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1 Algebra II – Chapter 12 Day #2 Topic: The Law of Sines Standards/Goals: G.SRT.11./G.3.a.: I can use the law of sines to find the lengths and measures of angles in triangles in mathematical and real-world problems. Today, we want to learn how to look at some situations where a triangle DOES NOT have a right triangle inside of it and we must find its missing sides and/or angles. The Law of Sines: In any triangle, the ratio of the sinde of each angle to its opposite sides is constant. In particular, for ΔABC is labeled as shown below: 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩 𝐬𝐢𝐧 𝑪 = = 𝒂 𝒃 𝒄 Let’s now try some example problems that require us to utilize the Law of Sines: EXAMPLES: #1. In ΔRST, m<T = 54 degrees, m<S = 67 degrees, and ST = 35 mm. What is RS? #2. In ΔJKL, j = 2 in, k = 4.1 in, and m<K = 101. What is m<L? 2 #3. In ΔMNP, m<M = 25, m<N = 61 and MP = 3.8 m. What is MN? #4. Two wildlife spotters are 2 miles apart on an east-west line. The spotter in the eastern spot sees a bear 62 degrees north of west, and the other spotter sees the bear 48 degrees north of east. How far is the bear from each spotter? #5. A contractor is checking the height of the frame for a building his company is constructing. He stands at some distance from the building and measures the angle of elevation to be 73 degrees. He then moves back 10 feet and measures the angle to be 68 degrees. How tall is the building? #6. A landscaper sights the top of a tree at a 68 degree angle. She then moves an additional 70 feet directly away from the tree and sights the top at a 43 degree angle. How tall is the tree to the nearest tenth of a foot? 3 HOMEWORK – Chapter 12 Day #2 Name _______________________________________ Date ________ In ∆ABC, C is a right angle. Find the remaining sides and angles. Round your answer to the nearest tenth. 1. mA = 29°, b = 8 2. a = 7, c = 9 3. mB = 52°, b = 10 4. a = 2, b = 4 5. mA = 37°, c = 12 Use the Law of Sines. Find the measure x to the nearest tenth. 7. 9. 8. 10. 6. b = 5, c = 8 4 11. In ΔGHJ, mJ = 39, h = 36 cm, and j = 42 cm. Find mH. 12. In ΔMNP, mP = 33, m = 54 ft, and p = 63 ft. Find mM. 13. A hot-air balloon is observed from two points, A and B, on the ground 800 ft apart as shown in the diagram. The angle of elevation of the balloon is 65° from point A and 37° from point B. Find the distance from point A to the balloon. FLASHBACK: 14. In ΔABC, find each value as a fraction and as a decimal. Round DECIMALS to the nearest ten-thousandth. a. cos A b. csc A c. tan B d. sec B e. cot A f. csc B