side length using
a primary trigonometric ratio in Lesson 7.5.
7. B; students who answered incorrectly may need to review solving problems that involve right
triangles in Lesson 7.6.
8. C; students who answered incorrectly may need to review solving problems that involve right
triangles in Lesson 7.6.
9. D; students who answered incorrectly may need to review solving problems that involve right
triangles in Lesson 7.6.
10. D; students who answered incorrectly may need to review solving problems that involve right
triangles in Lesson 7.6.
11. C; students who answered incorrectly may need to review solving problems that involve right
triangles in Lesson 7.6.
12. A; students who answered incorrectly may need to review the sine law in Lesson 8.2.
13. A; students who answered incorrectly may need to review the sine law in Lesson 8.2.
14. B; students who answered incorrectly may need to review the cosine law in Lesson 8.4.
15. B; students who answered incorrectly may need to review the cosine law in Lesson 8.4.
16. C; students who answered incorrectly may need to review solving acute triangles using the sine
law and the cosine law in Lesson 8.5.
17. A; students who answered incorrectly may need to review the sine law in Lesson 8.2 and the
cosine law in Lesson 8.4.
18. D; students who answered incorrectly may need to review solving problems that involve acute
triangles using the primary trigonometric ratios, the sine law, and the cosine law in Lesson 8.5.
19. B; students who answered incorrectly may need to review solving problems that involve acute
triangles using the sine law in Lesson 8.5.
20. Option B is less costly.
For option A, the cost of running a cable down the cliff is $276. The cost of running a cable
⎛ 23 ⎞
underwater is 33 ⎜
$3044.18. So, the total cost would be about $3320.18.
D⎟=
⎝ tan 14 ⎠
⎛ 8 ⎞
For option B, the change in elevation from the station to the first tower is sin −1 ⎜ ⎟ = 11.84 D
⎝ 39 ⎠
which means that three extra supports would be required. These supports would cost about $75.
The cost of running a cable from the station to the subdivision would be about
17(39 + 34 + 33 + 61 + 23) = $3230. So, the total cost would be about $3305.
Students who answered incorrectly may need to review solving problems that can be modelled
with right triangles using the primary trigonometric ratios in Lesson 7.6.
Chapters 7–8 Cumulative Review | 323
Copyright © 2011 by Nelson Education Ltd.
21. a) Since the base is a square, FG = GC = (0.5)(232.6) = 116.3. So FG is 116.3 m.
Using the Pythagorean theorem for+GFC, FC = 164.5. So FC is 164.5 m.
Using the Pythagorean theorem for+FCE, EF = 147.9. So EF is 147.9 m.
⎛ 147.9 ⎞
Using the tangent ratio for+EFG, ∠EGF = tan −1 ⎜
⎟ = 51.8205º.
⎝ 116.3 ⎠
Therefore, the angle that each face makes with the base is 52º to the nearest degree.
b) Since EB = AE = 221.2, or 221.2 m, and AB = 232.6 m, using the cosine law gives
∠AEB = 63.4400°. Therefore, the size of the apex angle of the face of the pyramid is 63° to
the nearest degree.
Students who answered incorrectly may need to review solving problems that involve acute
triangles choosing strategies such as the primary trigonometric ratios, the Pythagorean
theorem, the sine law, and the cosine law in Lesson 8.5.
324 |
Principles of Mathematics 10: Chapters 7–8 Cumulative Review