1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 5 Analytic Trigonometry

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.1 Fundamental Identities

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 4 Quick Review

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

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 6 What you’ll learn about Identities Basic Trigonometric Identities Pythagorean Identities Cofunction Identities Odd-Even Identities Simplifying Trigonometric Expressions Solving Trigonometric Equations … and why Identities are important when working with trigonometric functions in calculus.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 7 Basic Trigonometric Identities

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 8 Pythagorean Identities

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 9 Example Using Identities

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 10 Example Using Identities

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 11 Cofunction Identities

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 12 Cofunction Identities

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 13 Even-Odd Identities

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 14 Example Simplifying by Factoring and Using Identities

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 15 Example Simplifying by Factoring and Using Identities

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 16 Example Simplifying by Expanding and Using Identities

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 17 Example Simplifying by Expanding and Using Identities

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 18 Example Solving a Trigonometric Equation

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 19 Example Solving a Trigonometric Equation

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.2 Proving Trigonometric Identities

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 21 Quick Review

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 22 Quick Review Solutions

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 23 What you’ll learn about A Proof Strategy Proving Identities Disproving Non-Identities Identities in Calculus … and why Proving identities gives you excellent insights into the was mathematical proofs are constructed.

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 24 General Strategies I for Proving an Identity 1. The proof begins with the expression on one side of the identity. 2. The proof ends with the expression on the other side. 3. The proof in between consists of showing a sequence of expressions, each one easily seen to be equivalent to its preceding expression.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 25 General Strategies II for Proving an Identity 1. Begin with the more complicated expression and work toward the less complicated expression. 2. If no other move suggests itself, convert the entire expression to one involving sines and cosines. 3. Combine fractions by combining them over a common denominator.

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 26 Example Setting up a Difference of Squares

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 27 Example Setting up a Difference of Squares

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 28 General Strategies III for Proving an Identity 1. Use the algebraic identity (a+b)(a-b) = a 2 -b 2 to set up applications of the Pythagorean identities. 2. Always be mindful of the “target” expression, and favor manipulations that bring you closer to your goal.

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 29 Identities in Calculus

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.3 Sum and Difference Identities

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 31 Quick Review

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 32 Quick Review Solutions

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 33 What you’ll learn about Cosine of a Difference Cosine of a Sum Sine of a Difference or Sum Tangent of a Difference or Sum Verifying a Sinusoid Algebraically … and why These identities provide clear examples of how different the algebra of functions can be from the algebra of real numbers.

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 34 Cosine of a Sum or Difference

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 35 Example Using the Cosine-of-a- Difference Identity

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 36 Example Using the Cosine-of-a- Difference Identity

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 37 Sine of a Sum or Difference

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 38 Example Using the Sum and Difference Formulas

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 39 Example Using the Sum and Difference Formulas

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 40 Tangent of a Difference of Sum

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.4 Multiple-Angle Identities

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 42 Quick Review

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 43 Quick Review Solutions

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 44 What you’ll learn about Double-Angle Identities Power-Reducing Identities Half-Angle Identities Solving Trigonometric Equations … and why These identities are useful in calculus courses.

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 45 Double Angle Identities

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 46 Proving a Double-Angle Identity

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 47 Power-Reducing Identities

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 48 Example Reducing a Power of 4

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 49 Example Reducing a Power of 4

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 50 Half-Angle Identities

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 51 Example Using a Double Angle Identity

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 52 Example Using a Double Angle Identity

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.5 The Law of Sines

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 54 Quick Review

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 55 Quick Review Solutions

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 56 What you’ll learn about Deriving the Law of Sines Solving Triangles (AAS, ASA) The Ambiguous Case (SSA) Applications … and why The Law of Sines is a powerful extension of the triangle congruence theorems of Euclidean geometry.

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 57 Law of Sines

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 58 Example Solving a Triangle Given Two Angles and a Side

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 59 Example Solving a Triangle Given Two Angles and a Side

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 60 Example Solving a Triangle Given Two Sides and an Angle (The Ambiguous Case)

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 61 Example Solving a Triangle Given Two Sides and an Angle (The Ambiguous Case)

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 62 Example Finding the Height of a Pole x 15ft 15 º 65 º B A C

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 63 Example Finding the Height of a Pole x 15ft 15 º 65 º B A C

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

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 65 Quick Review

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 66 Quick Review Solutions

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 67 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.

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 68 Law of Cosines

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 69 Example Solving a Triangle (SAS)

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 70 Example Solving a Triangle (SAS)

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 71 Area of a Triangle

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 72 Heron’s Formula

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 73 Example Using Heron’s Formula Find the area of a triangle with sides 10, 12, 14.

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 74 Example Using Heron’s Formula Find the area of a triangle with sides 10, 12, 14.

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 75 Chapter Test

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 76 Chapter Test

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 77 Chapter Test 9. A hot-air balloon is seen over Tucson, Arizona, simultaneously by two observers at points A and B that are 1.75 mi apart on level ground and in line with the balloon. The angles of elevation are as shown here. How high above ground is the balloon? 10. A wheel of cheese in the shape of a right circular cylinder is 18 cm in diameter and 5 cm thick. If a wedge of cheese with a central angle of 15 º is cut from the wheel, find the volume of the cheese wedge.

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 78 Chapter Test Solutions

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 79 Chapter Test Solutions

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 80 Chapter Test Solutions 9. A hot-air balloon is seen over Tucson, Arizona, simultaneously by two observers at points A and B that are 1.75 mi apart on level ground and in line with the balloon. The angles of elevation are as shown here. How high above ground is the balloon? 10. A wheel of cheese in the shape of a right circular cylinder is 18 cm in diameter and 5 cm thick. If a wedge of cheese with a central angle of 15 º is cut from the wheel, find the volume of the cheese wedge. ≈0.6 mi 405π/24≈53.01