1. Slide 7.2 - 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Law of Cosines Learn the statement and the derivation of the Law of Cosines. Learn to use the Law of Cosines to solve SAS triangles. Learn to use the Law of Cosines to solve SSS triangles. Learn to state and derive Heron’s formula for the area of a triangle. SECTION 7.2 1 2 3 4

3. Slide 7.2 - 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley LAW OF COSINES The following diagrams illustrate the Law of Cosines.

4. Slide 7.2 - 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley LAW OF COSINES Let A, B, and C denote the measures of the angles of a triangle ABC, with opposite sides of lengths a, b, and c, respectively. Then In words, the square of any side of a triangle is equal to the sum of the squares of the length of the other two sides, less twice the product of the lengths of the other sides and the cosine of their included angle.

5. Slide 7.2 - 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SOLVING SAS TRIANGLES Step 1:Use the appropriate form of the Law of Cosines to find the side opposite the given angle. Step 2:Use the Law of Sines to find the angle opposite the shorter of the two given sides. Note that this angle is always an acute angle. Step 3:Use the angle sum formula to find the third angle. Step 4:Write the solution.

6. Slide 7.2 - 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Solving an SAS Triangle Solve triangle ABC with b = 16 meters, c = 12 meters, and A = 50º. Round each answer to the nearest tenths. Solution Step 1Find a, the length of the side opposite angle A.

7. Slide 7.2 - 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued EXAMPLE 1 Solving an SAS Triangle Step 2Find C, the measure of the angle opposite the shorter of the two given sides.

8. Slide 7.2 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued EXAMPLE 1 Solving an SAS Triangle Step 3Find the third angle measure, B. Step 4The solution of triangle ABC is: c = 12 metersC ≈ 48º b = 16 metersB ≈ 82º a ≈ 12.4 metersA = 50º

9. Slide 7.2 - 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Using the Law of Cosines Suppose that a Boeing 747 is flying over Disney World headed due south at 552 miles per hour. Twenty minutes later, an F-16 passes over Disney World with a bearing of N 37º E at a speed of 1250 miles per hour. Find the distance between the two planes 3 hours after the F-16 passes over Disney World. Round the answer to the nearest tenth.

10. Slide 7.2 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Using the Law of Cosines Solution Suppose the F-16 has been traveling for t hours after passing over Disney World. Then, because the Boeing 747 had a head start of 20 minutes hour, the Boeing 747 has been traveling hours due south. The distance between the two planes is d.

11. Slide 7.2 - 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Using the Law of Cosines Solution continued Using the Law of Cosines in triangle FDB, we have Substitute t = 3.

12. Slide 7.2 - 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SOLVING SSS TRIANGLES Step 1:Use the Law of Cosines to find the side angle opposite the longest side. Step 2:Use the Law of Sines to find either of the two remaining acute angles. Step 3:Use the angle sum formula to find the third angle. Step 4:Write the solution.

13. Slide 7.2 - 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solving an SSS Triangle Solve triangle ABC with a = 8, b = 5, and c = 7. Round each answer to the nearest tenths. Solution Step 1Find A, the angle opposite the largest side.

14. Slide 7.2 - 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued EXAMPLE 3 Solving an SSS Triangle Step 2Find B, using the Law of Sines.

15. Slide 7.2 - 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued Step 3Find C by using the angle sum formula. Step 4Write the solution. c = 7C ≈ 60º b = 5B ≈ 38.2º a = 8A ≈ 81.8º EXAMPLE 3 Solving an SSS Triangle

16. Slide 7.2 - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solving an SSS Triangle Solve triangle ABC with a = 2 meters, b = 9 meters, and c = 5 meters. Round each answer to the nearest tenths. Solution Step 1Find B, the angle opposite the longest side. Range of the cosine function is [–1, 1], there is no angle B, the triangle cannot exist.

17. Slide 7.2 - 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley HERON’S FORMULA FOR SSS TRIANGLES The area K of a triangle with sides of lengths a, b, and c is given by whereis the semiperimeter.

18. Slide 7.2 - 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Using Heron’s Formula A triangular swimming pool has side lengths 23 feet, 17 feet, and 26 feet. How many gallons of water will fill the pool to a depth of 5 feet? Round answer to the nearest whole number. Solution To calculate the volume of water in the swimming pool, we first calculate the area of the triangular surface. We have a = 23, b = 17, and c = 26.

19. Slide 7.2 - 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued By Heron’s formula, the area K of the triangula surface is EXAMPLE 6 Using Heron’s Formula Volume of water in pool = surface area  depth

20. Slide 7.2 - 20 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued One cubic foot contains approximately 7.5 gallons of water. EXAMPLE 6 Using Heron’s Formula So 961.25  7.5 ≈ 7209 gallons of water will fill the pool.