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6.5 The Law of Sines Copyright © Cengage Learning. All rights reserved. The Law of Sines The figure below describes the possible types of triangles that we will solve. (a) ASA or SAA (b) SSA (c) SAS (d) SSS Case 1 One side and two angles (ASA or AAS) Case 2 Two sides and the angle opposite one of those sides (SSA) Case 3 Two sides and the included angle (SAS) Case 4 Three sides (SSS) Cases 1 and 2 are solved by using the Law of Sines; Cases 3 and 4 require the Law of Cosines to initially start solving the triangles. 2 The Law of Sines The Law of Sines says that in any triangle the lengths of the sides are proportional to the sines of the corresponding opposite angles. 𝑎 𝑏 𝑐 = = Use to find a side. 𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝐵 𝑠𝑖𝑛𝐶 𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝐵 𝑠𝑖𝑛𝐶 Use to find an angle. = = 𝑎 𝑏 𝑐 We will follow the convention of labeling the angles of a triangle as A, B, and C and the lengths of the corresponding opposite sides as a, b, and c. 3 Example 1 – Tracking a Satellite (ASA) A satellite orbiting the earth passes directly overhead at observation stations in Phoenix and Los Angeles, 340 mi apart. At an instant when the satellite is between these two stations, its angle of elevation is simultaneously observed to be 60 at Phoenix and 75 at Los Angeles. How far is the satellite from Los Angeles? 4 Example 2 - AAS Given m∠A = 26°, m∠C = 56°, and c = 22, solve the triangle. 5 Example 3 - SAS Given m∠C = 37°, a = 8, b = 11, solve the triangle. 6 The Ambiguous Case 7 The Ambiguous Case In Example 1, there was only triangle possible from the information given. This is always true for ASA or AAS. But for SSA, there may be two triangles, one triangle, or no triangle with the given properties. For this reason, Case 2 is sometimes called the ambiguous case. 8 The Ambiguous Case To see why this is so, we show the possibilities when angle A and sides a and b are given. • In part (a) no solution is possible, since side a is too short to complete the triangle. • In part (b) the solution is a right triangle. • In part (c) two solutions are possible. • In part (d) there is a unique triangle with the given properties. (a) (b) (c) The ambiguous case (d) 9 Example 4 – SSA, the One-Solution Case Solve triangle ABC, where A = 45, a = 7 2, and b = 7. B is acute B is obtuse m∠B = _____________ m∠B = _____________ m∠C = _____________ m∠C = _____________ c = ________________ c = ________________ 10 Example 5 – SSA, the Two-Solution Case Solve triangle ABC if A = 43.1, a = 186.2, and b = 248.6. B is acute B is obtuse m∠B = _____________ m∠B = _____________ m∠C = _____________ m∠C = _____________ c = ________________ c = ________________ 11 Example 6 – SSA, the No-Solution Case Solve triangle ABC, where A = 42, a = 70, and b = 122. B is acute B is obtuse m∠B = _____________ m∠B = _____________ m∠C = _____________ m∠C = _____________ c = ________________ c = ________________ 12 Example 7 – SSS Given a = 10, b = 7, and c = 15, solve the triangle. 13 WebAssign #14 A tree on a hillside casts a shadow c = 210 ft down the hill. If the angle of inclination of the hillside is b = 18° to the horizontal and the angle of elevation of the sun is a = 48°, find the height of the tree. (Round your answer to the nearest foot.) 14