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Digital Lesson Law of Sines and Law of Cosines An oblique triangle is a triangle that has no right angles. C a b A c B To solve an oblique triangle, you need to know the measure of at least one side and the measures of any other two parts of the triangle – two sides, two angles, or one angle and one side. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The following cases are considered when solving oblique triangles. 1. Two angles and any side (AAS or ASA) A A c c B C 2. Two sides and an angle opposite one of them (SSA) c C 3. Three sides (SSS) b a c a 4. Two sides and their included angle (SAS) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. c B a 3 The first two cases can be solved using the Law of Sines. (The last two cases can be solved using the Law of Cosines.) Law of Sines If ABC is an oblique triangle with sides a, b, and c, then C C a b h h A c B Acute Triangle Copyright © by Houghton Mifflin Company, Inc. All rights reserved. a b c A Obtuse Triangle B 4 Example (ASA): Find the remaining angle and sides of the triangle. C The third angle in the triangle is A = 180 – C – B = 180 – 10 – 60 = 110. 10 a = 4.5 ft b 60 A c B Use the Law of Sines to find side b and c. a b sin A sin B a c sin A sin C 4.5 b sin 110 sin 60 4.5 c sin 110 sin 10 b 4.15 feet Copyright © by Houghton Mifflin Company, Inc. All rights reserved. c 0.83 feet 5 The Ambiguous Case (SSA) Law of Sines • In earlier examples, you saw that two angles and one side determine a unique triangle. • However, if two sides and one opposite angle are given, then three possible situations can occur: (1) no such triangle exists, (2) one such triangle exists, or (3) two distinct triangles satisfy the conditions. The Ambiguous Case (SSA) Example 3 – Single-Solution Case—SSA • For the triangle in Figure 6.5, a = 22 inches, b = 12 inches, and A = 42. Find the remaining side and angles. One solution: a b Figure 6.5 Example 3 – Solution • cont’d Multiply each side by b Substitute for A, a, and b. B is acute. • Check for the other “potential angle” • C 180 – 21.41 = 158.59 (158.59 + 42 = 200.59 which is more than 180 so there is only one triangle.) Now you can determine that • C 180 – 42 – 21.41 = 116.59 Example 3 – Solution cont’d • Then the remaining side is given by Law of Sines Multiply each side by sin C. Substitute for a, A, and C. Simplify. Example (SSA): Use the Law of Sines to solve the triangle. C A = 110, a = 125 inches, b = 100 inches a b sin A sin B 125 100 sin 110 sin B B 48.74 C 180 – 110 – 48.74 = 21.26 a = 125 in b = 100 in 110 A c B a c sin A sin C 125 c sin 110 sin 21.26 c 48.23 inches Since the “given angle is already obtuse, there will only be one triangle. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example (SSA): Use the Law of Sines to solve the triangle. A = 76, a = 18 inches, b = 20 inches a b sin A sin B 18 20 sin 76 sin B sin B 1.078 C b = 20 in a = 18 in 76 B A There is no angle whose sine is 1.078. There is no triangle satisfying the given conditions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 C Example (SSA): Use the Law of Sines to solve the triangle. A = 58, a = 11.4 cm, b = 12.8 cm a b sin A sin B 11.4 12.8 sin 58 sin B B1 72.2 C 180 – 58 – 72.2 = 49.8 b = 12.8 cm a = 11.4 cm 58 B1 c A c b sin C sin B c 12.8 Check for the other “potential angle” sin 49.8 sin 72.2 C 180 – 72.2 = 107.8 c 10.3 (107.8 + 58 = 165.8 which is less than 180 so there are two triangles.) Example continues. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Example (SSA) continued: Use the Law of Sines to solve the second triangle. A = 58, a = 11.4 cm, b = 12.8 cm B2 180 – 72.2 = 107.8 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 49.8 b = 12.8 cm a = 11.4 cm C 180 – 58 – 107.8 = 14.2 c b sin C sin B c 12.8 sin 14.2 sin 72.2 c 3.3 C B1 72.2 58 c A 10.3 cm C a = 11.4 cm b = 12.8 cm 58 B2 c A 14 Law of Cosines (SSS and SAS) can be solved using the Law of Cosines. (The first two cases can be solved using the Law of Sines.) Law of Cosines Standard Form Alternative Form a b c 2bc cos A 2 2 2 b c a cos A 2bc b a c 2ac cos B 2 2 2 a c b cos B 2ac 2 2 2 2 2 2 c a b 2ab cos C 2 2 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Example: Find the three angles of the triangle. C 8 6 A B 12 Find the angle opposite the longest side first. C 117.3 Law of Sines: 12 6 sin 117.3 sin B B 26.4 A 180 117.3 26.4 36.3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 C Example: Solve the triangle. 6.2 Law of Cosines: b 2 a 2 c 2 2ac cos B 75 A 9.5 B (6.2)2 (9.5)2 2(6.2)(9.5) cos 75 38.44 90.25 (117.8)(0.25882) 98.20 b 9.9 Law of Sines: 9.9 6.2 sin 75 sin A A 37.2 C 180 75 37.2 67.8 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17