Law of Sines and Law of Cosines An

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 a B 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 b A h C a c h B Acute Triangle Copyright © by Houghton Mifflin Company, Inc. All rights reserved. b a c A Obtuse Triangle B 4

Example (ASA): Find the remaining angle and sides of the triangle. The third angle in the triangle is A = 180 – C – B = 180 – 10 – 60 = 110. C 10 a = 4. 5 ft b 60 A c B Use the Law of Sines to find side b and c. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 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. A = 110 , a = 125 inches, b = 100 inches C a = 125 in b = 100 in 110 A c B C 180 – 110 – 48. 74 = 21. 26 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 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 = 11. 4 cm 58 B 1 c A C 180 – 58 – 72. 2 = 49. 8 Check for the other “potential angle” C 180 – 72. 2 = 107. 8 (107. 8 + 58 = 165. 8 which is less than 180 so there are two triangles. ) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example continues. 13

Example (SSA) continued: Use the Law of Sines to solve the second triangle. A = 58 , a = 11. 4 cm, b = 12. 8 cm B 2 180 – 72. 2 = 107. 8 C 49. 8 b = 12. 8 cm a = 11. 4 cm C 180 – 58 – 107. 8 = 14. 2 B 1 72. 2 58 c A 10. 3 cm C a = 11. 4 cm b = 12. 8 cm 58 B 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Alternative Form 15

Example: Find the three angles of the triangle. C 8 6 A 12 B Find the angle opposite the longest side first. Law of Sines: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16

C Example: Solve the triangle. 6. 2 Law of Cosines: A 75 9. 5 B Law of Sines: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17