1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 1 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.

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 2 Identities

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 3 Basic Trigonometric Identities

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 4 Pythagorean Identities

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 5 Example Using Identities

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 6 Example Using Identities

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 7 Cofunction Identities

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 8 Cofunction Identities

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 9 Even-Odd Identities

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 10 Example Simplifying by Factoring and Using Identities

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 11 Example Simplifying by Expanding and Using Identities

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 12 Example Simplifying by Combining Fractions and Using Identities

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 13 Example Solving a Trigonometric Equation

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 14 Example Solving a Trigonometric Equation by Factoring

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 15 Example Using Identities

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 16 Homework Homework Assignment #35 Review Section 5.1 Page 451, Exercises: 1 – 85 (EOO)

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

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 18 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.

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 19 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.

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 20 Example Proving an Algebraic Identity

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 21 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.

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 22 Example Proving a Trigonometric Identity

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 23 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.

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 24 Example Proving a Trigonometric Identity

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 25 Identities in Calculus