M2 PS 32 Solutions
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Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Exit Ticket Sample Solutions 1. Use the law of sines to find lengths 𝒃 and 𝒄 in the triangle below. Round answers to the nearest tenth as necessary. ∠𝑪 = 𝟖𝟐° 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩 𝐬𝐢𝐧 𝑪 = = 𝒂 𝒃 𝒄 𝐬𝐢𝐧 𝟒𝟐 𝐬𝐢𝐧 𝟓𝟔 𝐬𝐢𝐧 𝟖𝟐 = = 𝟏𝟖 𝒃 𝒄 2. 𝒃= 𝟏𝟖(𝐬𝐢𝐧 𝟓𝟔) ≈ 𝟐𝟐. 𝟑 𝐬𝐢𝐧 𝟒𝟐 𝒄= 𝟏𝟖(𝐬𝐢𝐧 𝟖𝟐) ≈ 𝟐𝟔. 𝟔 𝐬𝐢𝐧 𝟒𝟐 Given △ 𝑫𝑬𝑭, use the law of cosines to find the length of the side marked 𝒅 to the nearest tenth. 𝒅𝟐 = 𝟔𝟐 + 𝟗𝟐 − 𝟐(𝟔)(𝟗) (𝐜𝐨𝐬 𝟔𝟓) 𝒅𝟐 = 𝟑𝟔 + 𝟖𝟏 − 𝟏𝟎𝟖(𝐜𝐨𝐬 𝟔𝟓) 𝒅𝟐 = 𝟏𝟏𝟕 − 𝟏𝟎𝟖 (𝐜𝐨𝐬 𝟔𝟓) 𝒅 = √𝟏𝟏𝟕 − 𝟏𝟎𝟖(𝐜𝐨𝐬 𝟔𝟓) 𝒅 ≈ 𝟖. 𝟒 Problem Set Sample Solutions 1. Given △ 𝑨𝑩𝑪, 𝑨𝑩 = 𝟏𝟒, ∠𝑨 = 𝟓𝟕. 𝟐°, and ∠𝑪 = 𝟕𝟖. 𝟒°, calculate the measure of angle 𝑩 to the nearest tenth of a degree, and use the law of sines to find the lengths of 𝑨𝑪 and 𝑩𝑪 to the nearest tenth. By the angle sum of a triangle, ∠𝑩 = 𝟒𝟒. 𝟒°. 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩 𝐬𝐢𝐧 𝑪 = = 𝒂 𝒃 𝒄 𝐬𝐢𝐧 𝟓𝟕. 𝟐 𝐬𝐢𝐧 𝟒𝟒. 𝟒 𝐬𝐢𝐧 𝟕𝟖. 𝟒 = = 𝒂 𝒃 𝟏𝟒 𝒂= 𝟏𝟒 𝐬𝐢𝐧 𝟓𝟕. 𝟐 ≈ 𝟏𝟐. 𝟎 𝐬𝐢𝐧 𝟕𝟖. 𝟒 𝒃= 𝟏𝟒 𝐬𝐢𝐧 𝟒𝟒. 𝟒 ≈ 𝟏𝟎. 𝟎 𝐬𝐢𝐧 𝟕𝟖. 𝟒 Lesson 32: Date: Using Trigonometry to Find Side Lengths of an Acute Triangle 10/28/14 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 482 Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Calculate the area of △ 𝑨𝑩𝑪 to the nearest square unit. 𝟏 𝒃𝒄 𝐬𝐢𝐧 𝑨 𝟐 𝟏 𝑨𝒓𝒆𝒂 = (𝟏𝟎)(𝟏𝟒) 𝐬𝐢𝐧 𝟓𝟕. 𝟐 𝟐 𝑨𝒓𝒆𝒂 = 𝑨𝒓𝒆𝒂 = 𝟕𝟎 𝐬𝐢𝐧 𝟓𝟕. 𝟐 ≈ 𝟓𝟗 2. Given △ 𝑫𝑬𝑭, ∠𝑭 = 𝟑𝟗°, and 𝑬𝑭 = 𝟏𝟑, calculate the measure of ∠𝑬, and use the Law of Sines to find the lengths of ̅̅̅̅ 𝑫𝑭 and ̅̅̅̅ 𝑫𝑬 to the nearest hundredth. By the angle sum of a triangle, 𝒎∠𝑬 = 𝟓𝟓°. 𝐬𝐢𝐧 𝑫 𝐬𝐢𝐧 𝑬 𝐬𝐢𝐧 𝑭 = = 𝒅 𝒆 𝒇 𝐬𝐢𝐧 𝟖𝟔 𝐬𝐢𝐧 𝟓𝟓 = 𝟏𝟑 𝒆 𝟏𝟑 𝐬𝐢𝐧 𝟓𝟓 𝒆= ≈ 𝟏𝟎. 𝟔𝟕 𝐬𝐢𝐧 𝟖𝟔 𝐬𝐢𝐧 𝟖𝟔 𝐬𝐢𝐧 𝟑𝟗 = 𝟏𝟑 𝒇 𝒇= 3. 𝟏𝟑 𝐬𝐢𝐧 𝟑𝟗 ≈ 𝟖. 𝟐𝟎 𝐬𝐢𝐧 𝟖𝟔 Does the law of sines apply to a right triangle? Based on △ 𝑨𝑩𝑪, the following ratios were set up according to the law of sines. 𝐬𝐢𝐧 ∠𝑨 𝐬𝐢𝐧 ∠𝑩 𝐬𝐢𝐧 𝟗𝟎 = = 𝒂 𝒃 𝒄 Fill in the partially completed work below: 𝐬𝐢𝐧 ∠𝑨 𝐬𝐢𝐧 𝟗𝟎 = 𝒂 𝒄 𝐬𝐢𝐧 ∠𝑨 = 𝒂 𝟏 𝒄 𝐬𝐢𝐧 ∠𝑨 = 𝐚 𝒄 𝐬𝐢𝐧 ∠𝑩 𝐬𝐢𝐧 𝟗𝟎 = 𝒃 𝒄 𝐬𝐢𝐧 ∠𝑩 = 𝒃 𝐬𝐢𝐧 ∠𝑩 = 𝟏 𝒄 𝐛 𝒄 What conclusions can we draw? The law of sines does apply to a right triangle. We get the formulas that are equivalent to 𝐬𝐢𝐧 ∠𝑨 = 𝐬𝐢𝐧 ∠𝑩 = 𝒐𝒑𝒑 , where 𝑨 and 𝑩 are the measures of the acute angles of the right triangle. 𝒉𝒚𝒑 Lesson 32: Date: 𝒐𝒑𝒑 and 𝒉𝒚𝒑 Using Trigonometry to Find Side Lengths of an Acute Triangle 10/28/14 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 483 Lesson 32 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 4. Given quadrilateral 𝑮𝑯𝑲𝑱, ∠𝑯 = 𝟓𝟎°, ∠𝑯𝑲𝑮 = 𝟖𝟎°, ∠𝑲𝑮𝑱 = 𝟓𝟎°, ∠𝑱 is a right angle and 𝑮𝑯 = 𝟗 𝐢𝐧., use the Law ̅̅̅ and ̅̅̅̅ of Sines to find the length of 𝑮𝑲, and then find the lengths of 𝑮𝑱 𝑱𝑲 to the nearest tenth of an inch. By the angle sum of a triangle, ∠𝑯𝑮𝑲 = 𝟓𝟎°; therefore, △ 𝑮𝑯𝑲 is an isosceles triangle since its base ∠'s have equal measure. 𝐬𝐢𝐧 𝟓𝟎 𝐬𝐢𝐧 𝟖𝟎 = 𝒉 𝟗 𝟗 𝐬𝐢𝐧 𝟓𝟎 𝒉= ≈ 𝟕. 𝟎 𝐬𝐢𝐧 𝟖𝟎 𝒌 = 𝟕 𝒄𝒐𝒔 𝟓𝟎 ≈ 𝟒. 𝟓 𝒈 = 𝟕 𝒔𝒊𝒏 𝟓𝟎 ≈ 𝟓. 𝟒 5. ̅̅̅̅̅ to the Given triangle 𝑳𝑴𝑵, 𝑳𝑴 = 𝟏𝟎, 𝑳𝑵 = 𝟏𝟓, and ∠𝑳 = 𝟑𝟖°, use the Law of Cosines to find the length of 𝑴𝑵 nearest tenth. 𝒍𝟐 = 𝟏𝟎𝟐 + 𝟏𝟓𝟐 − 𝟐(𝟏𝟎)(𝟏𝟓) 𝐜𝐨𝐬 𝟑𝟖 𝒍𝟐 = 𝟏𝟎𝟎 + 𝟐𝟐𝟓 − 𝟑𝟎𝟎 𝐜𝐨𝐬 𝟑𝟖 𝒍𝟐 = 𝟑𝟐𝟓 − 𝟑𝟎𝟎 𝐜𝐨𝐬 𝟑𝟖 𝒍 = √𝟑𝟐𝟓 − 𝟑𝟎𝟎 𝐜𝐨𝐬 𝟑𝟖 𝒍 ≈ 𝟗. 𝟒 𝑴𝑵 = 𝟗. 𝟒 6. Given triangle 𝑨𝑩𝑪, 𝑨𝑪 = 𝟔, 𝑨𝑩 = 𝟖, and ∠𝑨 = 𝟕𝟖°. Draw a diagram of triangle 𝑨𝑩𝑪, and use the law of cosines to find the length of ̅̅̅̅ 𝑩𝑪. 𝒂𝟐 = 𝟔𝟐 + 𝟖𝟐 − 𝟐(𝟔)(𝟖)(𝐜𝐨𝐬 𝟕𝟖) 𝒂𝟐 = 𝟑𝟔 + 𝟔𝟒 − 𝟗𝟔(𝐜𝐨𝐬 𝟕𝟖) 𝒂𝟐 = 𝟏𝟎𝟎 − 𝟗𝟔 𝐜𝐨𝐬 𝟕𝟖 𝒂 = √𝟏𝟎𝟎 − 𝟗𝟔 𝐜𝐨𝐬 𝟕𝟖 𝒂 ≈ 𝟖. 𝟗 ̅̅̅̅ is approximately 𝟖. 𝟗. The length of 𝑩𝑪 Calculate the area of triangle 𝑨𝑩𝑪. 𝟏 𝒃𝒄(𝐬𝐢𝐧 𝑨) 𝟐 𝟏 𝑨𝒓𝒆𝒂 = (𝟔)(𝟖)(𝐬𝐢𝐧 𝟕𝟖) 𝟐 𝑨𝒓𝒆𝒂 = 𝑨𝒓𝒆𝒂 = 𝟐𝟑. 𝟓(𝐬𝐢𝐧 𝟕𝟖) 𝑨𝒓𝒆𝒂 ≈ 𝟐𝟑. 𝟓 The area of triangle 𝑨𝑩𝑪 is approximately 𝟐𝟑. 𝟓 square units. Lesson 32: Date: Using Trigonometry to Find Side Lengths of an Acute Triangle 10/28/14 © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 484
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