1. A right triangle has legs of 8 centimeters and 13 centimeters. Solve the triangle completely.

Transcription

1 9.7 Warmup 1. A right triangle has legs of 8 centimeters and 13 centimeters. Solve the triangle completely. 2. A right triangle has a leg length of 7 in. and a hypotenuse length of 14 in. Solve the triangle completely. 3. A right triangle has legs that both measure to be 10 meters. Solve the triangle completely. 1

2 Geometry 9.7 Law of Sines and Cosines

3 9.7 Essential Question When do you use the Law of Sines and the Law of Cosines? 3

4 What if you have a triangle that is not a right triangle? So far, you have used trigonometric ratios to solve right triangles. In this lesson, you will learn how to solve obtuse and acute triangles too. When the triangle is obtuse, you may need to find a trigonometric ratio for an obtuse angle. 4

5 Example 1 Use a calculator to find each trigonometric ratio. Round your answer to four decimal places. a. tan 150 b. sin 120 c. cos 95 5

6 Area of a Triangle 6

7 Example 2 Find the area of the triangle. Round your answer to the nearest tenth. 7

8 Your Turn Find the area of Δ ABC with the given side lengths and included angle. Round your answer to the nearest tenth. C = 29, a = 38, b = 31 8

9 Law of Sines 9

10 When do you use the Law of Sines? There are two cases for using the Law of Sines: Given 2 angles and any side AAS or ASA Find the side opposite an angle. OR Given 2 sides and 1 opposite angle Find the angle opposite a side. SSA 10

11 Example 3 Solve the triangle. Round decimal answers to the nearest tenth. Use the Law of Sines to find m B. sin B sin A = b a sin B sin 115 = sin115 sinb = 20 m B 29.9 By the Triangle Sum Theorem, m C m C

12 Example 3 (continued) Solve the triangle. Round decimal answers to the nearest tenth. Use the Law of Sines again to find the remaining side length, c, of the triangle. c sin C = a sin A c sin 35.1 = 20 sin 115 c = 20 sin 35.1 sin115 c 12.7 In ΔABC, m B 29.9, m C 35.1, and c

13 Your Turn Solve the triangle. Round decimal answers to the nearest tenth. 13

14 Example 4 Solve the triangle. Round decimal answers to the nearest tenth. By the Triangle Sum Theorem, m A = = 48 By the Law of Sines, you can write a sin 48 = 15 sin 25 a = 15 sin 48 sin 25 Write two equations, each with one variable. a = 15 = sin 48 sin 25 c sin 107 = 15 sin 25 c = 15 sin 107 sin 25 c sin 107 In ΔABC, m A = 48 a 26.4 c 33.9 a 26.4 c 33.9 April 1, Law of Sines and Cosines

15 Your Turn Solve the triangle. Round decimal answers to the nearest tenth. 15

16 Example 5 A surveyor makes the measurements shown to determine the length of a bridge to be built across a small lake from the North Picnic Area to the South Picnic Area. Find the length of the bridge. In the diagram, the bridge will be the length of c. By the Law of Sines, c sin C = b sin B By the Triangle Sum Theorem, m B = = 49 c = 150 sin 60 sin 49 c = 150 sin 60 sin 49 c The length of the bridge will be about meters. 16

17 Law of Cosines 17

18 When do you use the Law of Cosines? There are two cases for using the Law of Cosines: Given 2 sides and 1 included angle Find the third side or Given 3 sides Find any angle 18

19 Example 6 Solve the triangle. Round decimal answers to the nearest tenth. 19

20 Example 6 (continued) Solve the triangle. Round decimal answers to the nearest tenth. 20

21 Your Turn Solve the triangle. Round decimal answers to the nearest tenth. 21

22 Example 7 Solve the triangle. Round decimal answers to the nearest tenth. 22

23 Example 7 (continued) Solve the triangle. Round decimal answers to the nearest tenth. 23

24 Your Turn Solve the triangle. Round decimal answers to the nearest tenth. 24

25 Homework 25