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Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Lesson 26: The Definition of Sine, Cosine, and Tangent Student Outcomes ο§ Students define sine, cosine, and tangent of π, where π is the angle measure of an acute angle of a right triangle. Students denote sine, cosine, and tangent as sin, cos, and tan, respectively. ο§ If β π΄ is an acute angle whose measure in degrees is π, then we also say: sin β π΄ = sinπ, cos β π΄ = cosπ, and tan β π΄ = tanπ. ο§ Given the side lengths of a right triangle with acute angles, students find sine, cosine, and tangent of each acute angle. Lesson Notes It is convenient, as adults, to use the notation βsin2 π₯β to refer to the value of the square of the sine function. However, rushing too fast to this abbreviated notation for trigonometric functions leads to incorrect understandings of how functions are manipulated, which can lead students to think that sin π₯ is short for βsin β π₯β and to incorrectly divide out the variable, β sin π₯ π₯ = sin.β To reduce these types of common notation-driven errors later, this curriculum is very deliberate about how and when we use abbreviated function notation for sine, cosine, and tangent: 1. In geometry, sine, cosine, and tangent are thought of as the value of ratios of triangles, not as functions. No attempt is made to describe the trigonometric ratios as functions of the real number line. Therefore, the notation is just an abbreviation for the βsine of an angleβ (sin β π΄) or βsine of an angle measureβ (sin π). Parentheses are used more for grouping and clarity reasons than as symbols used to represent a function. 2. In Algebra II, to distinguish between the ratio version of sine in geometry, all sine functions are notated as functions: sin(π₯) is the value of the sine function for the real number π₯, just like π(π₯) is the value of the function π for the real number π₯. In this grade, we maintain function notation integrity and strictly maintain parentheses as part of π π 2 2 function notation, writing, for example, sin ( β π) = cos(π), instead of sin β π = cos π. 3. By pre-calculus, students have had two full years of working with sine, cosine, and tangent as both ratios and functions. It is finally in this year that we begin to blur the distinction between ratio and function notations and write, for example, sin2 π as the value of the square of the sine function for the real number π, which is how most calculus textbooks notate these functions. Lesson 26: Date: The Definition of Sine, Cosine, and Tangent 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 401 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Classwork Exercises 1β3 (6 minutes) The following exercises provide students with practice identifying specific ratios of sides based on the labels learned in Lesson 25. The third exercise leads into an understanding of the relationship between sine and cosine, which is observed in this lesson and formalized in Lesson 27. Exercises 1β3 1. Identify the For β π¨: For β π©: 2. For β π©: 3. ratios for angles β π¨ and β π©. ππ ππ π ππ Identify the For β π¨: πππ πππ Scaffolding: ππ π πππ ο§ For students who are struggling, model labeling opp, hyp, and adj for β π΄. ratios for angles β π¨ and β π©. π ππ ο§ Allow students to use the reference chart provided with Lesson 25. ππ ππ Describe the relationship between the ratios for angles β π¨ and β π©. The The πππ πππ πππ πππ ratio for β π¨ is equal to the ratio for β π© is equal to the ππ π πππ ππ π πππ ratio for β π©. ratio for β π¨. Discussion (6 minutes) The Discussion defines sine, cosine, and tangent. As the names πππ, πππ, and βπ¦π are relatively new to students, it is important that students have a visual of the triangle to reference throughout the discussion. Following the discussion is an exercise that can be used to informally assess studentsβ understanding of these definitions. ο§ ο§ For advanced students, include an extension: Draw another triangle π΄π΅πΆ with different side lengths that will have the same ratios for β π΄ and β π΅. Explain how you know the ratios will be the same, and justify your answer. In everyday life, we reference objects and people by name, especially when we use the object or see the person frequently. Imagine always calling your friend βheyβ or βhim/herβ or βfriendβ! We want to be able to easily distinguish and identify a person, so we use a name. The same reasoning can be applied to the fractional expressions that we have been investigating: πππ πππ , βπ¦π βπ¦π , and πππ πππ need names. Normally, we would say that these fractional expressions are values of the ratios. πππ However, to avoid saying βvalue of the ratio πππ: βπ¦π as β all of the time, we βπ¦π agree to call these fractional expressions, collectively, the βtrigonometric ratios,β based upon historical precedence, even though they really are not defined as ratios in the CCSS standards. Consider sharing this fact with students. Lesson 26: Date: The Definition of Sine, Cosine, and Tangent 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 402 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY ο§ These incredibly useful ratios were discovered long ago and have had several names. The names we currently use are translations of Latin words. You will learn more about the history behind these ratios in Algebra II. ο§ Presently, mathematicians have agreed upon the names sine, cosine, and tangent. ο§ If π is the angle measure of β π΄, as shown, then we define: The sine of π is the value of the ratio of the length of the opposite side to the length of the hypotenuse. As a formula, πππ sin π = . βπ¦π We also say sin β π΄ = sin π. Then, sin β π΄ = π΅πΆ π΄π΅ . The cosine of π is the value of the ratio of the length of the adjacent side to the length of the hypotenuse. As a formula, πππ cos π = . βπ¦π We also say cos β π΄ = cos π. Then, cos β π΄ = π΄πΆ π΄π΅ . The tangent of π is the value of the ratio of the length of the opposite side to the length of the adjacent side. As a formula, πππ tan π = . πππ We also say tan β π΄ = tan π . Then, tan β π΄ = ο§ π΅πΆ π΄πΆ . There are still three possible combinations of quotients of the side lengths; we briefly introduce them here. The secant of π is the value of the ratio of the length of the hypotenuse to the length of the adjacent side. As a formula, βπ¦π sec π = . πππ The cosecant of π is the value of the ratio of the length of the hypotenuse to the length of the opposite side. As a formula, βπ¦π csc π = . πππ The cotangent of π is the value of the ratio of the length of the adjacent side to the length of the opposite side. As a formula, πππ cot π = . πππ ο§ We have little need in this course for secant, cosecant, and cotangent because, given any two sides, it is possible to write the quotient so as to get sine, cosine, and tangent. In more advanced courses, secant, cosecant, and cotangent are more useful. Lesson 26: Date: The Definition of Sine, Cosine, and Tangent 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 403 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Exercises 4β9 (15 minutes) You can choose to have students complete Exercises 4β6 independently, or divide the work among the students and have them share their results. Once Exercises 4β6 have been completed, encourage students to discuss in small groups the relationships they notice between the sine of the angle and the cosine of its complement. Also, encourage them to discuss the relationship they notice about the tangent of both angles. Finally, have students share their observations with the whole class and then complete Exercises 7β8. Note that because we are not asking students to rationalize denominators, the relationships are clearer. Scaffolding: Exercises 4β9 4. ο§ Consider having students draw two right triangles and then color-code and/or label each with opp, adj, and hyp, relative to the angle they are looking at. In β³ π·πΈπΉ, πβ π· = ππ. π° and πβ πΈ = ππ. π°. Complete the following table. ο§ Consider having advanced students draw two right triangles, β³ π·πΈπΉ and β³ πΊπ»πΌ, such that they are not congruent, but so that sin πΈ = sin π» . Then, have students explain how they know. Measure of Angle Sine ( πππ ) πππ Cosine ππ π (πππ) Tangent πππ ( ππ π ) ππ. π π¬π’π§ ππ. π = π π ππ¨π¬ ππ. π = π π πππ§ ππ. π = π π ππ. π π¬π’π§ ππ. π = π π ππ¨π¬ ππ. π = π π πππ§ ππ. π = π π Lesson 26: Date: The Definition of Sine, Cosine, and Tangent 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 404 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 5. In the triangle below, πβ π¨ = ππ. π° and πβ π© = ππ. π°. Complete the following table. Measure of Angle 6. Sine ππ. π π¬π’π§ ππ. π = ππ. π π¬π’π§ ππ. π = Cosine π βππ π βππ ππ¨π¬ ππ. π = ππ¨π¬ ππ. π = Tangent π βππ π βππ π π π πππ§ ππ. π = π πππ§ ππ. π = In the triangle below, let π be the measure of β π¬ and π be the measure of β π«. Complete the following table. Measure of Angle π π¬π’π§ π = π Lesson 26: Date: Sine π¬π’π§ π = Cosine π π βππ π βππ π π ππ¨π¬ π = π ππ¨π¬ π = Tangent πππ§ π = π βππ βππ πππ§ π = π The Definition of Sine, Cosine, and Tangent 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 405 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 7. In the triangle below, let π be the measure of β πΏ and π be the measure of β π. Complete the following table. Measure of Angle 8. Sine π π¬π’π§ π = π π¬π’π§ π = π βππ π βππ ππ¨π¬ π = ππ¨π¬ π = Tangent π π π π πππ§ π = π πππ§ π = βππ π βππ Tamer did not finish completing the table below for a diagram similar to the previous problems that the teacher had on the board where π was the measure of β π· and π was the measure of β πΈ. Use any patterns you notice from Exercises 1β4 to complete the table for Tamer. Measure of Angle MP.8 9. Cosine Sine π π¬π’π§ π = π π¬π’π§ π = Cosine ππ βπππ π βπππ ππ¨π¬ π = Tangent π πππ§ π = βπππ ππ¨π¬ π = ππ ππ π πππ§ π = βπππ π ππ Explain how you were able to determine the sine, cosine, and tangent of β πΈ in Exercise 7. I was able to complete the table for Tamer by observing the patterns of previous problems. For example, I noticed that the sine of one angle was always equal to the ratio that represented the cosine of the other angle. Since I was given π¬π’π§ π, I knew the ratio ππ βπππ would be the ππ¨π¬ π. Similarly, ππ¨π¬ π = π¬π’π§ π = π . Finally, I noticed that the βπππ tangents of the angles were always reciprocals of each other. Since I was given the πππ§ π = πππ§ π must be equal to π ππ ππ , I knew that the π . Discussion (8 minutes) The sine, cosine, and tangent of an angle can be used to find unknown lengths of other triangles using within-figure ratios of similar triangles. The discussion that follows begins by posing a question to students. Provide time for students to discuss the answer in pairs or small groups, and then have them share their thoughts with the class. ο§ If 0 < π < 90, we can define sine, cosine, and tangent of π: take a right triangle that has an acute angle with angle degree measure π, and use the appropriate side lengths. ο§ If we use different right triangles, why will we get the same value for sin π, cos π, and tan π? Lesson 26: Date: The Definition of Sine, Cosine, and Tangent 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 406 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Provide time for students to talk to a partner or small group. If necessary, use the questions and diagrams below to MP.3 guide studentsβ thinking. The decimal values for the side lengths are used to make less obvious the fact that the ratios of the side lengths are equal. ο§ For example, consider the following two triangles. The triangles below contain approximations for all lengths. If the Pythagorean theorem is used to verify that the triangles are right triangles, then you will notice that the values are slightly off. For example, the length of the hypotenuse of β³ π·πΈπΉ contains 11 decimal digits, not the 5 shown. ο§ Find the sin π΄, cos π΄, and tan π΄. Compare those ratios to the sin π·, cos π·, and tan π·. What do you notice? The task of finding the ratios can be divided among the students and shared with the group. οΊ ο§ Students should notice that sin π΄ = sin π·, cos π΄ = cos π·, and tan π΄ = tan π· . Under what circumstances have we observed ratios within one figure being equal to ratios within another figure? οΊ Within-figure ratios are equal when the figures are similar. ο§ Two right triangles, each having an acute angle of angle measure π, are similar by the AA criterion. So, we know that the values of corresponding ratios of side lengths will be equal. That means sin π, cos π, and tan π do not depend on which right triangle we use. ο§ The ratios we write for the sine, cosine, and tangent of an angle are useful because they allow us to solve for two sides of a triangle when we know only the length of one side. Closing (5 minutes) Ask students the following questions. You may choose to have students respond in writing, to a partner, or to the whole class. ο§ Describe the ratios that we used to calculate sine, cosine, and tangent. οΊ ο§ Given an angle, π, sin π = πππ βπ¦π , cos π = πππ βπ¦π , and tan π = πππ πππ . Given any two right triangles that each have an acute angle with measure π, why would we get the same value for sin π, cos π, and tan π using either triangle? Lesson 26: Date: The Definition of Sine, Cosine, and Tangent 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 407 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 26 M2 GEOMETRY οΊ ο§ Since the two right triangles each have an acute angle with measure π, they are similar by the AA criterion. Similar triangles will have corresponding side lengths that are equal in ratio. Additionally, based on our investigations in Lesson 25, we know that the value of the ratios of corresponding sides for a particular angle size will be equal to the same constant. Given a right triangle, describe the relationship between the sine of one acute angle and the cosine of the other acute angle. οΊ The sine of one acute angle of a right triangle is equal to the cosine of the other acute angle in the triangle. Exit Ticket (5 minutes) Lesson 26: Date: The Definition of Sine, Cosine, and Tangent 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 408 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Name Date Lesson 26: The Definition of Sine, Cosine, and Tangent Exit Ticket 1. Given the diagram of the triangle, complete the following table. π¬π’π§ π½ Angle Measure ππ¨π¬ π½ πππ§ π½ π π 2. a. Which values are equal? b. How are tan π and tan π‘ related? 2 β21 5 2 If π’ and π£ are the measures of complementary angles such that sin π’ = and tan π£ = , label the sides and angles of the right triangle in the diagram below with possible side lengths. Lesson 26: Date: The Definition of Sine, Cosine, and Tangent 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 409 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Exit Ticket Sample Solutions 1. Given the diagram of the triangle, complete the following table. Angle π π a. π¬π’π§ π½ πβππ = ππ βππ π πβππ = ππ βππ π ππ¨π¬ π½ πβππ = ππ βππ π πβππ = ππ βππ πππ§ π½ π π π π π Which values are equal? π¬π’π§ π = ππ¨π¬ π and ππ¨π¬ π = π¬π’π§ π b. How are πππ§ π and πππ§ π related? They are reciprocals; 2. π π β π π = π. If π and π are the measures of complementary angles such that π¬π’π§ π = βππ π and πππ§ π = , label the sides and π π angles of the right triangle in the diagram below with possible side lengths: A possible solution is shown below; however, any similar triangle having a shorter leg with length of ππ, longer leg with length of πβππ, and hypotenuse with length of ππ, for some positive number π, is also correct. Problem Set Sample Solutions 1. Given the triangle in the diagram, complete the following table. Angle Measure πΆ π· Lesson 26: Date: π¬π’π§ πβπ π π π = π π ππ¨π¬ π π = π π πβπ π πππ§ πβπ βπ = π π π π = πβπ βπ The Definition of Sine, Cosine, and Tangent 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 410 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 2. Given the table of values below (not in simplest radical form), label the sides and angles in the right triangle. Angle Measure πΆ π· 3. ππ¨π¬ πβπ πππ§ π πβππ πβπ πβππ π πβπ πβππ πβππ πβπ π Given π¬π’π§ πΆ and π¬π’π§ π·, complete the missing values in the table. You may draw a diagram to help you. Angle Measure πΆ π· 4. π¬π’π§ π π¬π’π§ βπ ππ¨π¬ π πππ§ πβπ π πβπ βπ βπ π π πβπ πβπ βπ Given the triangle shown to the right, fill in the missing values in the table. Using the Pythagorean theorem: ππππ = ππ + ππ ππππ = π + ππ ππππ = ππ πππ = βππ πππ = πβππ Angle Measure πΆ π· 5. π¬π’π§ πβππ = ππ πβππ π βππ = ππ πβππ π ππ¨π¬ π βππ = ππ πβππ π πβππ = ππ πβππ πππ§ π =π π π π = π π Jules thinks that if πΆ and π· are two different acute angle measures, then π¬π’π§ πΆ β π¬π’π§ π·. Do you agree or disagree? Explain. I agree. If πΆ and π· are different acute angle measures, then either πΆ > π· or π· > πΆ. A right triangle with acute angle πΆ cannot be similar to a right triangle with acute angle π· (unless πΆ + π· = ππ) because the triangles fail the AA criterion. If the triangles are not similar, then their corresponding sides are not in proportion, meaning their within-figure ratios are not in proportion; therefore, π¬π’π§ πΆ β π¬π’π§ π·. In the case where πΆ + π· = ππ, the given right triangles are similar; however, πΆ and π· must be alternate acute angles, meaning π¬π’π§ πΆ = ππ¨π¬ π·, and π¬π’π§ π· = ππ¨π¬ πΆ, but π¬π’π§ πΆ β π¬π’π§ π·. Lesson 26: Date: The Definition of Sine, Cosine, and Tangent 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 411 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 6. Given the triangle in the diagram, complete the following table. Angle Measure πΆ π· π¬π’π§ ππ¨π¬ πβπ π πβπ π πβπ π πβπ π πππ§ πβπ πβπ πβπ πβπ Rewrite the values from the table in simplest terms. Angle Measure πΆ π· π¬π’π§ ππ¨π¬ βπ π βπ π βπ π βπ π πππ§ βπ = βπ βπ βπ βπ = π βπ Draw and label the sides and angles of a right triangle using the simplified values of the ratios π¬π’π§ and ππ¨π¬. How is the new triangle related to the original triangle? The triangles are similar by SSS criterion because the new triangle has sides π that are of the length of their corresponding sides in the original triangle. π 7. Given πππ§ πΆ and ππ¨π¬ π·, in simplest terms, find the missing side lengths of the right triangle if one leg of the triangle has a length of π. Draw and label the sides and angles of the right triangle. Angle Measure πΆ π· π¬π’π§ π½ βπ π π π ππ¨π¬ π½ π π βπ π πππ§ π½ βπ βπ π The problem does not specify which leg is π, so there are two possible solutions to this problem. The values given in the table do not represent the actual lengths of the sides of the triangles; however, they do represent the lengths of the sides of a similar triangle, which is a 30β60β90 right triangle with side lengths π, π, and βπ. Case 1: The short leg of the right triangle is π: Lesson 26: Date: Case 2: The long leg of the right triangle is π: The Definition of Sine, Cosine, and Tangent 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 412 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 26 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 8. Eric wants to hang a rope bridge over a small ravine so that it is easier to cross. To hang the bridge, he needs to know how much rope is needed to span the distance between two trees that are directly across from each other on either side of the ravine. Help Eric devise a plan using sine, cosine, and tangent to determine the approximate distance from tree A to tree B without having to cross the ravine. Student solutions will vary. Possible solution: If Eric walks a path parallel to the ravine to a point π· at a convenient distance from π¨, he could measure the angle formed by his line of sight to both trees. Using the measured angle and distance, he could use the value of the tangent ratio of the angle to determine the length of the opposite leg of the triangle. The length of the opposite leg of the triangle represents the distance between the two trees. 9. A fisherman is at point π on the open sea and has three favorite fishing locations. The locations are indicated by points π¨, π©, and πͺ. The fisherman plans to sail from π to π¨, then to π©, then to πͺ, then back to π. If the fisherman is Μ Μ Μ Μ , find the total distance that he will sail. ππ miles from π¨πͺ ππ· = ππ and can be considered the adjacent side to the ππ° angle shown in triangle π¨π·π. Using cosine: ππ¨π¬ ππ = ππ π¨π ππ πππ ππ π¨π β ππ. ππ π¨π = Using tangent: πππ§ ππ = π¨π· ππ π¨π· = ππ πππ§ ππ π¨π· β π. ππππ Μ Μ Μ Μ π·πͺ is the leg opposite angle π·ππͺ in triangle π·ππͺ and has a degree measure of ππ. π. Using tangent: πππ§ ππ. π = π·πͺ ππ π·πͺ = ππ πππ§ ππ. π π·πͺ β ππ. ππππ The total distance that the fisherman will sail: ππ’π¬πππ§ππ = π¨π + π¨π· + π·πͺ + ππͺ ππ’π¬πππ§ππ = Using cosine: ππ¨π¬ ππ. π = ππ ππͺ ππ ππ¨π¬ ππ + ππ πππ§ ππ + ππ πππ§ ππ. π + ππ ππ¨π¬ ππ.π ππ’π¬πππ§ππ β ππ. π ππ π π = ππ¨π¬ ππ.π ππͺ β ππ. ππππ The total distance that the fisherman will sail is approximately ππ. π miles. Lesson 26: Date: The Definition of Sine, Cosine, and Tangent 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 413 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
