Chapter 6. Additional Topics in Trigonometry. 6.1 The Law of Sines. Copyright 2014, 2010, 2007 Pearson Education, Inc.

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1 Chapter 6 dditional Topics in Trigonometry 6.1 The Law of Sines Copyright 2014, 2010, 2007 Pearson Education, Inc. 1

2 Objectives: Use the Law of Sines to solve oblique triangles. Use the Law of Sines to solve, if possible, the triangle or triangles in the ambiguous case. Find the area of an oblique triangle using the sine function. Solve applied problems using the Law of Sines. Copyright 2014, 2010, 2007 Pearson Education, Inc. 2

3 Oblique Triangles n oblique triangle is a triangle that does not contain a right angle. n oblique triangle has either three acute angles or two acute angles and one obtuse angle. The relationships among the sides and angles of right triangles defined by the trigonometric functions are not valid for oblique triangles. Copyright 2014, 2010, 2007 Pearson Education, Inc. 3

4 The Law of Sines Copyright 2014, 2010, 2007 Pearson Education, Inc. 4

5 Solving Oblique Triangles Solving an oblique triangle means finding the lengths of its sides and the measurements of its angles. The Law of Sines can be used to solve a triangle in which one side and two angles are known. The three known measurements can be abbreviated using S (a side and two angles are known) or S (two angles and the side between them are known). Copyright 2014, 2010, 2007 Pearson Education, Inc. 5

6 Example: Solving an S Triangle Using the Law of Sines Solve the triangle with = 64, C = 82, and c = 14 centimeters. Round lengths of sides to the nearest tenth. C a c sin sinc asinc csin a csin sinc 14sin 64 sin cm Copyright 2014, 2010, 2007 Pearson Education, Inc. 6

7 Example: Solving an S Triangle Using the Law of Sines (continued) Solve the triangle with = 64, C = 82, and c = 14 centimeters. Round lengths of sides to the nearest tenth. a cm b 7.4 cm b sin c sinc b csin sinc 14sin 34 sin cm bsinc csin Copyright 2014, 2010, 2007 Pearson Education, Inc. 7

8 Example: Solving an S Triangle Using the Law of Sines Solve triangle C if = 40, C = 22.5, and b = 12. Round measures to the nearest tenth. C C b a sin sin bsin 12sin 40 a 8.7 sin sin117.5 bsin asin Copyright 2014, 2010, 2007 Pearson Education, Inc. 8

9 Example: Solving an S Triangle Using the Law of Sines (continued) Solve triangle C if = 40, C = 22.5, and b = 12. Round measures to the nearest tenth C a 8.7 c 5.2 b c sin sinc csin bsinc c bsinc sin 12sin 22.5 sin Copyright 2014, 2010, 2007 Pearson Education, Inc. 9

10 The mbiguous Case (SS) If we are given two sides and an angle opposite one of the two sides (SS), the given information may result in one triangle, two triangles, or no triangle at all. SS is known as the ambiguous case when using the Law of Sines because the given information may result in one triangle, two triangles, or no triangle at all. Copyright 2014, 2010, 2007 Pearson Education, Inc. 10

11 The mbiguous Case (SS) (continued) Copyright 2014, 2010, 2007 Pearson Education, Inc. 11

12 Example: Solving an SS Triangle Using the Law of Sines (No Solution) Solve triangle C if = 50, a = 10, and b = 20. There is no angle for which the sine is greater than 1. There is no triangle with the given measurements. a sin b sin asin bsin sin bsin a 20sin Copyright 2014, 2010, 2007 Pearson Education, Inc. 12

13 Example: Solving an SS Triangle Using the Law of Sines (Two Solutions) Solve triangle C if = 35, a = 12, and b = 16. Round lengths of sides to the nearest tenth and angle measures to the nearest degree. a b sin sin asin bsin sin bsin a 1 sin sin There are two angles between 0 and 180 for which sin = Copyright 2014, 2010, 2007 Pearson Education, Inc. 13

14 Example: Solving an SS Triangle Using the Law of Sines (Two Solutions) (continued) Solve triangle C if = 35, a = 12, and b = 16. Round lengths of sides to the nearest tenth and angle measures to the nearest degree. 1 sin and 180, 1 2 there are two possible solutions Copyright 2014, 2010, 2007 Pearson Education, Inc. 14

15 Example: Solving an SS Triangle Using the Law of Sines (Two Solutions) (continued) Solve triangle C if = 35, a = 12, and b = 16. Round lengths of sides to the nearest tenth and angle measures to the nearest degree C C C C C C C C2 15 Copyright 2014, 2010, 2007 Pearson Education, Inc. 15

16 Example: Solving an SS Triangle Using the Law of Sines (Two Solutions) (continued) Solve triangle C if = 35, a = 12, and b = 16. Round lengths of sides to the nearest tenth and angle measures to the nearest degree C C1 95 b c2 b c1 sin 2 sinc2 sin 1 sinc1 bsinc2 c2 sin 2 bsinc1 c1 sin 1 bsinc2 16sin15 c2 5.4 bsinc1 16sin 95 sin c1 2 sin sin sin50 1 Copyright 2014, 2010, 2007 Pearson Education, Inc. 16

17 Example: Solving an SS Triangle Using the Law of Sines (Two Solutions) (continued) Solve triangle C if = 35, a = 12, and b = 16. Round lengths of sides to the nearest tenth and angle measures to the nearest degree. There are two triangles. In one triangle, the solution is 1 50 C1 95 c In the other triangle, the solution is C2 15 c2 5.4 Copyright 2014, 2010, 2007 Pearson Education, Inc. 17

18 The rea of an Oblique Triangle The area of a triangle equals one-half the product of the lengths of two sides times the sine of their included angle. In the figure, this wording can be expressed by the formulas rea bcsin absinc acsin Copyright 2014, 2010, 2007 Pearson Education, Inc. 18

19 Example: Finding the rea of an Oblique Triangle Find the area of a triangle having two sides of length 8 meters and 12 meters and an included angle of 135. Round to the nearest square meter. C 8 m 12 m 135 rea 1 sin 2 ab C 1 (12)(8)(sin135 ) 2 34 sq m Copyright 2014, 2010, 2007 Pearson Education, Inc. 19

20 Example: pplication Two fire-lookout stations are 13 miles apart, with station directly east of station. oth stations spot a fire. The bearing of the fire from station is N35 E and the bearing of the fire from station is N49 W. How far, to the nearest tenth of a mile, is the fire from station? C C C C 180 C 84 Copyright 2014, 2010, 2007 Pearson Education, Inc. 20

21 Example: pplication (continued) Two fire-lookout stations are 13 miles apart, with station directly east of station. oth stations spot a fire. The bearing of the fire from station is N35 E and the bearing of the fire from station is N49 W. How far, to the nearest tenth of a mile, is the fire from station? 35 C 49 c sinc a sin csin asinc a csin sinc 13sin 55 sin84 11 The fire is approximately 11 miles from station. Copyright 2014, 2010, 2007 Pearson Education, Inc. 21