Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 1 Homework, Page 451 Evaluate without using a calculator. Use the Pythagorean identities rather than reference angles. 1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 2 Homework, Page 451 Use identities to find the value of the expression. 5.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 3 Homework, Page 451 Use basic identities to simplify the expression. 9.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 4 Homework, Page 451 Use basic identities to simplify the expression. 13.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 5 Homework, Page 451 Simplify the expression to either 1 or –1. 17.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 6 Homework, Page 451 Simplify the expression to either 1 or –1. 21.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 7 Homework, Page 451 Simplify the expression to either a constant or a basic trigonometric function. Support your answer graphically. 25.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 8 Homework, Page 451 Use basic identities to change the expression to one involving only sines and cosines. Then simplify to a basic trigonometric function. 29.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 9 Homework, Page 451 Combine the fractions and simplify to a multiple of a power of a basic trigonometric function. 33.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 451 Combine the fractions and simplify to a multiple of a power of a basic trigonometric function. 37.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 451 Write each expression in factored form as an algebraic expression of a single trigonometric function. 41.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 451 Write each expression in factored form as an algebraic expression of a single trigonometric function. 45.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 451 Write each expression as an algebraic expression of a single trigonometric function. 49.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 451 Find all solutions to the equation in the interval (0, 2π). 53.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 451 Find all solutions to the equation. 57.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 451 Find all solutions to the equation. 61.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 451 Find all solutions to the equation. 65.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 451 Write the function as a multiple of a basic trigonometric function. 69.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 451 Write the function as a multiple of a basic trigonometric function. 73.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page Which of the following could not be set equal to sin x as an identity? a. b. c. d. e.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page Write all six trigonometric functions in terms of sin x.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page Because its orbit is elliptical, the distance from the Moon to the Earth in miles, measured from the center of the Moon to the center of the Earth, varies periodically. On Monday, January 18, 2002, the Moon was at its apogee. The distance from the Moon to the Earth each Friday from January 23 to March 27 is recorded in the table. DateDayDistanceDateDayDistance 1/230251,9662/ ,315 1/307238,3443/642226,101 2/614225,7843/ ,390 2/ ,3853/ ,333 2/ ,8073/ ,347
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page a. Draw a scatterplot of the data using day as x and distance as y. b. Do a sine regression and plot the resulting curve on the scatterplot.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page c. What is the approximate number of days from apogee to apogee? d. Approximately how far is the Moon from the Earth at perigee? e. Use a cofunction to write a cosine curve that fits the data.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.2 Proving Trigonometric Identities
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Proving an Algebraic Identity
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Proving a Trigonometric Identity
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Proving a Trigonometric Identity
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Identities in Calculus
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework Homework Assignment #36 Read Section 5.3 Page 460, Exercises: 1 – 77 (EOO)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.3 Sum and Difference Identities
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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 Cosine of a Sum or Difference
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Using the Cosine-of-a- Difference Identity
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Sine of a Sum or Difference
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Using the Sum and Difference Formulas
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Tangent of a Difference of Sum
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Proving an Identity
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Proving a Reduction Formula
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Expressing a Function as a Sinusoid
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Proving an Identity