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Lesson 31 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Lesson 31: Using Trigonometry to Determine Area Student Outcomes ๏ง Students prove that the area of a triangle is one-half times the product of two side lengths times the sine of the included angle and solve problems using this formula. ๏ง Students find the area of an isosceles triangle given the base length and the measure of one angle. Lesson Notes Students discover how trigonometric ratios can help with area calculations in cases where the measurement of the height is not provided. In order to determine the height in these cases, students must draw an altitude to create right triangles within the larger triangle. With the creation of the right triangles, students can then set up the necessary trigonometric ratios to express the height of the triangle (G.SRT.8). Students carefully connect the meanings of formulas 1 to the diagrams they represent (MP.2 and 7). In addition, this lesson introduces the formula Area = ๐๐ sin ๐ถ as 2 described by G-SRT.D.9. Classwork Opening Exercise (5 minutes) Opening Exercise Three triangles are presented below. Determine the areas for each triangle, if possible. If it is not possible to find the area with the provided information, describe what is needed in order to determine the area. The area of โณ ๐จ๐ฉ๐ช is ๐ ๐ ๐ (๐)(๐๐) = ๐๐ square units, and the area of โณ ๐ซ๐ฌ๐ญ is (๐)(๐๐) = ๐๐. ๐ There is not enough information to find the height of โณ ๐ฎ๐ฏ๐ฐ and, therefore, the area of the triangle. Is there a way to find the missing information? Without further information, there is no way to calculate the area. Lesson 31: Date: Using Trigonometry to Determine Area 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 462 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 31 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Example 1 (13 minutes) ๏ง What if the third side length of the triangle were provided? Is it possible to determine the area of the triangle now? Example 1 Find the area of โณ ๐ฎ๐ฏ๐ฐ. Allow students the opportunity and the time to determine what they must find (the height) and how to locate it (one MP.1 option is to drop an altitude from vertex ๐ป to side ๐บ๐ผ). For students who are struggling, consider showing just the 3 altitude and allowing them to label the newly divided segment lengths and the height. ๏ง How can the height be calculated? ๏บ By applying the Pythagorean theorem to both of the created right triangles to find ๐ฅ, โ2 = 49 โ ๐ฅ 2 โ2 = 144 โ (15 โ ๐ฅ)2 49 โ ๐ฅ 2 = 144 โ (15 โ ๐ฅ)2 49 โ ๐ฅ 2 = 144 โ 225 + 30๐ฅ โ ๐ฅ 2 130 = 30๐ฅ 13 ๐ฅ= . 3 ๐ป๐ฝ = ๏บ 13 3 , ๐ผ๐ฝ = 32 3 The value of ๐ฅ can then be substituted into either of the expressions equivalent to โ2 to find โ. 13 2 ) 3 169 โ2 = 49 โ 9 4โ17 โ= 3 โ2 = 49 โ ( ๏ง What is the area of the triangle? 1 4โ17 ๐ด๐๐๐ = ( ) (15) ( ) 2 3 ๐ด๐๐๐ = 10โ17 Lesson 31: Date: Using Trigonometry to Determine Area 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 463 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 31 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Discussion (10 minutes) Scaffolding: Now consider โณ ๐ด๐ต๐ถ which is set up similarly to the triangle in Example 1: ๐ ๏ง Write an equation that describes the area of this triangle. ๏บ 1 ๏ง For students that are ready for a challenge, pose the unstructured question, โHow could you write a formula for area that uses trigonometry, if we didnโt know โ?โ ๏ง For students that are struggling, use a numerical example, such as the 6โ8 โ10 triangle in Lesson 24 to demonstrate how โ ๐an be found using sin ๐. ๐ด๐๐๐ = ๐โ 2 Write the left-hand column on the board, and elicit the anticipated student response on the right-hand side after writing the second line; then elicit the anticipated student response after writing the third line. ๏ง We will rewrite this equation. Describe what you see. ๐ด๐๐๐ = 1 ๐โ 2 1 ๐ ๐โ ( ) 2 ๐ 1 โ ๐ด๐๐๐ = ๐๐ ( ) 2 ๐ The statement is multiplied by 1. ๐ด๐๐๐ = MP.2 & MP.7 The last statement is rearranged, but the value remains the same. Create a discussion around the third line of the newly written formula. Modify โณ ๐ด๐ต๐ถ: Add in an arc mark at vertex ๐ถ, and label it ๐. ๏ง โ What do you notice about the structure of ? Can we think ๐ of this newly written area formula in a different way using trigonometry? ๏บ โ The value of is equivalent to sin ๐; the newly ๐ 1 written formula can be written as ๐๐๐๐ = ๐๐ sin ๐. 2 ๏ง 1 If the area can be expressed as ๐๐ sin ๐, which part of the 2 expression represents the height? ๏บ ๏ง โ = ๐ sin ๐. Compare the information provided to find the area of โณ ๐บ๐ป๐ผ in the Opening Exercise, how the information changed in Example 1, and then changed again in the triangle above, โณ ABC. ๏บ Only two side lengths were provided in the Opening Exercise, and the area could not be found because there was not enough information to determine the height. In Example 1, all three side lengths were provided, and then the area could be determined because the height could be determined by applying the Pythagorean theorem in two layers. In the most recent figure, we only needed two sides and the included angle to find the area. Lesson 31: Date: Using Trigonometry to Determine Area 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 464 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 31 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY If you had to determine the area of a triangle, and were given the option to have three side lengths of a triangle or two side lengths and the included measure, which set of measurements would you rather have? Why? The response to this question is a matter of opinion. Considering the amount of work needed to find the area when provided three side lengths, our guess is they will opt for the briefer, trigonometric solution! Example 3 (5 minutes) Example 3 A farmer is planning how to divide his land for planting next yearโs crops. A triangular plot of land is left with two known side lengths measuring ๐๐๐ ๐ฆ and ๐, ๐๐๐ ๐ฆ. What could the farmer do next in order to find the area of the plot? ๏ง With just two side lengths known of the plot of land, what are the farmerโs options to determine the area of his plot of land? ๏บ ๏ง He can either measure the third side length, apply the Pythagorean theorem to find the height of the triangle, and then calculate the area, or he can find the measure of the included angle between the known side lengths and use trigonometry to express the height of the triangle and then determine the area of the triangle. Suppose the included angle measure between the known side lengths is 30ห. What is the area of the plot of land? Sketch a diagram of the plot of land. 1 ๏บ ๐ด๐๐๐ = (1,700)(500) sin 30 ๏บ ๐ด๐๐๐ = 212,500 ๏บ The area of the plot of land is 212,500 square meters. 2 Exercise 1 (5 minutes) Exercise 1 1. A real estate developer and her surveyor are searching for their next piece of land to build on. They each examine a plot of land in the shape of โณ ๐จ๐ฉ๐ช. The real estate developer measures the length of ๐จ๐ฉ and ๐จ๐ช and finds them to both be approximately ๐, ๐๐๐ feet, and the included angle has a measure of approximately ๐๐ห. The surveyor measures the length of ๐จ๐ช and ๐ฉ๐ช and finds the lengths to be approximately ๐, ๐๐๐ feet and ๐, ๐๐๐ feet, respectively, and measures the angle between the two sides to be approximately ๐๐°. a. Draw a diagram that models the situation, labeling all lengths and angle measures. Lesson 31: Date: Using Trigonometry to Determine Area 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 465 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 31 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY b. The real estate developer and surveyor each calculate the area of the plot of land and both find roughly the same area. Show how each person calculated the area; round to the nearest hundred. Redraw the diagram with only the relevant labels for both the real estate agent and surveyor. ๐ (๐๐๐๐)(๐๐๐๐) ๐ฌ๐ข๐ง ๐๐ ๐ ๐จ โ ๐, ๐๐๐, ๐๐๐ ๐ (๐๐๐๐)(๐๐๐๐) ๐ฌ๐ข๐ง ๐๐ ๐ ๐จ โ ๐, ๐๐๐, ๐๐๐ ๐จ= ๐จ= The area is approximately ๐, ๐๐๐, ๐๐๐ square feet. c. The area is approximately ๐, ๐๐๐, ๐๐๐ square feet. What could possibly explain the difference between the real estate agent and surveyorโs calculated areas? The difference in the area of measurements can be accounted for by the approximations of the measurements taken, instead of exact measurements. Closing (2 minutes) Ask students to summarize the key points of the lesson. Additionally, consider asking students to respond to the following questions independently in writing, to a partner, or to the whole class. ๏ง For a triangle with side lengths ๐ and ๐ and included angle of measure ๐, when will we need to use the area 1 formula ๐ด๐๐๐ = ๐๐ sin ๐ ? 2 ๏บ ๏ง We will need it when we are determining the area of a triangle and are not provided a height. 1 1 2 2 Recall how to transform the equation ๐ด๐๐๐ = ๐โ to ๐ด๐๐๐ = ๐๐ sin ๐. Exit Ticket (5 minutes) Lesson 31: Date: Using Trigonometry to Determine Area 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 466 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 31 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Name Date Lesson 31: Using Trigonometry to Determine Area Exit Ticket 1. Given two sides of the triangle shown, having lengths of 3 and 7, and their included angle of 49°, find the area of the triangle to the nearest tenth. 2. In isosceles triangle ๐๐๐ , the base ๐๐ = 11, and the base angles have measures of 71.45°. Find the area of โณ ๐๐๐ . Lesson 31: Date: Using Trigonometry to Determine Area 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 467 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 31 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Exit Ticket Sample Solutions 1. Given two sides of the triangle shown, having lengths of ๐ and ๐, and their included angle of ๐๐°, find the area of the triangle to the nearest tenth. ๐ (๐)(๐)(๐ฌ๐ข๐ง ๐๐) ๐ ๐จ๐๐๐ = ๐๐. ๐(๐ฌ๐ข๐ง ๐๐) โ ๐. ๐ ๐จ๐๐๐ = The area of the triangle is approximately ๐. ๐ square units. 2. In isosceles triangle ๐ท๐ธ๐น, the base ๐ธ๐น = ๐๐, and the base angles have measures of ๐๐. ๐๐°. Find the area of โณ ๐ท๐ธ๐น to the nearest tenth. Drawing an altitude from ๐ท to midpoint ๐ด on ฬ ฬ ฬ ฬ ๐ธ๐น cuts the isosceles triangle into two right triangles with ๐ธ๐ด = ๐ด๐น = ๐. ๐. Using tangent solve the following: ๐ญ๐๐ง ๐๐. ๐๐ = ๐ท๐ด ๐. ๐ ๐ท๐ด = ๐. ๐(๐ญ๐๐ง ๐๐. ๐๐) ๐ ๐๐ ๐ ๐ ๐จ๐๐๐ = (๐๐)(๐. ๐(๐ญ๐๐ง ๐๐. ๐๐)) ๐ ๐จ๐๐๐ = ๐จ๐๐๐ = ๐๐. ๐๐(๐ญ๐๐ง ๐๐. ๐๐) โ ๐๐. ๐ The area of the isosceles triangle is approximately ๐๐. ๐ square units. Problem Set Sample Solutions Find the area of each triangle. Round each answer to the nearest tenth. 1. ๐ (๐๐)(๐) (๐ฌ๐ข๐ง ๐๐) ๐ ๐จ๐๐๐ = ๐๐(๐ฌ๐ข๐ง ๐๐) โ ๐๐. ๐ ๐จ๐๐๐ = The area of the triangle is approximately ๐๐. ๐ square units. 2. ๐ (๐)(๐๐)(๐ฌ๐ข๐ง ๐๐) ๐ ๐จ๐๐๐ = ๐๐(๐ฌ๐ข๐ง ๐๐) โ ๐. ๐ ๐จ๐๐๐ = The area of the triangle is approximately ๐. ๐ square units. Lesson 31: Date: Using Trigonometry to Determine Area 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 468 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 31 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 3. ๐ ๐ (๐) (๐ ) (๐ฌ๐ข๐ง ๐๐) ๐ ๐ ๐จ๐๐๐ = ๐๐(๐ฌ๐ข๐ง ๐๐) โ ๐๐. ๐ ๐จ๐๐๐ = The area of the triangle is approximately ๐๐. ๐ square units. 4. The included angle is ๐๐° by the angle sum of a triangle. ๐ (๐๐)(๐ + ๐โ๐) ๐ฌ๐ข๐ง ๐๐ ๐ โ๐ ๐จ๐๐๐ = ๐(๐ + ๐โ๐) ( ) ๐ ๐จ๐๐๐ = โ๐ ๐จ๐๐๐ = (๐๐ + ๐๐โ๐) ( ) ๐ ๐จ๐๐๐ = ๐๐โ๐ + ๐๐(๐) ๐จ๐๐๐ = ๐๐โ๐ + ๐๐ โ ๐๐. ๐ The area of the triangle is approximately ๐๐. ๐ square units. 5. In โณ ๐ซ๐ฌ๐ญ, ๐ฌ๐ญ = ๐๐, ๐ซ๐ญ = ๐๐, ๐๐ง๐ โ ๐ญ = ๐๐. Determine the area of the triangle. Round to the nearest tenth. ๐ ๐ ๐จ๐๐๐ = (๐๐)(๐๐)๐ฌ๐ข๐ง(๐๐) โ ๐๐๐. ๐ ๐ฎ๐ง๐ข๐ญ๐ฌ ๐ Lesson 31: Date: Using Trigonometry to Determine Area 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 469 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 31 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 6. A landscape designer is designing a flower garden for a triangular area that is bounded on two sides by the clientโs house and driveway. The length of the edges of the garden along the house and driveway are ๐๐ ๐๐ญ. and ๐ ๐๐ญ. respectively, and the edges come together at an angle of ๐๐°. Draw a diagram, and then find the area of the garden to the nearest square foot. The garden is in the shape of a triangle in which the lengths of two sides and the included angle have been provided. ๐จ๐๐๐(๐จ๐ฉ๐ช) = ๐ (๐ ๐๐ญ. )(๐๐ ๐๐ญ. ) ๐ฌ๐ข๐ง ๐๐ ๐ ๐จ๐๐๐(๐จ๐ฉ๐ช) = (๐๐ ๐ฌ๐ข๐ง ๐๐) ๐๐ญ.๐ ๐จ๐๐๐(๐จ๐ฉ๐ช) โ ๐๐ ๐๐ญ.๐ 7. A right rectangular pyramid has a square base with sides of length ๐. Each lateral face of the pyramid is an isosceles triangle. The angle on each lateral face between the base of the triangle and the adjacent edge is ๐๐°. Find the surface area of the pyramid to the nearest tenth. Using tangent, the altitude of the triangle to the base of length ๐ is equal to ๐. ๐ ๐ญ๐๐ง ๐๐. ๐ ๐๐ ๐ ๐ ๐จ๐๐๐ = (๐)(๐. ๐ ๐ฌ๐ข๐ง ๐๐) ๐ ๐จ๐๐๐ = ๐จ๐๐๐ = ๐. ๐๐(๐ฌ๐ข๐ง ๐๐) The total surface area of the pyramid is the sum of the four lateral faces and the area of the square base: ๐บ๐จ = ๐(๐. ๐๐(๐ฌ๐ข๐ง ๐๐)) + ๐๐ ๐บ๐จ = ๐๐ ๐ฌ๐ข๐ง ๐๐ + ๐๐ ๐บ๐จ โ ๐๐. ๐ The surface area of the right rectangular pyramid is approximately ๐๐. ๐ square units. Lesson 31: Date: Using Trigonometry to Determine Area 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 470 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 31 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 8. The Pentagon Building in Washington D.C. is built in the shape of a regular pentagon. Each side of the pentagon measures ๐๐๐ ๐๐ญ. in length. The building has a pentagonal courtyard with the same center. Each wall of the center courtyard has a length of ๐๐๐ ๐๐ญ. What is the approximate area of the roof of the Pentagon Building? Let ๐จ๐ represent the area within the outer perimeter of the Pentagon Building in square feet. ๐จ๐ = ๐๐๐ ๐๐๐ ๐ ๐ญ๐๐ง ( ) ๐ ๐ โ (๐๐๐)๐ ๐๐๐ ๐ ๐ญ๐๐ง ( ) ๐ ๐, ๐๐๐, ๐๐๐ ๐จ๐ = โ ๐, ๐๐๐, ๐๐๐ ๐ ๐ญ๐๐ง(๐๐) ๐จ๐ = The area within the outer perimeter of the Pentagon Building is approximately ๐, ๐๐๐, ๐๐๐ ๐๐ญ ๐. Let ๐จ๐ represent the area within the perimeter of the courtyard of the Pentagon Building in square feet. ๐จ๐ = Let ๐จ๐ป represent the total area of the roof of the Pentagon Building in square feet. ๐๐๐ ๐ ๐ญ๐๐ง ๐๐ ๐จ๐ป = ๐จ๐ โ ๐จ๐ ๐จ๐ป = ๐(๐๐๐)๐ ๐จ๐ = ๐ ๐ญ๐๐ง ๐๐ ๐๐๐๐๐๐ ๐จ๐ = ๐ ๐ญ๐๐ง ๐๐ ๐๐๐๐๐๐ ๐จ๐ = โ ๐๐๐, ๐๐๐ ๐ญ๐๐ง ๐๐ ๐๐๐๐๐๐๐ ๐๐๐๐๐๐ โ ๐ ๐ญ๐๐ง(๐๐) ๐ญ๐๐ง ๐๐ ๐๐๐๐๐๐๐ ๐๐๐๐๐๐ โ ๐ ๐ญ๐๐ง ๐๐ ๐ ๐ญ๐๐ง ๐๐ ๐๐๐๐๐๐๐ ๐จ๐ป = โ ๐, ๐๐๐, ๐๐๐ ๐ ๐ญ๐๐ง ๐๐ ๐จ๐ป = The area of the roof of the Pentagon Building is approximately ๐, ๐๐๐, ๐๐๐ ๐๐ญ ๐. 9. A regular hexagon is inscribed in a circle with a radius of ๐. Find the perimeter and area of the hexagon. The regular hexagon can be divided into six equilateral triangular regions, with each side of the triangles having a length of ๐. To find the perimeter of the hexagon, solve the following: ๐ โ ๐ = ๐๐ , so the perimeter of the hexagon is ๐๐ units. To find the area of one equilateral triangle: ๐ ๐จ๐๐๐ = (๐)(๐) ๐ฌ๐ข๐ง ๐๐ ๐ ๐จ๐๐๐ = ๐๐ โ๐ ( ) ๐ ๐ ๐จ๐๐๐ = ๐๐โ๐ ๐ The area of the hexagon is six times the area of the equilateral triangle. ๐๐โ๐ ๐ป๐๐๐๐ ๐๐๐๐ = ๐ ( ) ๐ ๐ป๐๐๐๐ ๐จ๐๐๐ = ๐๐๐โ๐ โ ๐๐๐. ๐ ๐ The total area of the regular hexagon is approximately ๐๐๐. ๐ square units. Lesson 31: Date: Using Trigonometry to Determine Area 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 471 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 31 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY ๐ ๐ 10. In the figure below, โ ๐จ๐ฌ๐ฉ is acute. Show that ๐จ๐๐๐ (โณ ๐จ๐ฉ๐ช) = ๐จ๐ช โ ๐ฉ๐ฌ โ ๐ฌ๐ข๐ง โ ๐จ๐ฌ๐ฉ. Let ๐ฝ represent the degree measure of angle ๐จ๐ฌ๐ฉ, and let ๐ represent the altitude of โณ ๐จ๐ฉ๐ช (and โณ ๐จ๐ฉ๐ฌ). ๐จ๐๐๐(โณ ๐จ๐ฉ๐ช) = ๐ฌ๐ข๐ง ๐ฝ = ๐ โ ๐จ๐ช โ ๐ ๐ ๐ , which implies that ๐ = ๐ฉ๐ฌ โ ๐ฌ๐ข๐ง ๐ฝ. ๐ฉ๐ฌ Therefore, by substitution: ๐ ๐ ๐จ๐๐๐(โณ ๐จ๐ฉ๐ช) = ๐จ๐ช โ ๐ฉ๐ฌ โ ๐ฌ๐ข๐ง โ ๐จ๐ฌ๐ฉ. ฬ ฬ ฬ ฬ and ๐ฉ๐ซ ฬ ฬ ฬ ฬ ฬ . Show that 11. Let ๐จ๐ฉ๐ช๐ซ be a quadrilateral. Let ๐ be the measure of the acute angle formed by diagonals ๐จ๐ช ๐ ๐ ๐จ๐๐๐(๐จ๐ฉ๐ช๐ซ) = ๐จ๐ช โ ๐ฉ๐ซ โ ๐ฌ๐ข๐ง ๐. (Hint: Apply the result from Problem 10 to โณ ๐จ๐ฉ๐ช and โณ ๐จ๐ช๐ซ.) ฬ ฬ ฬ ฬ and ๐ฉ๐ซ ฬ ฬ ฬ ฬ ฬ be called point ๐ท. Let the intersection of ๐จ๐ช Using the results from Problem 10, solve the following: ๐ and ๐ ๐ ๐จ๐๐๐(โณ ๐จ๐ซ๐ช) = ๐จ๐ช โ ๐ท๐ซ โ ๐๐๐ ๐ ๐ ๐ ๐ ๐จ๐๐๐(๐จ๐ฉ๐ช๐ซ) = [ ๐จ๐ช โ ๐ฉ๐ท โ ๐๐๐ ๐] + [ ๐จ๐ช โ ๐ท๐ซ โ ๐๐๐ ๐] Area is additive; ๐ ๐ ๐ ๐จ๐๐๐(๐จ๐ฉ๐ช๐ซ) = [ ๐จ๐ช โ ๐๐๐ ๐] โ [๐ฉ๐ท + ๐ท๐ซ] Distributive property; ๐ ๐ ๐จ๐๐๐(๐จ๐ฉ๐ช๐ซ) = [ ๐จ๐ช โ ๐๐๐ ๐] โ [๐ฉ๐ซ] Distance is additive; ๐ ๐จ๐๐๐(โณ ๐จ๐ฉ๐ช) = ๐จ๐ช โ ๐ฉ๐ท โ ๐๐๐ ๐ And commutative addition gives us ๐จ๐๐๐(๐จ๐ฉ๐ช๐ซ) = Lesson 31: Date: ๐ โ ๐จ๐ช โ ๐ฉ๐ซ โ ๐ฌ๐ข๐ง ๐. ๐ Using Trigonometry to Determine Area 5/5/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org 472 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.