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Homework, Page 468 Use a sum or difference identity to find an exact value. 1. sin15 sin15 sin 45 30 sin 45 cos30 cos 45 sin 30 2 3 2 1 6 2 4 2 2 2 2 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6 2 4 Slide 5- 1 Homework, Page 468 Use a sum or difference identity to find an exact value. 5. cos 12 cos cos cos cos sin sin 12 3 4 3 4 3 4 1 2 3 2 2 6 4 2 2 2 2 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 6 4 Slide 5- 2 Homework, Page 468 Use a sum or difference identity to find an exact value. 7 9. cos 12 7 cos cos cos cos sin sin 12 3 4 3 4 3 4 1 2 3 2 2 6 4 2 2 2 2 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 6 4 Slide 5- 3 Homework, Page 468 Write the expression as the sine, cosine, or tangent of an angle. 5 2 13. sin cos sin 5 cos 2 sin sin 2 2 cos cos 5 5 sin 5 cos cos sin 2 5 2 sin 5 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 4 Homework, Page 468 Write the expression as the sine, cosine, or tangent of an angle. 7 7 17. cos cos x sin sin x cos cos x sin sin x cos x 7 7 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 5 Homework, Page 468 Write the expression as the sine, cosine, or tangent of an angle. tan 2 y tan 3 x 21. 1 tan 2 y tan 3 x tan 2 y tan 3 x tan 2 y 3 x 1 tan 2 y tan 3 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 6 Homework, Page 468 Prove the identity. 25. cos x sin x 2 sin x cos x 2 cos x cos sin x sin 2 cos x 0 sin x 1 2 sin x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 7 Homework, Page 468 Prove the identity. 29. 1 tan tan 4 1 tan 1 tan tan 1 tan 4 tan tan 1 tan tan tan 1 1 tan 1 4 4 1 tan 1 tan Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 8 Homework, Page 468 Match each graph with a pair of the equations. 33. d . y sin 2 x 5 h. y sin 2 x cos5 cos 2 x sin 5 sin 2 x 5 sin 2 x cos5 cos 2 x sin 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 9 Homework, Page 468 Prove the reduction formula. 37. sin u cos u 2 cos u sin u 2 sin cos u cos sin u 2 2 1 cos u 0 sin u cos u Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 10 Homework, Page 468 Prove the reduction formula. 41. csc u sec u 2 1 1 cos u sin u 2 cos u sin u 2 sin cos u cos 2 2 1 cos u 0 sin u sin u cos u Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 11 Homework, Page 468 Express the function as a sinusoid in the form y = a sin (bx + c). 45. y cos3x 2sin 3x y cos3 x 2sin 3 x y a sin bx c a sin bx c a sin bx cos c cos bx sin c 2sin 3 x cos3 x a cos c sin bx a sin c cos bx b 3 a cos c 2 a sin c 1 2 2 2 a cos c a sin c 2 1 a 5a 5 2 2 2 4 1 2 cos c ;sin c c cos 0.464 5 5 5 1 2 cos3 x 2sin 3 x 5 sin 3 x cos 2.236sin 3 x 0.464 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 12 Homework, Page 468 Prove the identity. 49. cos3x cos3 x 3sin 2 x cos x cos3 x 3sin 2 x cos x cos3 x cos 2 x cos x sin 2 x sin x cos x cos x cos x sin x sin x sin x sin x cos x cos x sin x cos3 x cos x sin 2 x sin 2 x cos x sin 2 x cos x cos3 x 3sin 2 x cos x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 13 Homework, Page 468 Prove the identity. tan 2 x tan 2 y 53. tan x y tan x y 1 tan 2 x tan 2 y tan x tan y tan x tan y tan x y tan x y 1 tan x tan y 1 tan x tan y tan 2 x tan 2 y 1 tan 2 x tan 2 y Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 14 Homework, Page 468 57. If cos A + cos B = 0, then A and B are supplementary angles. False. If A and B are supplementary angles, then A B 180º. For example, cos cos 2 0, but + 2 180º. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 15 Homework, Page 468 f 1 f 2 61. A function with the property f 1 2 is 1 f 1 f 2 A. f x sin x f x tan x B. C. f x sec x D. E. f x ex f x 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 16 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- 18 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- 19 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- 20 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- 21 Double Angle Identities sin 2u 2sin u cos u cos 2 u sin 2 u 2 cos 2u 2cos u 1 1 2sin 2 u 2 tan u tan 2u 2 1 tan u Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 22 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- 23 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- 24 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- 25 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- 26 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- 27 Homework Homework Assignment #13 Read Section 5.5 Page 475, Exercises: 1 – 57 (EOO) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 28 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. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 29 Deriving the Law of Sines Consider the two triangles ABC. C b a h A B c C b a A Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley c h B Slide 5- 30 Law of Sines In ABC with angles A, B, and C opposite sides a, b, and c, respectively, the following equation is true: sin A sin B sin C . a b c Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 31 Example Solving a Triangle Given Two Angles and a Side Solve ABC given that A 38, B 46, and a 9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 32 Example Solving a Triangle Given Two Sides and an Angle (The Ambiguous Case) Solve ABC given that a 5, b 6, and A 30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 33 Example Finding the Height of a Pole A road slopes 15 above the horizontal, and a vertical telephone pole stands beside the road. The angle of elevation of the Sun is 65, and the pole casts a 15 foot shadow downhill along the road. A 65 Find the height of the pole. º x B 15º C Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 34