CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY


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1 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. 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, a 2 = b 2 + c 2 2 bc cos A b 2 = a 2 + c 2 2 ac cos B c 2 = a 2 + b 2 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 realworld 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
2 Chapter 8: Planning Chart Lesson Title Getting Started, pp Lesson 8.1: Exploring the Sine Law, pp Lesson 8.2: Applying the Sine Law, pp Lesson 8.3: Exploring the Cosine Law, pp Lesson 8.4: Applying the Cosine Law, pp Lesson 8.5: Solving Acute Triangle Problems, pp MidChapter Review, pp Chapter Review, pp Chapter SelfTest, p. 454 Curious Math, p. 439 Chapter Task, p. 455 Lesson Goal Use concepts and skills developed prior to this chapter. Explore the relationship between each side in an acute triangle and the sine of its opposite angle. Use the sine law to calculate unknown side lengths and angle measures in acute triangles. Explore the relationship between side lengths and angle measures in a triangle using the cosines of angles. Use the cosine law to calculate unknown measures of sides and angles in acute triangles. Solve problems using the primary trigonometric ratios and the sine and cosine laws. Pacing 10 days Materials/Masters Needed 2 days ruler; protractor; Diagnostic Test 1 day dynamic geometry software, or ruler and protractor 1 day ruler; Lesson 8.2 Extra Practice 1 day dynamic geometry software 1 day ruler; Lesson 8.4 Extra Practice 1 day ruler; Lesson 8.5 Extra Practice 3 days MidChapter Review Extra Practice; Chapter Review Extra Practice; Chapter Test; Chapters 7 8 Cumulative Review 290 Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
3 CHAPTER OPENER Using the Chapter Opener 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
4 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 Preparation and Planning Pacing 5 10 min Words You Need to Know min Skills and Concepts You Need min Applying What You Know Materials ruler protractor Nelson Website Using the Applying What You Know 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 , 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
5 C. Draw h from a goalpost perpendicular to the opposite side. h D. sin 75º = 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 = cos 75º = x 1.68 = x Then calculate y. y = 5.5 x y = y = 3.82 Let the width of the soccer net, in metres, be n. Use the Pythagorean theorem to calculate n. n 2 = y 2 + h 2 n 2 = n 2 = n = n = 7.35 The width of the soccer net is about 7.4 m. Initial Assessment When students understand Students articulately explain why a primary trigonometric ratio cannot be used to calculate the width of the soccer net directly. Students draw the height of the triangle correctly. Students apply the sine ratio, the cosine ratio, and then the Pythagorean theorem to determine the width of the net. What You Will See Students Doing If students misunderstand 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 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 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
6 8.1 EXPLORING THE SINE LAW Lesson at a Glance GOAL Explore the relationship between each side in an acute triangle and the sine of its opposite angle. 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])]. Mathematical Process Focus Reasoning and Proving Connecting Student Book Pages Preparation and Planning Pacing 5 10 min Introduction min Teaching and Learning min Consolidation Materials dynamic geometry software, or ruler and protractor Recommended Practice Questions 1, 2, 3, 4 Nelson Website 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 a = b = c. sin A sin B sin C length of opposite sin (angle) side is 294 Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
7 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. 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
8 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., Angle Side Sine length of opposite side sin (angle) A = 52.3º a = 2.6 sin A = 0.79 B = 55.5º b = 2.8 sin B = 0.82 C = 72.2º c = 3.2 sin C = 0.95 a sin A b sin B c sin C = 3.3 = 3.3 = 3.3 C. The ratio is the same for all three angles in my triangle. D. 296 Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
9 Angle Side Sine length of opposite side sin (angle) A = 26.3º a = 2.6 sin A = 0.44 B = 65.2º b = 5.4 sin B = 0.91 C = 88.5º c = 6.0 sin C = 1.00 a sin A b sin B c sin C = 6.0 = 6.0 = 6.0 length of opposite side The ratio 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 sin (angle) in a triangle. F. is always the same for all three angles Angle Side Sine A = 48.2º a = 3.9 sin A = 0.75 B = 41.8º b = 3.5 sin B = 0.67 C = 90.0º c = 5.2 sin C = 1.00 length of opposite side sin (angle) a sin A b sin B c sin C = 5.2 = 5.2 = 5.2 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
10 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. Students should be able to answer the Further Your Understanding questions independently. 298 Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
11 8.2 APPLYING THE SINE LAW Lesson at a Glance GOAL Use the sine law to calculate unknown side lengths and angle measures in acute triangles. Prerequisite Skills/Concepts length of opposite side Understand that the ratio 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. 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. Student Book Pages Preparation and Planning Pacing 5 10 min Introduction min Teaching and Learning min Consolidation Materials 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 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 sin (angle) side 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
12 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. Answers to Reflecting 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 would be able to show that the ratios and are equal. 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 equation by a and then by c will result in =. Therefore, 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
13 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. Answer to the Key Assessment Question Students should draw a diagram to help them visualize the parallelogram in the multistep 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
14 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 Students write the sine law correctly, with the value to be determined in the numerator. Students solve an equation correctly and state the results in a concluding sentence. If students misunderstand 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 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. Key Assessment Question 7 Students create and label a parallelogram correctly to model the situation. 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 cannot create or label a parallelogram correctly to model the situation. They may label the given sides incorrectly. Students may have difficulty with multistep 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. Differentiating Instruction How You Can Respond EXTRA SUPPORT 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. 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 realworld 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
15 MIDCHAPTER REVIEW Big Ideas Covered So Far The ratio length of opposite sin (angle) side 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 = c. 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. 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 MidChapter 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? What have you learned about the sine law? Chapter 8 MidChapter Review 303
16 8.3 EXPLORING THE COSINE LAW 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 Preparation and Planning Pacing 5 10 min Introduction min Teaching and Learning min Consolidation Materials dynamic geometry software Recommended Practice Questions 1, 2, 3, 4, 5 New Vocabulary/Symbols cosine law Nelson Website MATH BACKGROUND LESSON OVERVIEW Students should understand the Pythagorean theorem. In this lesson, students explore the cosine law for an acute triangle,+abc: a 2 = b 2 + c 2 2 bc cos A b 2 = a 2 + c 2 2 ac cos B c 2 = a 2 + b 2 2 ab cos C The cosine law is an extension of the Pythagorean theorem to triangles that do not have a right angle. 304 Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
17 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. 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
18 Answers to Explore the Math A. B. C. c 2 = a 2 + b 2, or = D. E., G. Triangle a b c C c 2 a 2 + b 2 a 2 + b 2 c 2 2ab cos C º º º º º 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 a 2 + b 2 c 2 and 2 ab cos C are equivalent. H. I can conclude that 2 ab cos C = a 2 + b 2 c 2 in an acute triangle. 306 Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
19 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 a 2 to b 2 + c 2 by a 2 = b 2 + c 2 2 bc cos A. ii) The cosine law can be used to relate b 2 to a 2 + c 2 by b 2 = a 2 + c 2 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: a 2 = b 2 + c 2 2 bc cos A b 2 = a 2 + c 2 2 ac cos B c 2 = a 2 + b 2 2 ab cos C Students may notice that the values of a 2 + b 2 c 2 and 2 ab cos C are very close but not exactly equivalent. Ask students to suggest reasons for this. Students should be able to answer the Further Your Understanding questions independently. 8.3: Exploring the Cosine Law 307
20 308 Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry 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., 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. The ratios b a b a + and + ) ( 2 1 tan ) ( 2 1 tan B A B A are equal. 5. The tangent law is + = + ) ( 2 1 tan ) ( 2 1 tan B A B A b a b a. 6. a) + = + ) ( 2 1 tan ) ( 2 1 tan C B C B c b c b b) + = + ) ( 2 1 tan ) ( 2 1 tan C A C A c a c a
21 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 Preparation and Planning Pacing 10 min Introduction min Teaching and Learning min Consolidation Materials 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 Mathematical Process Focus 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
22 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, a 2 = b 2 + c 2 2 bc cos A b 2 = a 2 + c 2 2 ac cos B c 2 = a 2 + b 2 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) Learn About the Math 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
23 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 c 2 x 2 = b 2 (a x) 2 c 2 x 2 + (a x) 2 = b 2 c 2 x 2 + a 2 2 ax + x 2 = b 2 c 2 + a 2 2 ax = b 2 (substituting c cos B for x) c 2 + a 2 2 ac cos B = b 2 or b 2 = a 2 + c 2 2 ac cos B 3 Consolidation (30 to 35 min) Apply the Math Using the Solved Examples 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 a 2, 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 reallife problem involving the cosine law. Ask students for examples of reallife 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
24 Assessment and Differentiating Instruction What You Will See Students Doing When students understand Students solve problems by applying the cosine law. Students correctly solve equations for an unknown side or angle measure. 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. If students misunderstand Students may have difficulty choosing the relevant version of the cosine law. They may have difficulty drawing or interpreting diagrams. 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 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 solve the equation correctly to determine the diameter of the top of the cone. Students write an appropriate concluding sentence, with the length expressed to the nearest centimetre. Students may write the cosine law incorrectly. They may forget to square one or all of the sides. Students may write the cosine law correctly but make errors when solving the equation. 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 EXTRA SUPPORT 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. 312 Principles of Mathematics 10: Chapter 8: Acute Triangle Trigonometry
25 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 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 Problem Solving Reasoning and Proving Selecting Tools and Computational Strategies Representing Student Book Pages Preparation and Planning Pacing 5 10 min Introduction min Teaching and Learning min Consolidation Materials 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 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
26 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 multistep problem involving an acute triangle divided into two right triangles. As you discuss the problem with the class, ask these questions: 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. 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 sinθ 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
27 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. 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
28 Assessment and Differentiating Instruction What You Will See Students Doing When students understand Students apply appropriate strategies, including angle relationships, primary trigonometric ratios, the sine law, and the cosine law to solve problems. Students solve multistep problems. Students accurately solve problems that involve more than one triangle. If students misunderstand 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 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 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 determine the angle opposite the given distance and use the sine law to determine the distance from Fred to the airplane. Students use the sine ratio to calculate the height of the airplane. 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 may be unable to devise a strategy to solve the problem. Students may set up the proportion correctly but make errors when solving it. Differentiating Instruction How You Can Respond 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 troubleshooting 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
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