2. Copyright © 2009 Pearson Education, Inc. CHAPTER 8: Applications of Trigonometry 8.1The Law of Sines 8.2The Law of Cosines 8.3Complex Numbers: Trigonometric Form 8.4Polar Coordinates and Graphs 8.5Vectors and Applications 8.6Vector Operations

3. Copyright © 2009 Pearson Education, Inc. 8.2 The Law of Cosines  Use the law of cosines to solve triangles.  Determine whether the law of sines or the law of cosines should be applied to solve a triangle.

4. Slide 8.2 - 4 Copyright © 2009 Pearson Education, Inc. Law of Cosines Thus, in any triangle, the square of a side is the sum of the squares of the other two sides, minus twice the product of those sides and the cosine of the included angle. When the included angle is 90º, the law of cosines reduces to the Pythagorean theorem. The Law of Cosines In any triangle ABC, A B C a b c

5. Slide 8.2 - 5 Copyright © 2009 Pearson Education, Inc. When to use the Law of Cosines The Law of Cosines is used to solve triangles given two sides and the included angle (SAS) or given three sides (SSS).

6. Slide 8.2 - 6 Copyright © 2009 Pearson Education, Inc. Example In ! ABC, a = 32, c = 48, and B = 125.2º. Solve the triangle. Solution: Draw and label a triangle.

7. Slide 8.2 - 7 Copyright © 2009 Pearson Education, Inc. Example Solution continued Use the law of cosines to find the third side, b. We need to find the other two angle measures. We can use either the law of sines or law of cosines. Using the law of cosines avoids the possibility of the ambiguous case. So use the law of cosines.

8. Slide 8.2 - 8 Copyright © 2009 Pearson Education, Inc. Example Solution continued Find angle A. Now find angle C. C ≈ 180º – (125.2º + 22º) C ≈ 32.8º

9. Slide 8.2 - 9 Copyright © 2009 Pearson Education, Inc. Example Solve ! RST, r = 3.5, s = 4.7, and t = 2.8. Solution: Draw and label a triangle.

10. Slide 8.2 - 10 Copyright © 2009 Pearson Education, Inc. Example Solution continued Similarly, find angle R.

11. Slide 8.2 - 11 Copyright © 2009 Pearson Education, Inc. Example Solution continued Now find angle T. T ≈ 180º – (95.86º + 47.80º) ≈ 36.34º

12. Slide 8.2 - 12 Copyright © 2009 Pearson Education, Inc. Example Knife makers know that the bevel of the blade (the angle formed at the cutting edge of the blade) determines the cutting characteristics of the knife. A small bevel like that of a straight razor makes for a keen edge, but is impractical for heavy-duty cutting because the edge dulls quickly and is prone to chipping. A large bevel is suitable for heavy-duty work like chopping wood. Survival knives, being universal in application, are a compromise between small and large bevels. The diagram illustrates the blade of a hand-made Randall Model 18 survival knife. What is its bevel?

13. Slide 8.2 - 13 Copyright © 2009 Pearson Education, Inc. Example Solution: Use the law of cosines to find angle A. The bevel is approximately 14.36º.