1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 1 Homework, Page 484 Solve the triangle. 1.

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 2 Homework, Page 484 Solve the triangle. 5.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 3 Homework, Page 484 Solve the triangle. 9.

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 4 Homework, Page 484 State whether the given measurements determine zero, one, or two triangles. 13.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 5 Homework, Page 484 State whether the given measurements determine zero, one, or two triangles. 17.

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 6 Homework, Page 484 Two triangles can be formed using the given measurements. Find both triangles. 21.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 7 Homework, Page 484 Decide whether the triangle can be solved using the Law of Sines. If so, solve it, if not, explain why not. 25. Neither triangle can be solved using the Law of Sines, for the one on the left we need to know the length of the side opposite the known angle and for the one on the right, we have the same problem.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 8 Homework, Page 484 Respond in one of the following ways: (a) State: “Cannot be solved with Law of Sines.” (b) State: “No triangle is formed.” (c) solve the triangle. 29. No triangle is formed. The largest side of a triangle is opposite the largest angle and angle A must be the largest angle and side a is no the largest side.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 9 Homework, Page 484 Respond in one of the following ways: (a) State: “Cannot be solved with Law of Sines.” (b) State: “No triangle is formed.” (c) solve the triangle. 33.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 10 Homework, Page 484 37. Two markers A and B on the same side of a canyon rim are 56 ft apart. A third marker C, located on the opposite rim, is positioned so that (a) Find the distance between C and A. (b) Find the distance between the canyon rims.

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 11 Homework, Page 484 41. A 4-ft airfoil attached to the cab of a truck makes an 18º angle with the roof and angle β is 10º. Find the length of the vertical brace positioned as shown.

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 12 Homework, Page 484 45. Two lighthouses A and B are known to be exactly 20 mi apart. A ship’s captain at S measures the angle S at 33º. A radio operator measures the angle B at 52º. Find the distance from the ship to each lighthouse.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 13 Homework, Page 484 49. The length x in the triangle is (A) 8.6 (B) 15.0 (C) 18.1 (D) 19.2 (E) 22.6

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 14 Homework, Page 484 53. (a) Show that there are infinitely many triangles with AAA given if the sum of the three positive angles is 180º. Consider the triangle formed with its base on a radius that is one-half the diameter of a semi-circle. If the opposite ends of the radius are connected to a point on the semi-circle, a triangle is formed. Since there are an infinite number of possible values of the radius, there must be an infinite number of possible triangles.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 15 Homework, Page 484 53. (b) Give three examples of triangles where A = 30º, B = 60º, and C = 90º. (c) Give three examples where A = B = C = 60º.

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 16 Homework, Page 484 57. Towers A and B are known to be 4.1 mi apart on level ground. A pilot measures the angles of depression to the towers at 36.5º and 25º, respectively. Find distances AC and BC and the height of the aircraft.

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.6 The Law of Cosines

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 18 Quick Review

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 19 Quick Review Solutions

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 20 What you’ll learn about Deriving the Law of Cosines Solving Triangles (SAS, SSS) Triangle Area and Heron’s Formula Applications … and why The Law of Cosines is an important extension of the Pythagorean theorem, with many applications.

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 21 Deriving the Law of Cosines

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 22 Law of Cosines

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 23 Example Solving a Triangle (SAS)

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 24 Example Solving a Triangle (SSS)

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 25 Area of a Triangle

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 26 Heron’s Formula

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 27 Example Using Heron’s Formula Find the area of a triangle with sides 10, 12, 14.

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 28 Example Finding the Area of a Regular Circumscribed Polygon Find the area of a regular nonagon (9-sided) circumscribed about a circle of radius 10 in.

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 29 Example Surveyor’s Problem Tony must find the distance from point A to point B on opposite sides of a lake. He finds point C which is 860 ft from point A and 175 ft from point B. If he measures the angle at point C between points A and B as 78º, what is the distance between points A and B.

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 30 Homework Homework Assignment #1 Review Section 5.6 Page 494, Exercises: 1 – 53 (EOO)

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 31 What you’ll learn about Two-Dimensional Vectors Vector Operations Unit Vectors Direction Angles Applications of Vectors … and why These topics are important in many real-world applications, such as calculating the effect of the wind on an airplane’s path.

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 32 Directed Line Segment

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 33 Two-Dimensional Vector

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 34 Two-Dimensional Vector

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 35 Initial Point, Terminal Point, Equivalent

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 36 Magnitude

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 37 Example Finding Magnitude of a Vector

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 38 Vector Addition and Scalar Multiplication

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 39 Example Performing Vector Operations

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 40 Unit Vectors

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 41 Example Finding a Unit Vector

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 42 Standard Unit Vectors

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 43 Resolving the Vector

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 44 Example Finding the Components of a Vector

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 45 Example Finding the Direction Angle of a Vector

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 46 Velocity and Speed The velocity of a moving object is a vector because velocity has both magnitude and direction. The magnitude of velocity is speed.

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 47 Example Writing Velocity as a Vector

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 48 Example Calculating the Effects of Wind Velocity

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 49 Example Finding the Direction and Magnitude of the Resultant Force