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Homework, Page 460 Prove 3the algebraic identity. 2 x x x 1 x 1 1 x 1. x x3 x 2 1 x x 1 x 1 x x x2 x x x2 1 x2 x x2 1 x2 x x2 1 x 1 1 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 1 Homework, Page 460 Tell whether or not f (x) = sin x is an identity. 5. sin 2 x cos 2 x f x csc x sin 2 x cos 2 x f x csc x 1 csc x 1 1 sin x sin x Yes, f x sin x is an identity. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 2 Homework, Page 460 Tell whether or not f (x) = sin x is an identity. 3 2 f x sin x 1 cot x 9. f x sin 3 x 1 cot 2 x 2 cos x 2 sin x sin x 1 2 sin x sin x sin 2 x cos 2 x sin x 1 sin x Yes, f x sin x is an identity. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 3 Homework, Page 460 Prove the identity. 2 2 1 tan x sec x 2 tan x 13. 2 1 tan x sec x 2 tan x 2 1 tan 2 x 2 tan x 1 2 tan x tan 2 x 1 tan x 1 tan x 1 tan x 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 4 Homework, Page 460 Prove the identity. cos 2 x 1 tan x sin x 17. cos x cos 2 x 1 tan x sin x cos x sin x sin x cos x sin 2 x cos x 1 cos 2 x cos x cos 2 x 1 cos x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 5 Homework, Page 460 Prove the identity. 21. cos t sin t cos t sin t 2 2 2 2 cos t sin t cos t sin t 2 2 cos 2 t 2cos t sin t sin 2 t cos 2 t 2cos t sin t sin 2 t 2cos 2 t 2cos t sin t 2sin 2 t 2cos t sin t 2 cos 2 cos 2 t sin 2 t cos t sin t cos t sin t 2 t sin 2 t 21 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 6 Homework, Page 460 Prove the identity. cos 1 sin 25. 1 sin cos cos 1 sin 1 sin 1 sin cos 1 sin 1 sin 2 cos 1 sin cos 2 cos 1 sin cos 1 sin Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 7 Homework, Page 460 Prove the identity. 29. cot 2 x cos2 x cos2 x cot 2 x cot 2 x cos 2 x cos 2 x cot 2 x cos 2 x csc 2 x 1 1 cos x 2 1 sin x cos 2 x 2 cos x 2 sin x cot 2 x cos 2 x 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 8 Homework, Page 460 Prove the identity. 2 2 x sin y cos x cos y sin x 2 y 2 33. x 2 y 2 x sin y cos x cos y sin 2 2 x 2 sin 2 2 xy sin cos y 2 cos 2 x 2 cos 2 2 xy cos sin y 2 sin 2 y cos sin x 2 sin 2 cos 2 2 xy sin cos cos sin 2 2 2 x2 y 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 9 Homework, Page 460 Prove the identity. sin x cos x 2sin 2 x 1 37. sin x cos x 1 2sin x cos x 2sin 2 x sin 2 x cos 2 x sin 2 x cos 2 x 2sin x cos x sin 2 x cos 2 x sin 2 x 2sin x cos x cos 2 x sin x cos x sin x cos x sin x cos x sin x cos x sin x cos x sin x cos x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 10 Homework, Page 460 Prove the identity. 2 3 2 4 sin x cos x sin x sin x cos x 41. sin 2 x cos3 x sin 2 x sin 4 x cos x x cos x cos x sin 2 x 1 sin 2 x cos x sin 2 2 sin 2 x cos3 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 11 Homework, Page 460 Prove the identity. tan x cot x 45. 1 cot x 1 tan x 1 sec x csc x tan x sin x cot x cos x 1 sec x csc x 1 cot x sin x 1 tan x cos x sin 2 x cos 2 x cos x sin x sin x cos x cos x sin x sin 2 x cos 2 x cos x sin x sin x cos x sin x cos x sin 2 x cos 2 x sin x cos x cos x sin x sin x cos x sin x cos x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 12 Homework, Page 460 Prove the identity. 45. sin 2 x cos 2 x sin x cos x 1 sec x csc x cos x sin x sin x cos x sin x cos x sin 3 x cos3 x sin x cos x sin x cos x sin 2 x sin x cos x cos 2 x sin x cos x 1 sin x cos x sin x cos x 1 sec x csc x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 13 Homework, Page 460 Prove the identity. 49. cos3 x 1 sin 2 x cos x cos x cos x cos3 x 1 sin 2 x cos x 2 cos3 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 14 Homework, Page 460 Match the function with an equivalent expression, then confirm the matching with a proof. 53. 1 sec x 1 cos x d. 1 sec x 1 cos x 1 cos x sec x 1 1 cos x cos x cos 2 x 1 cos x sin 2 x cos x tan x sin x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 15 Homework, Page 460 Match the function with an equivalent expression, then confirm the matching with a proof. 1 1 1 1 sin x sec x tan x 57. b. sec x tan x cos x cos x cos x 1 sin x 1 sin x 1 sin x cos x 1 sin x 1 sin 2 x cos x 1 sin x cos 2 x 1 sin x cos x sec x tan x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 16 Homework, Page 460 61. Which of the following is a good first step in proving: sin x 1 cos x 1 cos x sin x cos x sin x sin x A. sin x B. 2 1 cos x sin 2 x cos 2 x cos x 1 cos x 1 cos x sin x sin x csc x sin x sin x 1 cos x C. 1 cos x 1 cos x csc x D. 1 cos x 1 cos x 1 cos x sin x sin x 1 cos x E. 1 cos x 1 cos x 1 cos x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 17 Homework, Page 460 Identify a simple function that has the same graph. Then confirm your answer with a proof. 65. cos x tan x y sin x sin x cos x tan x cos x cos x sin x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 18 Homework, Page 460 Identify a simple function that has the same graph. Then confirm your answer with a proof. 2 2 sec x 1 sin x 69. y 1 1 2 sec x 1 sin x cos x 2 cos x 1 2 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 19 Homework, Page 460 Confirm the identity. 73. 1 sin t 1 sin t 1 sin t cos t 1 sin t 1 sin t 1 sin t 1 sin t 1 sin t 1 sin t 1 sin t 2 1 sin 2 t 1 sin t 2 cos 2 t 1 sin t cos t Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 20 Homework, Page 460 Confirm the identity. 77. ln tan x ln sin x ln cos x ln tan x ln sin x ln cos x sin x ln cos x ln tan x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 21 5.3 Sum and Difference Identities Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review Express the angle as a sum or difference of special angles (multiples of 30 , 45 , /6, or /4). Answers are not unique. 1. 15 2. /12 3. 75 Tell whether or not the identity f ( x y ) f ( x) f ( y ) holds for the function f . 4. f ( x) 16 x 5. f ( x) 2e x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 23 Quick Review Solutions Express the angle as a sum or difference of special angles (multiples of 30 , 45 , /6, or /4). Answers are not unique. 1. 15 45 30 2. /12 3 4 3. 75 30 45 Tell whether or not the identity f ( x y ) f ( x) f ( y ) holds for the function f . 4. f ( x) 16 x yes 5. f ( x) 2e no x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 24 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. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 25 Cosine of a Sum or Difference cos u v cos u cos v sin u sin v cos u v cos u cos v sin u sin v Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 26 Example Using the Cosine-of-aDifference Identity Find the exact value of cos 75 without using a calculator. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 27 Sine of a Sum or Difference sin u v sin u cos v cos u sin v sin u v sin u cos v cos u sin v Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 28 Example Using the Sum and Difference Formulas Write the following expression as the sine or cosine of an angle: sin 3 cos 4 sin 4 cos 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 29 Tangent of a Difference of Sum sin u v sin u cos v sin v cos u sec u sec v tan u v cos u v cos u cos v sin u sin v sec u sec v tan u tan v 1 tan u tan v sin u v sin u cos v sin v cos u sec u sec v tan u v cos u v cos u cos v sin u sin v sec u sec v tan u tan v 1 tan u tan v Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 30 Example Proving an Identity Prove the identity: cos x y sin x y 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 31 Example Proving a Reduction Formula sec u csc u 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 32 Example Expressing a Function as a Sinusoid y 5sin x 12cos x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 33 Example Proving an Identity 2 3 sin 3u 3cos u sin u sin u Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 34 Homework Homework Assignment #12 Read Section 5.4 Page 468, Exercises: 1 – 61 (EOO) Quiz next time Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 35 5.4 Multiple-Angle Identities Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review Find the general solution of the equation. 1. cot x 1 0 2. (sin x)(1 cos x) 0 3. cos x sin x 0 4. 2sin x 2 2sin x 1 0 5. Find the height of the isosceles triangle with base length 6 and leg length 4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 37 Quick Review Solutions Find the general solution of the equation. 3 1. cot x 1 0 x n 4 2. (sin x)(1 cos x) 0 x n 3. cos x sin x 0 x 4 n 4. 2sin x 2 2sin x 1 0 x 5 2 n 6 6 5. Find the height of the isosceles triangle with base length 6 x 5 7 2 n, x 2 n, 4 4 2 n, x and leg length 4. 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 38 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. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 39 cos 2 x Deriving Double-Angle Identities sin 2 x tan 2 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 40 Double Angle Identities sin 2u 2sin u cos u cos u sin u cos 2u 2 cos u 1 1 2sin u 2 2 2 2 2 tan u tan 2u 1 tan u 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 41 Example Solving a Problem Using double Angle Identities Find all solutions in the interval 0,2 sin 2 x sin x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 42 Power-Reducing Identities 1 cos 2u sin u 2 1 cos 2u 2 cos u 2 1 cos 2u 2 tan u 1 cos 2u 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 43 Example Reducing a Power of 4 Rewrite sin 4 x in terms of trigonometric functions with no power greater than 1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 44 Half-Angle Identities u 1 cos u sin 2 2 u 1 cos u cos 2 2 1 cos u 1 cos u u 1 cos u tan 2 sin u sin u 1 cos u Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 45 Example Using a Double Angle Identity Solve algebraically in the interval [0, 2 ) : sin 2 x sin x. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 46