Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle Student Outcomes ๏ง Students find missing side lengths of an acute triangle given one side length and the measures of two angles. ๏ง Students find the missing side length of an acute triangle given two side lengths and the measure of the included angle. Lesson Notes In Lesson 32, students learn how to determine unknown lengths in acute triangles. Once again, they drop an altitude in the given triangle to create right triangles and use trigonometric ratios and the Pythagorean theorem to solve triangle problems (G-SRT.C.8). This lesson and the next introduce the law of sines and cosines (G-SRT.D.10 and G-SRT.D.11). Based on availability of time in the module, teachers may want to divide the lesson into two parts by addressing everything until Exercises 1โ2 on one day and the remaining content on the following day. Classwork Opening Exercise (3 minutes) The objective for part (b) is that students realize that ๐ฅ and ๐ฆ cannot be found using the method they know with trigonometric ratios. Opening Exercise a. Find the lengths of ๐ and ๐. ๐ ๐๐ ;๐ = ๐ โ๐ ๐ ๐ ๐๐จ๐ฌ ๐๐ = ; ๐ = ๐๐ โ๐ โ๐ ๐ฌ๐ข๐ง ๐๐ = b. Find the lengths of ๐ and ๐. How is this different from part (a)? Accept any reasonable answer explaining that the triangle is not a right triangle; therefore, the trigonometric ratios used in part (a) are not applicable here. Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 477 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Discussion (10 minutes) Lead students through an explanation of the law of sines for acute triangles. Scaffolding: ๏ง Today we will show how two facts in trigonometry aid us to find unknown measurements in triangles that are not right triangles; we can use these facts for acute and obtuse triangles, but today we will specifically study acute triangles. The facts are called the law of sines and the law of cosines. We begin with the law of sines. ๏ง LAW OF SINES: For an acute triangle โณ ๐ด๐ต๐ถ with โ ๐ด, โ ๐ต, and โ ๐ถ and the sides opposite them ๐, ๐, and ๐, the law of sines states: sin โ ๐ด sin โ ๐ต sin โ ๐ถ = = ๐ ๐ ๐ Restate the law of sines in your own words with a partner. ๏ง Show students that the law of sines holds true by calculating the ratios using the following triangles: This may be difficult for students to articulate generally, without reference to a specific angle. State it for students if they are unable to state it generally. ๏บ ๏ง The ratio of the sine of an angle in a triangle to the side opposite the angle is the same for each angle in the triangle. ฬ ฬ ฬ ฬ . What is sin โ ๐ถ? Consider โณ ๐ด๐ต๐ถ with an altitude drawn from ๐ต to ๐ด๐ถ ๏บ โ ๐ ๏ง Therefore, โ = ๐ sin โ ๐ถ. ๏ง What is sin ๐ด? ๏บ sin โ ๐ด = โ ๐ ๏ง Therefore, โ = ๐ sin โ ๐ด. ๏ง What can we conclude so far? ๏บ MP.3 sin โ ๐ถ = Since โ = ๐ sin โ ๐ถ and โ = ๐ sin โ ๐ด, then ๐ sin โ ๐ถ = ๐ sin โ ๐ด. sin โ ๐ด sin โ ๐ถ ๏ง With a little algebraic manipulation, we can rewrite ๐ sin โ ๐ถ = ๐ sin โ ๐ด as ๏ง We have partially shown why the law of sines is true. What do we need to show in order to complete the proof, and how can we go about determining this? ๐ = ๐ . Allow students a few moments to try and develop this argument independently. Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 478 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY ๏บ We need to show that sin โ ๐ต ๐ is equal to sin โ ๐ด ๐ and to sin โ ๐ถ ๐ . If we draw a different altitude, we can achieve this drawing: โ ๐ ๏บ An altitude from ๐ด gives us sin โ ๐ต = or โ = ๐ sin โ ๐ต. ๏บ Also, sin โ ๐ถ = ๏บ Since sin โ ๐ด ๐ = sin โ ๐ต sin โ ๐ถ โ or โ = ๐ sin โ ๐ถ. Therefore, ๐ sin โ ๐ต = ๐ sin โ ๐ถ or = . ๐ ๐ ๐ sin โ ๐ถ sin โ ๐ต sin โ ๐ถ sin โ ๐ด sin โ ๐ต sin โ ๐ถ ๐ As soon as it has been established that and sin โ ๐ต ๐ ๐ = = sin โ ๐ถ ๐ ๐ , then ๐ = ๐ = ๐ . , the proof is really done, as the angles selected are arbitrary and, therefore, apply to any angle within the triangle. This can be explained for students if they are ready for the explanation. Example 1 (4 minutes) Students apply the law of sines to determine unknown measurements within a triangle. Example 1 A surveyor needs to determine the distance between two points ๐จ and ๐ฉ that lie on opposite banks of a river. A point ๐ช is chosen ๐๐๐ meters from point ๐จ, on the same side of the river as ๐จ. The measures of โ ๐ฉ๐จ๐ช and โ ๐จ๐ช๐ฉ are ๐๐° and ๐๐°, respectively. Approximate the distance from ๐จ to ๐ฉ to the nearest meter. Allow students a few moments to begin the problem before assisting them. Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 479 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY ๏ง What measurement can we add to the diagram based on the provided information? ๏บ ๏ง Use the law of sines to set up all possible ratios applicable to the diagram. ๏บ ๏ง sin 41 ๐ = sin 84 160 = sin 55 ๐ Which ratios will be relevant to determining the distance from ๐ด to ๐ต? ๏บ ๏ง The measurement of โ ๐ต must be 84° by the triangle sum theorem. sin 84 160 = sin 55 ๐ Solve for ๐. ๏บ ๐= 160 sin 55 sin 84 ๐ = 132 The distance from ๐ด to ๐ต is 132 m. Exercises 1โ2 (6 minutes) Depending on the time available, consider having students move directly to Exercise 2. Exercises 1โ2 1. In โณ ๐จ๐ฉ๐ช, ๐โ ๐จ = ๐๐°, ๐ = ๐๐, and ๐ = ๐๐. Find ๐ฌ๐ข๐งโ ๐ฉ. Include a diagram in your answer. ๐ฌ๐ข๐ง๐๐ ๐ฌ๐ข๐งโ ๐ฉ = ๐๐ ๐๐ ๐ ๐ฌ๐ข๐งโ ๐ฉ = ๐๐ 2. A car is moving toward a tunnel carved out of the base of a hill. As the accompanying diagram shows, the top of the hill, ๐ฏ, is sighted from two locations, ๐จ and ๐ฉ. The distance between ๐จ and ๐ฉ is ๐๐๐ ๐๐ญ. What is the height, ๐, of the hill to the nearest foot? Let ๐ represent ๐ฉ๐ฏ, in feet. Applying the law of sines, ๐ฌ๐ข๐ง ๐๐ ๐ฌ๐ข๐ง ๐๐ = ๐๐๐ ๐ ๐๐๐ ๐ฌ๐ข๐ง ๐๐ ๐= ๐ฌ๐ข๐ง ๐๐ ๐๐๐ ๐= ๐ฌ๐ข๐ง ๐๐ ๐ โ ๐๐๐. ๐๐ Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 480 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY ฬ ฬ ฬ ฬ ฬ ๐ฉ๐ฏ is the hypotenuse of a ๐๐โ๐๐โ๐๐ triangle whose sides are in the ratio ๐: ๐: โ๐, or ๐: ๐: ๐โ๐. ๐โ๐ = ๐ ๐๐๐ ๐โ๐ = ๐ฌ๐ข๐ง ๐๐ ๐๐๐ ๐= ๐ฌ๐ข๐ง ๐๐ โ โ๐ ๐ โ ๐๐๐ The height of the hill is approximately ๐๐๐ feet. Discussion (10 minutes) Lead students through an explanation of the law of cosines for acute triangles. If more time is needed to cover the following Discussion, skip Exercise 1 as suggested earlier to allow for more time here. ๏ง The next fact we will examine is the law of cosines. ๏ง LAW OF COSINES: For an acute triangle โณ ๐ด๐ต๐ถ with โ ๐ด, โ ๐ต, and โ ๐ถ and the sides opposite them ๐, ๐, and ๐, the law of cosines states ๐ 2 = ๐2 + ๐ 2 โ 2๐๐ cos โ ๐ถ. ๏ง State the law of cosines in your own words. ๏บ The square of one side of the triangle is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the angle between them. The objective of being able to state the law in words is to move the focus away from specific letters and generalize the formula for any situation. ๏ง The law of cosines is a generalization of the Pythagorean theorem, which can only be used with right triangles. ๏ง Substitute 90° for ๐โ ๐ถ into the law of cosines formula, and observe the result. ๏บ ๏ง What is the value of cos 90? What happens to the equation? ๏บ ๏ง Scaffolding: ๐ 2 = ๐2 + ๐ 2 โ 2๐๐ cos 90 2 2 2 cos 90 = 0; the equation simplifies to ๐ = ๐ + ๐ . This explains why the law of cosines is a generalization of the Pythagorean theorem. We use the law of cosines for acute triangles in order to determine the side length of the third side of a triangle, provided two side lengths and the included angle measure. Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 For students ready for a challenge, instead of asking them to substitute 90° for โ ๐ถ, ask them what case results with the law of cosines being reduced to the Pythagorean theorem. 481 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY ๏ง We return to โณ ๐ด๐ต๐ถ from Example 1, with an altitude drawn from ๐ต to ฬ ฬ ฬ ฬ ๐ด๐ถ , but this time the point where the ฬ ฬ ฬ ฬ , point ๐ท, divides ๐ด๐ถ ฬ ฬ ฬ ฬ into lengths ๐ and ๐. altitude meets ๐ด๐ถ ๏ง Express ๐ and โ using trigonometry with respect to โ ๐ถ. ๏ง ๏บ โ = ๐ sinโ ๐ถ ๏บ ๐ = ๐ cosโ ๐ถ Now we turn to right triangle โณ ๐ด๐ต๐ท. What length relationship can be concluded between the sides of the triangle? ๏บ ๏ง Substitute the trigonometric expressions for ๐ and โ into this statement. Notice you will need length ๐. ๏บ ๏ง By the Pythagorean theorem, the length relationship in โณ ๐ด๐ต๐ท is ๐ ๐ = ๐ ๐ + โ๐ . Then, the statement becomes ๐ ๐ = (๐ โ ๐ cosโ ๐ถ)๐ + (๐ sinโ ๐ถ)2. Simplify this statement as much as possible. ๏บ The statement becomes: ๐ 2 = ๐ 2 โ 2๐๐cosโ ๐ถ + ๐2 cos 2 โ ๐ถ + ๐2 sin2 โ ๐ถ ๐ 2 = ๐ 2 โ 2๐๐cosโ ๐ถ + ๐2 ((cos โ ๐ถ)2 + (sin โ ๐ถ)2 ) ๐ 2 = ๐ 2 โ 2๐๐cosโ ๐ถ + ๐2 (1) ๐ 2 = ๐2 + ๐ 2 โ 2๐๐cosโ ๐ถ ๏ง Notice that the right-hand side is composed of two side lengths and the cosine of the included angle. This will remain true if the labeling of the triangle is rearranged. ๏ง What are all possible arrangements of the law of cosines for โณ ๐ด๐ต๐ถ? ๏บ ๐2 = ๐ 2 + ๐ 2 โ 2(๐๐)cosโ ๐ด ๏บ b2 = a2 + c 2 โ 2(๐๐)cosโ ๐ต Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 482 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Example 2 (4 minutes) Example 2 Our friend the surveyor from Example 1 is doing some further work. He has already found the distance between points ๐จ and ๐ฉ (from Example 1). Now he wants to locate a point ๐ซ that is equidistant from both ๐จ and ๐ฉ and on the same side of the river as ๐จ. He has his assistant mark the point ๐ซ so that โ ๐จ๐ฉ๐ซ and โ ๐ฉ๐จ๐ซ both measure ๐๐°. What is the distance between ๐ซ and ๐จ to the nearest meter? ๏ง What do you notice about โณ ๐ด๐ต๐ท right away? ๏บ ๏ง โณ ๐ด๐ต๐ท must be an isosceles triangle since it has two angles of equal measure. We must keep this in mind going forward. Add all relevant labels to the diagram. Students should add the distance of 132 m between ๐ด and ๐ต and add the label of ๐ and ๐ to the appropriate sides. ๏ง Set up an equation using the law of cosines. Remember, we are trying to find the distance between ๐ท and ๐ด or, as we have labeled it, ๐. ๏บ ๏ง ๐ 2 = 1322 + ๐2 โ 2(132)(๐) cos 75 Recall that this is an isosceles triangle; we know that ๐ = ๐. To reduce variables, we will substitute ๐ for ๐. Rewrite the equation, and solve for ๐. ๏บ Sample solution: ๐ 2 = 1322 + (๐)2 โ 2(132)(๐) cos 75 ๐ 2 = 1322 + (๐)2 โ 264(๐) cos 75 0 = 1322 โ 264(๐) cos 75 264(๐) cos 75 = 1322 ๐= 1322 264 cos 75 ๐ โ 255 m Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 483 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Exercise 3 (2 minutes) Exercise 3 Parallelogram ๐จ๐ฉ๐ช๐ซ has sides of lengths ๐๐ ๐ฆ๐ฆ and ๐๐ ๐ฆ๐ฆ, and one of the angles has a measure of ๐๐๐°. Approximate the length of diagonal ฬ ฬ ฬ ฬ ๐จ๐ช to the nearest millimeter. 3. In parallelogram ๐จ๐ฉ๐ช๐ซ, ๐โ ๐ช = ๐๐๐°; therefore, ๐โ ๐ซ = ๐๐°. Let ๐ represent the length of ฬ ฬ ฬ ฬ ๐จ๐ช. ๐ ๐ = ๐๐๐ + ๐๐๐ โ ๐(๐๐)(๐๐) ๐๐จ๐ฌ ๐๐ ๐ = ๐๐ ฬ ฬ ฬ ฬ is ๐๐ millimeters. The length of ๐จ๐ช Closing (1 minute) Ask students to summarize the key points of the lesson. Additionally, consider asking students to answer the following questions independently in writing, to a partner, or to the whole class. ๏ง In what kinds of cases are we applying the laws of sines and cosines? ๏บ ๏ง We apply the laws of sines and cosines when we do not have right triangles to work with. We used the laws of sines and cosines for acute triangles. State the law of sines. State the law of cosines. ๏บ For an acute triangle โณ ๐ด๐ต๐ถ with โ ๐ด, โ ๐ต, and โ ๐ถ and the sides opposite them ๐, ๐, and ๐, - The law of cosines states: ๐ 2 = ๐2 + ๐ 2 โ 2๐๐ cos โ ๐ถ. - The law of sines states: sin โ ๐ด ๐ = sin โ ๐ต ๐ = sin โ ๐ถ ๐ . Exit Ticket (5 minutes) Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 484 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Name Date Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle 1. Use the law of sines to find lengths ๐ and ๐ in the triangle below. Round answers to the nearest tenth as necessary. 2. Given โณ ๐ท๐ธ๐น, use the law of cosines to find the length of the side marked ๐ to the nearest tenth. Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 485 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Exit Ticket Sample Solutions Use the law of sines to find lengths ๐ and ๐ in the triangle below. Round answers to the nearest tenth as necessary. ๐โ ๐ช = ๐๐° ๐ฌ๐ข๐ง ๐จ ๐ฌ๐ข๐ง ๐ฉ ๐ฌ๐ข๐ง ๐ช = = ๐ ๐ ๐ ๐ฌ๐ข๐ง ๐๐ ๐ฌ๐ข๐ง ๐๐ ๐ฌ๐ข๐ง ๐๐ = = ๐๐ ๐ ๐ ๐= ๐๐(๐ฌ๐ข๐ง ๐๐) โ ๐๐. ๐ ๐ฌ๐ข๐ง ๐๐ ๐= ๐๐(๐ฌ๐ข๐ง ๐๐) โ ๐๐. ๐ ๐ฌ๐ข๐ง ๐๐ Given โณ ๐ซ๐ฌ๐ญ, use the law of cosines to find the length of the side marked ๐ to the nearest tenth. ๐ ๐ = ๐๐ + ๐๐ โ ๐(๐)(๐) (๐๐จ๐ฌ ๐๐) ๐ ๐ = ๐๐ + ๐๐ โ ๐๐๐(๐๐จ๐ฌ ๐๐) ๐ ๐ = ๐๐๐ โ ๐๐๐ (๐๐จ๐ฌ ๐๐) ๐ = โ๐๐๐ โ ๐๐๐(๐๐จ๐ฌ ๐๐) ๐ โ ๐. ๐ Problem Set Sample Solutions 1. Given โณ ๐จ๐ฉ๐ช, ๐จ๐ฉ = ๐๐, ๐โ ๐จ = ๐๐. ๐°, and ๐โ ๐ช = ๐๐. ๐°, calculate the measure of angle ๐ฉ to the nearest tenth of a degree, and use the law of sines to find the lengths of ฬ ฬ ฬ ฬ ๐จ๐ช and ฬ ฬ ฬ ฬ ๐ฉ๐ช to the nearest tenth. By the angle sum of a triangle, ๐โ ๐ฉ = ๐๐. ๐°. ๐ฌ๐ข๐ง ๐จ ๐ฌ๐ข๐ง ๐ฉ ๐ฌ๐ข๐ง ๐ช = = ๐ ๐ ๐ ๐ฌ๐ข๐ง ๐๐. ๐ ๐ฌ๐ข๐ง ๐๐. ๐ ๐ฌ๐ข๐ง ๐๐. ๐ = = ๐ ๐ ๐๐ ๐= ๐๐ ๐ฌ๐ข๐ง ๐๐. ๐ โ ๐๐. ๐ ๐ฌ๐ข๐ง ๐๐. ๐ ๐= ๐๐ ๐ฌ๐ข๐ง ๐๐. ๐ โ ๐๐. ๐ ๐ฌ๐ข๐ง ๐๐. ๐ Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 486 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Calculate the area of โณ ๐จ๐ฉ๐ช to the nearest square unit. ๐ ๐๐ ๐ฌ๐ข๐ง ๐จ ๐ ๐ ๐๐ซ๐๐ = (๐๐)(๐๐) ๐ฌ๐ข๐ง ๐๐. ๐ ๐ ๐๐ซ๐๐ = ๐๐ซ๐๐ = ๐๐ ๐ฌ๐ข๐ง ๐๐. ๐ ๐๐ซ๐๐ โ ๐๐ 2. Given โณ ๐ซ๐ฌ๐ญ, ๐โ ๐ญ = ๐๐°, and ๐ฌ๐ญ = ๐๐, calculate the measure of โ ๐ฌ, and use the law of sines to find the lengths ฬ ฬ ฬ ฬ and ๐ซ๐ฌ ฬ ฬ ฬ ฬ to the nearest hundredth. of ๐ซ๐ญ By the angle sum of a triangle, ๐โ ๐ฌ = ๐๐°. ๐ฌ๐ข๐ง ๐ซ ๐ฌ๐ข๐ง ๐ฌ ๐ฌ๐ข๐ง ๐ญ = = ๐ ๐ ๐ ๐ฌ๐ข๐ง ๐๐ ๐ฌ๐ข๐ง ๐๐ = ๐๐ ๐ ๐ฌ๐ข๐ง ๐๐ ๐ฌ๐ข๐ง ๐๐ = ๐๐ ๐ ๐๐ ๐ฌ๐ข๐ง ๐๐ ๐= ๐ฌ๐ข๐ง ๐๐ ๐= ๐๐ ๐ฌ๐ข๐ง ๐๐ ๐ฌ๐ข๐ง ๐๐ ๐ โ ๐. ๐๐ ๐ โ ๐๐. ๐๐ 3. Does the law of sines apply to a right triangle? Based on โณ ๐จ๐ฉ๐ช, the following ratios were set up according to the law of sines. ๐ฌ๐ข๐ง โ ๐จ ๐ฌ๐ข๐ง โ ๐ฉ ๐ฌ๐ข๐ง ๐๐ = = ๐ ๐ ๐ Fill in the partially completed work below: ๐ฌ๐ข๐ง โ ๐จ ๐ฌ๐ข๐ง ๐๐ = ๐ ๐ ๐ฌ๐ข๐ง โ ๐ฉ ๐ฌ๐ข๐ง ๐๐ = ๐ ๐ ๐ฌ๐ข๐ง โ ๐จ = ๐ ๐ ๐ ๐ฌ๐ข๐ง โ ๐ฉ = ๐ ๐ฌ๐ข๐ง โ ๐จ = ๐ ๐ ๐ฌ๐ข๐ง โ ๐ฉ = ๐ ๐ ๐ ๐ What conclusions can we draw? The law of sines does apply to a right triangle. We get the formulas that are equivalent to ๐ฌ๐ข๐ง โ ๐จ = ๐จ๐ฉ๐ฉ ๐ฌ๐ข๐ง โ ๐ฉ = , where ๐จ and ๐ฉ are the measures of the acute angles of the right triangle. ๐ก๐ฒ๐ฉ Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 ๐จ๐ฉ๐ฉ and ๐ก๐ฒ๐ฉ 487 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 4. Given quadrilateral ๐ฎ๐ฏ๐ฒ๐ฑ, ๐โ ๐ฏ = ๐๐°, mโ ๐ฏ๐ฒ๐ฎ = ๐๐°, ๐โ ๐ฒ๐ฎ๐ฑ = ๐๐°, โ ๐ฑ is a right angle, and ๐ฎ๐ฏ = ๐ ๐ข๐ง., use ฬ ฬ ฬ ฬ ฬ , and then find the lengths of ๐ฎ๐ฑ ฬ ฬ ฬ and ฬ ฬ ฬ ฬ the law of sines to find the length of ๐ฎ๐ฒ ๐ฑ๐ฒ to the nearest tenth of an inch. By the angle sum of a triangle, ๐โ ๐ฏ๐ฎ๐ฒ = ๐๐°; therefore, โณ ๐ฎ๐ฏ๐ฒ is an isosceles triangle since its base โ โs have equal measure. ๐ฌ๐ข๐ง ๐๐ ๐ฌ๐ข๐ง ๐๐ = ๐ ๐ ๐ ๐ฌ๐ข๐ง ๐๐ ๐= ๐ฌ๐ข๐ง ๐๐ ๐ โ ๐. ๐ ๐ = ๐ ๐๐จ๐ฌ ๐๐ โ ๐. ๐ ๐ = ๐ ๐ฌ๐ข๐ง ๐๐ โ ๐. ๐ 5. ฬ ฬ ฬ ฬ ฬ to the Given triangle ๐ณ๐ด๐ต, ๐ณ๐ด = ๐๐, ๐ณ๐ต = ๐๐, and ๐โ ๐ณ = ๐๐°, use the law of cosines to find the length of ๐ด๐ต nearest tenth. ๐๐ = ๐๐๐ + ๐๐๐ โ ๐(๐๐)(๐๐) ๐๐จ๐ฌ ๐๐ ๐๐ = ๐๐๐ + ๐๐๐ โ ๐๐๐ ๐๐จ๐ฌ ๐๐ ๐๐ = ๐๐๐ โ ๐๐๐ ๐๐จ๐ฌ ๐๐ ๐ = โ๐๐๐ โ ๐๐๐ ๐๐จ๐ฌ ๐๐ ๐ โ ๐. ๐ ๐ด๐ต = ๐. ๐ ฬ ฬ ฬ ฬ ฬ is approximately ๐. ๐ units. The length of ๐ด๐ต 6. Given triangle ๐จ๐ฉ๐ช, ๐จ๐ช = ๐, ๐จ๐ฉ = ๐, and ๐โ ๐จ = ๐๐°, draw a diagram of triangle ๐จ๐ฉ๐ช, and use the law of cosines ฬ ฬ ฬ ฬ . to find the length of ๐ฉ๐ช ๐๐ = ๐๐ + ๐๐ โ ๐(๐)(๐)(๐๐จ๐ฌ ๐๐) ๐๐ = ๐๐ + ๐๐ โ ๐๐(๐๐จ๐ฌ ๐๐) ๐๐ = ๐๐๐ โ ๐๐ ๐๐จ๐ฌ ๐๐ ๐ = โ๐๐๐ โ ๐๐ ๐๐จ๐ฌ ๐๐ ๐ โ ๐. ๐ ฬ ฬ ฬ ฬ is approximately ๐. ๐ units. The length of ๐ฉ๐ช Calculate the area of triangle ๐จ๐ฉ๐ช. ๐ ๐๐(๐ฌ๐ข๐ง ๐จ) ๐ ๐ ๐๐ซ๐๐ = (๐)(๐)(๐ฌ๐ข๐ง ๐๐) ๐ ๐๐ซ๐๐ = ๐๐ซ๐๐ = ๐๐. ๐(๐ฌ๐ข๐ง ๐๐) ๐๐ซ๐๐ โ ๐๐. ๐ The area of triangle ๐จ๐ฉ๐ช is approximately ๐๐. ๐ square units. Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 488 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.