ANALYTIC TRIGONOMETRY

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1 7 Son Pictures Classics/Courtes of Everett Collection C H A P T E R ANALYTIC TRIGONOMETRY 7. Trigonometric Identities 7. Addition and Subtraction Formulas 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas 7.4 Basic Trigonometric Equations 7.5 More Trigonometric Equations FOCUS ON MODELING Traveling and Standing Waves In Chapters 5 and 6 we studied graphical and geometric properties of the trigonometric functions. In this chapter we stud algebraic properties of these functions, that is, simplifing and factoring epressions and solving equations that involve trigonometric functions. We have used the trigonometric functions to model different real-world phenomena, including periodic motion (such as the motion of an ocean wave). To obtain information from a model, we often need to solve equations. If the model involves trigonometric functions, we need to solve trigonometric equations. Solving trigonometric equations often involves using trigonometric identities. We've alread encountered some basic trigonometric identities in the preceding chapters. We begin this chapter b finding man new identities. 493 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

2 494 CHAPTER 7 Analtic Trigonometr 7. TRIGONOMETRIC IDENTITIES Simplifing Trigonometric Epressions Proving Trigonometric Identities We begin b listing some of the basic trigonometric identities. We studied most of these in Chapters 5 and 6; ou are asked to prove the cofunction identities in Eercise 0. FUNDAMENTAL TRIGONOMETRIC IDENTITIES Reciprocal Identities Pthagorean Identities Even-Odd Identities csc sin Cofunction Identities tan sin cos sec cos cot cos sin cot tan sin cos tan sec cot csc sin sin cos cos tan tan sin a p u b cos u tan a p u b cot u sec a p u b csc u cos a p u b sin u cot a p u b tan u csc a p u b sec u Simplifing Trigonometric Epressions Identities enable us to write the same epression in different was. It is often possible to rewrite a complicated-looking epression as a much simpler one. To simplif algebraic epressions, we used factoring, common denominators, and the Special Product Formulas. To simplif trigonometric epressions, we use these same techniques together with the fundamental trigonometric identities. EXAMPLE Simplifing a Trigonometric Epression Simplif the epression cos t tan t sin t. SOLUTION We start b rewriting the epression in terms of sine and cosine: cos t tan t sin t cos t a sin t cos t b sin t cos t sin t cos t cos t sec t NOW TRY EXERCISE 3 Reciprocal identit Common denominator Pthagorean identit Reciprocal identit Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

3 SECTION 7. Trigonometric Identities 495 EXAMPLE Simplifing b Combining Fractions sin u Simplif the epression. cos u cos u sin u SOLUTION We combine the fractions b using a common denominator: sin u cos u cos u sin u sin u sin u cos u cos u sin u NOW TRY EXERCISE sin u sin u cos u cos u sin u sin u cos u sin u sec u cos u Common denominator Distribute sin u Pthagorean identit Cancel and use reciprocal identit Proving Trigonometric Identities Man identities follow from the fundamental identities. In the eamples that follow, we learn how to prove that a given trigonometric equation is an identit, and in the process we will see how to discover new identities. First, it s eas to decide when a given equation is not an identit. All we need to do is show that the equation does not hold for some value of the variable (or variables). Thus the equation is not an identit, because when p/4, we have sin p 4 cos p 4 sin cos To verif that a trigonometric equation is an identit, we transform one side of the equation into the other side b a series of steps, each of which is itself an identit. GUIDELINES FOR PROVING TRIGONOMETRIC IDENTITIES. Start with one side. Pick one side of the equation and write it down. Your goal is to transform it into the other side. It s usuall easier to start with the more complicated side.. Use known identities. Use algebra and the identities ou know to change the side ou started with. Bring fractional epressions to a common denominator, factor, and use the fundamental identities to simplif epressions. 3. Convert to sines and cosines. If ou are stuck, ou ma find it helpful to rewrite all functions in terms of sines and cosines. Warning: To prove an identit, we do not just perform the same operations on both sides of the equation. For eample, if we start with an equation that is not an identit, such as () and square both sides, we get the equation () sin sin sin sin which is clearl an identit. Does this mean that the original equation is an identit? Of course not. The problem here is that the operation of squaring is not reversible in the sense Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

4 496 CHAPTER 7 Analtic Trigonometr that we cannot arrive back at () from () b taking square roots (reversing the procedure). Onl operations that are reversible will necessaril transform an identit into an identit. EXAMPLE 3 Proving an Identit b Rewriting in Terms of Sine and Cosine Consider the equation cos u sec u cos u sin u. (a) Verif algebraicall that the equation is an identit. (b) Confirm graphicall that the equation is an identit. SOLUTION (a) The left-hand side looks more complicated, so we start with it and tr to transform it into the right-hand side: LHS cos u sec u cos u cos u a cos u b cos u Reciprocal identit cos u Epand sin u RHS Pthagorean identit _3.5 FIGURE (b) We graph each side of the equation to see whether the graphs coincide. From Figure we see that the graphs of cos u sec u cos u and sin u are identical. This confirms that the equation is an identit. NOW TRY EXERCISE 7 In Eample 3 it isn t eas to see how to change the right-hand side into the left-hand side, but it s definitel possible. Simpl notice that each step is reversible. In other words, if we start with the last epression in the proof and work backward through the steps, the right-hand side is transformed into the left-hand side. You will probabl agree, however, that it s more difficult to prove the identit this wa. That s wh it s often better to change the more complicated side of the identit into the simpler side. EXAMPLE 4 Proving an Identit b Combining Fractions Verif the identit tan sec sin sin SOLUTION Finding a common denominator and combining the fractions on the right-hand side of this equation, we get RHS sin sin sin sin Common denominator sin sin sin Simplif sin sin Pthagorean identit cos sin Factor cos a cos b tan sec LHS Reciprocal identities NOW TRY EXERCISE 79 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

5 SECTION 7. Trigonometric Identities 497 See the Prologue: Principles of Problem Solving, pages P P4. We multipl b sin u because we know b the difference of squares formula that sin u sin u sin u, and this is just cos u,a simpler epression. EUCLID (circa 300 B.C.) taught in Aleandria. His Elements is the most widel influential scientific book in histor. For 000 ears it was the standard introduction to geometr in the schools, and for man generations it was considered the best wa to develop logical reasoning. Abraham Lincoln, for instance, studied the Elements as a wa to sharpen his mind.the stor is told that King Ptolem once asked Euclid if there was a faster wa to learn geometr than through the Elements. Euclid replied that there is no roal road to geometr meaning b this that mathematics does not respect wealth or social status. Euclid was revered in his own time and was referred to b the title The Geometer or The Writer of the Elements. The greatness of the Elements stems from its precise, logical, and sstematic treatment of geometr. For dealing with equalit, Euclid lists the following rules, which he calls common notions.. Things that are equal to the same thing are equal to each other.. If equals are added to equals, the sums are equal. 3. If equals are subtracted from equals, the remainders are equal. 4. Things that coincide with one another are equal. 5. The whole is greater than the part. In Eample 5 we introduce something etra to the problem b multipling the numerator and the denominator b a trigonometric epression, chosen so that we can simplif the result. EXAMPLE 5 Proving an Identit b Introducing Something Etra cos u Verif the identit sec u tan u. sin u SOLUTION We start with the left-hand side and multipl the numerator and denominator b sin u: NOW TRY EXERCISE 53 Multipl numerator and denominator b sin u Epand denominator Pthagorean identit Cancel common factor Separate into two fractions Reciprocal identities Here is another method for proving that an equation is an identit. If we can transform each side of the equation separatel, b wa of identities, to arrive at the same result, then the equation is an identit. Eample 6 illustrates this procedure. EXAMPLE 6 Proving an Identit b Working with Both Sides Separatel cos u Verif the identit tan u. cos u sec u SOLUTION We prove the identit b changing each side separatel into the same epression. Suppl the reasons for each step: LHS RHS LHS cos u sin u cos u # sin u sin u sin u cos u cos u tan u sec u sec u sec u sec u sec u sec u sec u It follows that LHS RHS, so the equation is an identit. NOW TRY EXERCISE 8 cos u sin u sin u cos u sin u cos u sin u cos u sin u cos u cos u sec u tan u cos u sec u cos u cos u We conclude this section b describing the technique of trigonometric substitution, which we use to convert algebraic epressions to trigonometric ones. This is often useful in calculus, for instance, in finding the area of a circle or an ellipse. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

6 498 CHAPTER 7 Analtic Trigonometr EXAMPLE 7 Trigonometric Substitution Substitute sin u for in the epression and simplif. Assume that 0 u p/. SOLUTION Setting sin u, we have sin u Substitute sin u cos u Pthagorean identit cos u Take square root The last equalit is true because cos u 0 for the values of u in question. NOW TRY EXERCISE 9 7. EXERCISES CONCEPTS. An equation is called an identit if it is valid for values of the variable. The equation is an algebraic identit, and the equation sin cos is a trigonometric identit.. For an it is true that cos has the same value as cos. We epress this fact as the identit. SKILLS 3 Write the trigonometric epression in terms of sine and cosine, and then simplif. 3. cos t tan t 4. cos t csc t 5. sin u sec u 6. tan u csc u 7. tan sec 8. sec csc 9. sin u cot u cos u 0. cos u tan u. sec u cos u cot u. sin u csc u sin u 3 6 Simplif the trigonometric epression. 3. sin sec tan 4. cos 3 sin cos 5. cos tan 6. sec sec 7. sec sec cos 8. sec tan 9. csc sin cos 0. cos cot csc sec. sin u cos u cos u sin u. tan cos csc 3. tan cot A 4. sec csc A 5. tan u cosu tanu 6. cos sec tan 7 8 Consider the given equation. (a) Verif algebraicall that the equation is an identit. (b) Confirm graphicall that the equation is an identit. cos tan 7. csc sin 8. sec cos sec sin csc 9 90 Verif the identit. sin u tan 9. cos u 30. sin tan u sec cos u sec u cot sec 3. cot u 3. tan u csc 33. sin B cos B cot B csc B 34. cos sin cos sin 35. cota cosa sina csc a 36. csc 3csc sin4 cot 37. tan u cot u sec u csc u 38. sin cos sin cos 39. cos b cos b csc b cos sin 40. sec csc sin cos 4. sin cos sin cos sin cos 4. sin cos 4 sin cos sec t cos t 43. sin t sec t sin 44. sec tan sin csc sin cos cot sin tan 47. cot csc cos sin 48. sin 4 u cos 4 u sin u cos u 49. cos cot Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

7 SECTION 7. Trigonometric Identities cos sin cos 5. cos sin 5. tan cot sin cos sin a cos a tan a sec a 55. tan u sin u tan u sin u 56. cot u cos u cot u cos u sec tan csc cot 69. cot 70. sec 7. tan u sin u tan u sin u tan sin tan sin 7. tan sin tan sin 73. sec 4 tan 4 sec tan cos a sin a cos sin sin a cos a sin 57. sin cos sin sin tan 58. sin cos tan sin t cos t 59. sec t csc t sin t cos t 60. sec t csc t tan t cot t sec t csc t tan u tan u cos u sin u sec tan cos sec sec sec tan sec tan sec csc sin cos tan cot sec tan sin A cot A csc A cos A sin cos sin cos sec csc sin csc cos csc cot sec cos u sec u tan u sin u cos u sin u csc u sin u cos u cot u tan cos sin tan cos sin cos t tan t tan t sin t sin sec tan sin sec tan sec sec tan cos sin sin tan sec sin sin 8. tan cot sec csc 8. tan cot sec csc sec u cos u cot tan sec u cos u cot tan sin 3 cos 3 sin cos sin cos tan cot sin cos tan cot sin tan sec sin tan tan tan tan cot cot 89. tan cot 4 csc 4 sec sin a tan acos a cot a cos a sin a 9 96 Make the indicated trigonometric substitution in the given algebraic epression and simplif (see Eample 7). Assume that 0 u p/. 9., sin u 9., tan u 93., sec u 94. tan u 4, 95. 9, 3 sin u Graph f and g in the same viewing rectangle. Do the graphs suggest that the equation f g is an identit? Prove our answer f cos sin, g sin sin cos f tan sin, g sin f sin cos, g f cos 4 sin 4, g cos 0. Show that the equation is not an identit. (a) sin sin (b) sin sin sin (c) sec csc (d) csc sec sin cos DISCOVERY DISCUSSION WRITING 0. Cofunction Identities In the right triangle shown, eplain wh p/ u. Eplain how ou can obtain all si cofunction identities from this triangle for 0 u p/. u 5, 5 sec u Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

8 500 CHAPTER 7 Analtic Trigonometr 03. Graphs and Identities Suppose ou graph two functions, f and g, on a graphing device and their graphs appear identical in the viewing rectangle. Does this prove that the equation f g is an identit? Eplain. 04. Making Up Your Own Identit If ou start with a trigonometric epression and rewrite it or simplif it, then setting the original epression equal to the rewritten epression ields a trigonometric identit. For instance, from Eample we get the identit cos t tan t sin t sec t Use this technique to make up our own identit, then give it to a classmate to verif. 7. ADDITION AND SUBTRACTION FORMULAS Addition and Subtraction Formulas Evaluating Epressions Involving Inverse Trigonometric Functions Epressions of the form A sin B cos Addition and Subtraction Formulas We now derive identities for trigonometric functions of sums and differences. ADDITION AND SUBTRACTION FORMULAS Formulas for sine: Formulas for cosine: Formulas for tangent: sins t sin s cos t cos s sin t sins t sin s cos t cos s sin t coss t cos s cos t sin s sin t coss t cos s cos t sin s sin t tans t tans t tan s tan t tan s tan t tan s tan t tan s tan t P O s s+t Q t P _s Q PROOF OF ADDITION FORMULA FOR COSINE To prove the formula coss t cos s cos t sin s sin t, we use Figure. In the figure, the distances t, s t, and s have been marked on the unit circle, starting at P 0, 0 and terminating at Q, P, and Q 0, respectivel. The coordinates of these points are P 0, 0 P coss t, sins t Q 0 coss, sins Q cos t, sin t Since coss cos s and sins sin s, it follows that the point Q 0 has the coordinates Q 0 cos s, sin s. Notice that the distances between P 0 and P and between Q 0 and Q measured along the arc of the circle are equal. Since equal arcs are subtended b equal chords, it follows that dp 0, P dq 0, Q. Using the Distance Formula, we get FIGURE 3coss t 4 3sins t 04 cos t cos s sin t sin s Squaring both sides and epanding, we have These add to cos s t coss t sin s t cos t cos s cos t cos s sin t sin s sin t sin s These add to These add to Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

9 SECTION 7. Addition and Subtraction Formulas 50 JEAN BAPTISTE JOSEPH FOURIER ( ) is responsible for the most powerful application of the trigonometric functions (see the margin note on page 394). He used sums of these functions to describe such phsical phenomena as the transmission of sound and the flow of heat. Orphaned as a oung bo, Fourier was educated in a militar school, where he became a mathematics teacher at the age of 0. He was later appointed professor at the École Poltechnique but resigned this position to accompan Napoleon on his epedition to Egpt, where Fourier served as governor. After returning to France he began conducting eperiments on heat. The French Academ refused to publish his earl papers on this subject because of his lack of rigor. Fourier eventuall became Secretar of the Academ and in this capacit had his papers published in their original form. Probabl because of his stud of heat and his ears in the deserts of Egpt, Fourier became obsessed with keeping himself warm he wore several laers of clothes, even in the summer, and kept his rooms at unbearabl high temperatures. Evidentl, these habits overburdened his heart and contributed to his death at the age of 6. Using the Pthagorean identit sin u cos u three times gives Finall, subtracting from each side and dividing both sides b, we get which proves the Addition Formula for Cosine. PROOF OF SUBTRACTION FORMULA FOR COSINE Addition Formula for Cosine, we get This proves the Subtraction Formula for Cosine. Replacing t with t in the Addition Formula for Cosine Even-odd identities See Eercises 70 and 7 for proofs of the other Addition Formulas. EXAMPLE coss t cos s cos t sin s sin t coss t coss t Using the Addition and Subtraction Formulas Find the eact value of each epression. (a) cos 75 (b) cos p SOLUTION (a) Notice that Since we know the eact values of sine and cosine at 45 and 30, we use the Addition Formula for Cosine to get cos p cos a p 4 p 6 b NOW TRY EXERCISES 3 AND 9 coss t cos s cos t sin s sin t cos s cost sin s sint cos s cos t sin s sin t cos 75 cos45 30 cos 45 cos 30 sin 45 sin p (b) Since, the Subtraction Formula for Cosine gives p 4 p 6 cos p 4 cos p 6 sin p 4 sin p EXAMPLE Using the Addition Formula for Sine Find the eact value of the epression sin 0 cos 40 cos 0 sin 40. SOLUTION We recognize the epression as the right-hand side of the Addition Formula for Sine with s 0 and t 40. So we have sin 0 cos 40 cos 0 sin 40 sin0 40 sin 60 3 NOW TRY EXERCISE 5 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

10 50 CHAPTER 7 Analtic Trigonometr EXAMPLE 3 Proving a Cofunction Identit Prove the cofunction identit cos a p u b sin u. SOLUTION B the Subtraction Formula for Cosine, we have u r a π u cosa p ub b r sin u b NOW TRY EXERCISE The cofunction identit in Eample 3, as well as the other cofunction identities, can also be derived from the figure in the margin. EXAMPLE 4 Proving an Identit tan Verif the identit. tan tan a p 4 b Starting with the right-hand side and using the Addition Formula for Tan- SOLUTION gent, we get cos a p u b cos p cos u sin p sin u 0 # cos u # sin u sin u p RHS tan a p tan tan 4 b 4 tan p 4 tan tan LHS tan NOW TRY EXERCISE 5 The net eample is a tpical use of the Addition and Subtraction Formulas in calculus. EXAMPLE 5 If f sin, show that SOLUTION An Identit from Calculus f h f sin a h cos h h b cos a sin h h b f h f sin h sin h h sin a NOW TRY EXERCISE 6 sin cos h cos sin h sin h sin cos h cos sin h h cos h h b cos a sin h h b Definition of f Addition Formula for Sine Factor Separate the fraction Evaluating Epressions Involving Inverse Trigonometric Functions Epressions involving trigonometric functions and their inverses arise in calculus. In the net eamples we illustrate how to evaluate such epressions. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

11 SECTION 7. Addition and Subtraction Formulas 503 œ + ƒ FIGURE cos = tan ƒ= œ - EXAMPLE 6 Simplifing an Epression Involving Inverse Trigonometric Functions Write sincos tan as an algebraic epression in and, where and is an real number. SOLUTION Let u cos and f tan. Using the methods of Section 6.4, we sketch triangles with angles u and f such that cos u and tan f (see Figure ). From the triangles we have sin u cos f sin f From the Addition Formula for Sine we have sincos tan sinu f sin u cos f cos u sin f NOW TRY EXERCISES 43 AND 47 Addition Formula for Sine From triangles Factor EXAMPLE 7 Evaluating an Epression Involving Trigonometric Functions Evaluate sinu f, where sin u with u in Quadrant II and tan f with f in Quadrant III. SOLUTION We first sketch the angles u and f in standard position with terminal sides in the appropriate quadrants as in Figure 3. Since sin u /r 3 we can label a side and the hpotenuse in the triangle in Figure 3(a). To find the remaining side, we use the Pthagorean Theorem: r Pthagorean Theorem 3, r Solve for Because 0 Similarl, since tan f / 3 4, we can label two sides of the triangle in Figure 3(b) and then use the Pthagorean Theorem to find the hpotenuse. P (, ) _5 3 _3 _4 ƒ 5 P (, ) FIGURE 3 (a) (b) Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

12 504 CHAPTER 7 Analtic Trigonometr Now, to find sinu f, we use the Addition Formula for Sine and the triangles in Figure 3: sinu f sin u cos f cos u sin f Addition Formula A 3BA 4 5B A 5 3BA 3 5B From triangles Calculate NOW TRY EXERCISE 5 Epressions of the Form A sin B cos We can write epressions of the form A sin B cos in terms of a single trigonometric function using the Addition Formula for Sine. For eample, consider the epression 3 sin cos If we set f p/3, then cos f and sin f 3/, and we can write 3 sin cos cos f sin sin f cos sin f sin a p 3 b B A +B œ (A, B) We are able to do this because the coefficients and 3/ are precisel the cosine and sine of a particular number, in this case, p/3. We can use this same idea in general to write A sin B cos in the form k sin f. We start b multipling the numerator and denominator b A B to get A sin B cos A B A B a sin A B We need a number f with the propert that A B cos b 0 ƒ A cos f A A B and sin f B A B FIGURE 4 Figure 4 shows that the point A, B this propert. With this f we have in the plane determines a number f with precisel A sin B cos A B cos f sin sin f cos We have proved the following theorem. A B sin f SUMS OF SINES AND COSINES If A and B are real numbers, then A sin B cos k sin f where k A B and f satisfies cos f A A B and sin f B A B Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

13 SECTION 7. Addition and Subtraction Formulas 505 EXAMPLE 8 A Sum of Sine and Cosine Terms Epress 3 sin 4 cos in the form k sin f. SOLUTION B the preceding theorem, k A B The angle f has the propert that sin f 4 and cos f Using a calculator, we find f 53.. Thus NOW TRY EXERCISE 55 3 sin 4 cos 5 sin 53. EXAMPLE 9 Graphing a Trigonometric Function Write the function f sin 3 cos in the form k sin f, and use the new form to graph the function. π _ 3 SOLUTION Since A and B 3, we have k A B 3. The angle f satisfies cos f and sin f 3/. From the signs of these quantities we conclude that f is in Quadrant II. Thus f p/3. B the preceding theorem we can write f sin 3 cos sin a p 3 b _π π _ 0 π π Using the form FIGURE 5 _ π = 3 f sin a p 3 b we see that the graph is a sine curve with amplitude, period p/ p, and phase shift p/3. The graph is shown in Figure 5. NOW TRY EXERCISE EXERCISES CONCEPTS. If we know the values of the sine and cosine of and, we can find the value of sin b using the Formula for Sine. State the formula: sin.. If we know the values of the sine and cosine of and, we can find the value of cos b using the Formula for Cosine. State the formula: cos. SKILLS 3 4 Use an Addition or Subtraction Formula to find the eact value of the epression, as demonstrated in Eample. 3. sin sin 5 5. cos cos tan 5 8. tan 65 9p 9. sin 0.. tan a p. b 3. cos p Use an Addition or Subtraction Formula to write the epression as a trigonometric function of one number, and then find its eact value. 5. sin 8 cos 7 cos 8 sin 7 6. cos 0 cos 80 sin 0 sin cos 3p 7 8. cos p sin 3p 7 tan p 8 tan p 9 tan p 8 tan p 9 sin p cos 7p 5p sin a b tan 7p Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

14 506 CHAPTER 7 Analtic Trigonometr 9. 4 Prove the cofunction identit using the Addition and Subtraction Formulas.. tan a p u b cot u. cot a p u b tan u 3. sec a p u b csc u 4. csc a p u b sec u 5 4 Prove the identit. 5. tan 73 tan 3 tan 73 tan 3 0. cos 3p 5 cos a p 3p b sin 5 5 sin a p 5 b sin a p b cos 6. cos a p b sin 7. sin p sin 8. cos p cos 9. tan p tan 30. sin a p b sin a p b 3. cos a p 6 b sin a p 3 b 0 3. tan a p 4 b tan tan 33. sin sin cos sin 34. cos cos cos cos cot cot 35. cot cot cot cot cot 36. cot cot cot sin 37. tan tan cos cos cos 38. tan tan cos cos sin sin 39. tan cos cos 40. cos cos cos sin 4. sin z sin cos cos z cos sin cos z cos cos sin z sin sin sin z 4. tan tan z tanz tan tan z tanz Write the given epression in terms of and onl. 43. cossin tan 44. tansin cos 45. sintan tan 46. sinsin cos Find the eact value of the epression. 47. sinacos tan B cosasin cot 3B 49. tanasin 3 4 cos 3B 50. sinacos 3 tan B 5 54 Evaluate each epression under the given conditions. 5. cosu f; cos u 3 5, u in Quadrant IV, tan f 3, f in Quadrant II. 5. sinu f; tan u 4 3, u in Quadrant III, sin f 0/0, f in Quadrant IV 53. sinu f; sin u 5 3, u in Quadrant I, cos f 5/5, f in Quadrant II 54. tanu f; cos u, u in Quadrant III, sin f 3 4, f in Quadrant II Write the epression in terms of sine onl sin cos 56. sin cos 57. 5sin cos sin p 33 cos p (a) Epress the function in terms of sine onl. (b) Graph the function. 59. g cos 3 sin Let g cos. Show that g h g cos a h 6. Show that if b a p/, then sin a cos b Refer to the figure. Show that a b g, and find tang. 6 4 å (a) If L is a line in the plane and u is the angle formed b the line and the -ais as shown in the figure, show that the slope m of the line is given b 0 m tan u cos h h L (b) Let L and L be two nonparallel lines in the plane with slopes m and m, respectivel. Let c be the acute angle formed b the two lines (see the following figure). Show that tan c m m m m f sin cos b sin a sin h h b Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

15 SECTION 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas 507 (c) Find the acute angle formed b the two lines (d) Show that if two lines are perpendicular, then the slope of one is the negative reciprocal of the slope of the other. [Hint: First find an epression for cot c.] (a) Graph the function and make a conjecture, then (b) prove that our conjecture is true. 65. L 0 L sin a p 4 b sin a p 4 b ψ= - 3 and cos p cos p4 67. Find A B C in the figure. [Hint: First use an addition formula to find tana B.] 69. Interference Two identical tuning forks are struck, one a fraction of a second after the other. The sounds produced are modeled b f t C sin vt and f t C sinvt a. The two sound waves interfere to produce a single sound modeled b the sum of these functions ft C sin vt C sinvt a (a) Use the Addition Formula for Sine to show that f can be written in the form ft A sin vt B cos vt, where A and B are constants that depend on a. (b) Suppose that C 0 and a p/3. Find constants k and f so that ft k sinvt f. DISCOVERY DISCUSSION WRITING 70. Addition Formula for Sine In the tet we proved onl the Addition and Subtraction Formulas for Cosine. Use these formulas and the cofunction identities sin cos a p b cos sin a p b A B C to prove the Addition Formula for Sine. [Hint: To get started, use the first cofunction identit to write sins t cos a p s tb APPLICATIONS 68. Adding an Echo A digital dela device echoes an input signal b repeating it a fied length of time after it is received. If such a device receives the pure note f t 5 sin t and echoes the pure note f t 5 cos t, then the combined sound is ft f t f t. (a) Graph ft and observe that the graph has the form of a sine curve k sint f. (b) Find k and f. cos aa p s b t b and use the Subtraction Formula for Cosine.] 7. Addition Formula for Tangent Use the Addition Formulas for Cosine and Sine to prove the Addition Formula for Tangent. [Hint: Use sins t tans t coss t and divide the numerator and denominator b cos s cos t.] 7.3 DOUBLE-ANGLE,HALF-ANGLE, AND PRODUCT-SUM FORMULAS Double-Angle Formulas Half-Angle Formulas Simplifing Epressions Involving Inverse Trigonometric Functions Product-Sum Formulas The identities we consider in this section are consequences of the addition formulas. The Double-Angle Formulas allow us to find the values of the trigonometric functions at from their values at. The Half-Angle Formulas relate the values of the trigonometric functions at to their values at. The Product-Sum Formulas relate products of sines and cosines to sums of sines and cosines. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

16 508 CHAPTER 7 Analtic Trigonometr Double-Angle Formulas The formulas in the following bo are immediate consequences of the addition formulas, which we proved in the preceding section. DOUBLE-ANGLE FORMULAS Formula for sine: Formulas for cosine: sin sin cos cos cos sin sin cos Formula for tangent: tan tan tan The proofs for the formulas for cosine are given here. You are asked to prove the remaining formulas in Eercises 35 and 36. PROOF OF DOUBLE-ANGLE FORMULAS FOR COSINE cos cos cos cos sin sin cos sin The second and third formulas for cos are obtained from the formula we just proved and the Pthagorean identit. Substituting cos sin gives cos cos sin sin sin sin The third formula is obtained in the same wa, b substituting sin cos. EXAMPLE Using the Double-Angle Formulas 3 If cos and is in Quadrant II, find cos and sin. SOLUTION Using one of the Double-Angle Formulas for Cosine, we get To use the formula sin sin cos, we need to find sin first. We have where we have used the positive square root because sin is positive in Quadrant II. Thus NOW TRY EXERCISE 3 cos cos a 3 b sin cos A 3B 5 3 sin sin cos a 5 3 ba 3 b 45 9 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

17 SECTION 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas 509 EXAMPLE Write cos 3 in terms of cos. A Triple-Angle Formula SOLUTION cos 3 cos cos cos sin sin cos cos sin cos sin cos 3 cos sin cos cos 3 cos cos cos cos 3 cos cos cos 3 4 cos 3 3 cos NOW TRY EXERCISE 0 Addition formula Double-Angle Formulas Epand Pthagorean identit Epand Simplif Eample shows that cos 3 can be written as a polnomial of degree 3 in cos. The identit cos cos shows that cos is a polnomial of degree in cos. In fact, for an natural number n, we can write cos n as a polnomial in cos of degree n (see the note following Eercise 0). The analogous result for sin n is not true in general. EXAMPLE 3 Proving an Identit sin 3 Prove the identit 4 cos sec. sin cos SOLUTION sin 3 sin sin cos sin cos We start with the left-hand side: sin cos cos sin sin cos sin cos cos sin cos sin cos sin cos sin cos cos cos cos cos cos cos 4 cos sec NOW TRY EXERCISE 8 cos sin cos sin cos Addition Formula Double-Angle Formulas Separate fraction Cancel Separate fraction Reciprocal identit Half-Angle Formulas The following formulas allow us to write an trigonometric epression involving even powers of sine and cosine in terms of the first power of cosine onl. This technique is important in calculus. The Half-Angle Formulas are immediate consequences of these formulas. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

18 50 CHAPTER 7 Analtic Trigonometr FORMULAS FOR LOWERING POWERS sin cos tan cos cos cos cos PROOF The first formula is obtained b solving for sin in the double-angle formula cos sin. Similarl, the second formula is obtained b solving for cos in the Double-Angle Formula cos cos. The last formula follows from the first two and the reciprocal identities: EXAMPLE 4 Lowering Powers in a Trigonometric Epression Epress sin cos in terms of the first power of cosine. SOLUTION tan sin cos We use the formulas for lowering powers repeatedl: sin cos a cos cos cos cos ba 8 8 cos 4 cos 4 8 Another wa to obtain this identit is to use the Double-Angle Formula for Sine in the form sin cos sin. Thus sin cos 4 sin 4 cos 4 a b 4 cos cos 4 8 cos b 4 4 cos cos cos cos 4 a b NOW TRY EXERCISE HALF-ANGLE FORMULAS sin u cos u B tan u cos u sin u cos u cos u B The choice of the or sign depends on the quadrant in which u/ lies. sin u cos u Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

19 SECTION 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas 5 PROOF We substitute u/ in the formulas for lowering powers and take the square root of each side. This gives the first two Half-Angle Formulas. In the case of the Half- Angle Formula for Tangent we get u tan cos u B cos u cos u cos u a ba B cos u cos u b B cos u cos u 0 cos u 0 0 sin u 0 Multipl numerator and denominator b cos u Simplif A 0 A 0 and cos u sin u Now, cos u is nonnegative for all values of u. It is also true that sin u and tanu/ alwas have the same sign. (Verif this.) It follows that tan u cos u sin u The other Half-Angle Formula for Tangent is derived from this b multipling the numerator and denominator b cos u. EXAMPLE 5 Find the eact value of sin.5. Using a Half-Angle Formula SOLUTION Since.5 is half of 45, we use the Half-Angle Formula for Sine with u 45. We choose the sign because.5 is in the first quadrant: sin 45 cos 45 B B / B 4 Half-Angle Formula cos 45 / Common denominator NOW TRY EXERCISE 7 3 Simplif EXAMPLE 6 Using a Half-Angle Formula Find tanu/ if sin u 5 and u is in Quadrant II. SOLUTION To use the Half-Angle Formula for Tangent, we first need to find cos u. Since cosine is negative in Quadrant II, we have Thus NOW TRY EXERCISE 37 cos u sin u tan A 5B 5 u cos u sin u /5 5 5 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

20 5 CHAPTER 7 Analtic Trigonometr Evaluating Epressions Involving Inverse Trigonometric Functions Epressions involving trigonometric functions and their inverses arise in calculus. In the net eamples we illustrate how to evaluate such epressions. FIGURE œ - EXAMPLE 7 Simplifing an Epression Involving an Inverse Trigonometric Function Write sin cos as an algebraic epression in onl, where. SOLUTION Let u cos, and sketch a triangle as in Figure. We need to find sin u, but from the triangle we can find trigonometric functions of u onl, not u. So we use the Double-Angle Formula for Sine. sin cos sin u sin u cos u NOW TRY EXERCISES 43 AND 47 cos u Double-Angle Formula From the triangle P (, ) 5 EXAMPLE 8 Evaluating an Epression Involving Inverse Trigonometric Functions Evaluate sin u, where cos u 5 with u in Quadrant II. SOLUTION We first sketch the angle u in standard position with terminal side in Quadrant II as in Figure. Since cos u /r 5, we can label a side and the hpotenuse of the triangle in Figure. To find the remaining side, we use the Pthagorean Theorem: _ FIGURE Pthagorean Theorem, r 5 Solve for Because 0 We can now use the Double-Angle Formula for Sine: NOW TRY EXERCISE 5 r 5 sin u sin u cos u a b a 5 5 b 4 5 Double-Angle Formula From the triangle Simplif Product-Sum Formulas It is possible to write the product sin u cos as a sum of trigonometric functions. To see this, consider the addition and subtraction formulas for the sine function: sinu sin u cos cos u sin sinu sin u cos cos u sin Adding the left- and right-hand sides of these formulas gives sinu sinu sin u cos Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

21 SECTION 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas 53 Dividing b gives the formula sin u cos 3sinu sinu 4 The other three Product-to-Sum Formulas follow from the addition formulas in a similar wa. PRODUCT-TO-SUM FORMULAS sin u cos 3sinu sinu 4 cos u sin 3sinu sinu 4 cos u cos 3cosu cosu 4 sin u sin 3cosu cosu 4 EXAMPLE 9 Epressing a Trigonometric Product as a Sum Epress sin 3 sin 5 as a sum of trigonometric functions. SOLUTION Using the fourth Product-to-Sum Formula with u 3 and 5 and the fact that cosine is an even function, we get sin 3 sin 5 3cos3 5 cos3 54 cos cos 8 cos cos 8 NOW TRY EXERCISE 55 The Product-to-Sum Formulas can also be used as Sum-to-Product Formulas. This is possible because the right-hand side of each Product-to-Sum Formula is a sum and the left side is a product. For eample, if we let in the first Product-to-Sum Formula, we get so u sin cos and sin sin sin sin sin cos The remaining three of the following Sum-to-Product Formulas are obtained in a similar manner. SUM-TO-PRODUCT FORMULAS sin sin sin sin sin cos cos cos cos cos cos sin cos sin cos sin Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

22 54 CHAPTER 7 Analtic Trigonometr EXAMPLE 0 Write sin 7 sin 3 as a product. SOLUTION Epressing a Trigonometric Sum as a Product The first Sum-to-Product Formula gives NOW TRY EXERCISE 6 sin 7 sin 3 sin 7 3 sin 5 cos cos 7 3 EXAMPLE Proving an Identit sin 3 sin Verif the identit tan. cos 3 cos SOLUTION We appl the second Sum-to-Product Formula to the numerator and the third formula to the denominator: LHS cos sin 3 sin cos 3 cos cos cos sin cos cos sin tan RHS cos NOW TRY EXERCISE sin 3 cos 3 Sum-to-Product Formulas Simplif Cancel 7.3 EXERCISES CONCEPTS. If we know the values of sin and cos, we can find the value of sin b using the Formula for Sine. State the formula: sin.. If we know the value of cos and the quadrant in which / lies, we can find the value of sin/ b using the Formula for Sine. State the formula: sin/ SKILLS 3 0 Find sin, cos, and tan from the given information. 3. sin 5, in Quadrant I tan, in Quadrant II 5. cos 4 5, csc 0 6. csc 4, tan sin, in Quadrant III 8. sec, in Quadrant IV 3 9. tan, cos 0 0. cot 3, sin 0 6 Use the formulas for lowering powers to rewrite the epression in terms of the first power of cosine, as in Eample 4.. sin 4. cos 4 3. cos sin 4 4. cos 4 sin 5. cos 4 sin 4 6. cos Use an appropriate Half-Angle Formula to find the eact value of the epression. 7. sin 5 8. tan 5 9. tan.5 0. sin 75. cos 65. cos.5 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

23 SECTION 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas tan p 4. cos 3p cos p 6. tan 5p 7. sin 9p 8. sin p Simplif the epression b using a Double-Angle Formula or a Half-Angle Formula. 9. (a) sin 8 cos 8 (b) sin 3u cos 3u tan 7 tan 7u 30. (a) (b) tan 7 tan 7u 3. (a) cos 34 sin 34 (b) cos 5u sin 5u 3. (a) cos u u sin (b) sin u cos u 33. (a) sin 8 cos 4u (b) cos 8 sin 4u 34. (a) cos 30 cos 8u (b) B B 35. Use the Addition Formula for Sine to prove the Double-Angle Formula for Sine. 36. Use the Addition Formula for Tangent to prove the Double- Angle Formula for Tangent Find sin, cos, and tan from the given information. 37. sin 3 5, cos, csc 3, tan, sec 3, cot 5, Write the given epression as an algebraic epression in. 43. sin tan 44. tan cos 45. sina cos B 46. cos sin Find the eact value of the given epression. 47. sina cos B 49. seca sin 4B 50. tana cos 3B 5 54 Evaluate each epression under the given conditions. 5. cos u; sin u 3 5, u in Quadrant III 5. sinu/; tan u 5, u in Quadrant IV 53. sin u; sin u 7, u in Quadrant II 54. tan u; cos u 3 5, u in Quadrant I Write the product as a sum. cosa tan 5 B 55. sin cos sin sin cos sin cos 5 cos cos 4 cos sin cos Write the sum as a product. 6. sin 5 sin 3 6. sin sin cos 4 cos cos 9 cos 65. sin sin sin 3 sin Find the value of the product or sum. 67. sin 5.5 sin cos 37.5 cos cos 37.5 sin sin 75sin 5 7. cos 55 cos cos p Prove the identit. 73. cos 5 sin 5 cos sin 8 sin 4 cos sin cos sin tan 76. sin tan sin cos cos sin sin 78. sec csc sin tan cot 79. sin tan cot 80. cot tan tan 8. tan 3 3 tan tan3 3 tan 8. 4sin 6 cos sin 83. cos 4 sin 4 cos 84. tan a p cos 5p sin sin 5 tan 3 cos cos 5 sin 3 sin 7 cot cos 3 cos 7 sin 0 cos 5 sin 9 sin cos 4 sin sin 3 sin 5 tan 3 cos cos 3 cos 5 sin sin cos cos b sin sin tan a b sin sin 90. tan cos cos 9. Show that sin 30 sin 0 sin Show that cos 00 cos 00 sin Show that sin 45 sin 5 sin Show that cos 87 cos 33 sin 63. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

24 56 CHAPTER 7 Analtic Trigonometr 95. Prove the identit sin sin sin 3 sin 4 sin 5 tan 3 cos cos cos 3 cos 4 cos Use the identit n times to show that sin 3 cos (a) Graph f and make a conjecture. sin cos (b) Prove the conjecture ou made in part (a). 98. (a) Graph f cos sin and make a conjecture. (b) Prove the conjecture ou made in part (a). 99. Let f sin 6 sin 7. (a) Graph f. (b) Verif that f cos 3 sin. (c) Graph cos and cos, together with the graph in part (a), in the same viewing rectangle. How are these graphs related to the graph of f? 00. Let 3 p/3 and let cos. Use the result of Eample to show that satisfies the equation NOTE This equation has roots of a certain kind that are used to show that the angle p/3 cannot be trisected b using a ruler and compass onl. 0. (a) Show that there is a polnomial Pt of degree 4 such that cos 4 Pcos (see Eample ). (b) Show that there is a polnomial Qt of degree 5 such that cos 5 Qcos. NOTE In general, there is a polnomial P n t of degree n such that cos n P n cos. These polnomials are called Tchebcheff polnomials, after the Russian mathematician P. L. Tchebcheff (8 894). 0. In triangle ABC (see the figure) the line segment s bisects angle C. Show that the length of s is given b ab cos s a b [Hint: Use the Law of Sines.] B a C sin sin cos sin n n sin cos cos cos 4... cos n s 03. If A, B, and C are the angles in a triangle, show that sin A sin B sin C 4 sin A sin B sin C 04. A rectangle is to be inscribed in a semicircle of radius 5 cm as shown in the following figure. (a) Show that the area of the rectangle is modeled b the function Au 5 sin u (b) Find the largest possible area for such an inscribed rectangle. b A (c) Find the dimensions of the inscribed rectangle with the largest possible area. APPLICATIONS 05. Sawing a Wooden Beam A rectangular beam is to be cut from a clindrical log of diameter 0 in. (a) Show that the cross-sectional area of the beam is modeled b the function where u is as shown in the figure. (b) Show that the maimum cross-sectional area of such a beam is 00 in. [Hint: Use the fact that sin u achieves its maimum value at u p/.] 0 in. Au 00 sin u 06. Length of a Fold The lower right-hand corner of a long piece of paper 6 in. wide is folded over to the left-hand edge as shown. The length L of the fold depends on the angle u. Show that L 3 sin u cos u 07. Sound Beats When two pure notes that are close in frequenc are plaed together, their sounds interfere to produce beats; that is, the loudness (or amplitude) of the sound alternatel increases and decreases. If the two notes are given b f t cos t and f t cos 3t the resulting sound is ft f t f t. (a) Graph the function ft. (b) Verif that ft cos t cos t. L 6 in. 5 cm 0 in. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

25 SECTION 7.4 Basic Trigonometric Equations 57 (c) Graph cos t and cos t, together with the graph in part (a), in the same viewing rectangle. How do these graphs describe the variation in the loudness of the sound? 08. Touch-Tone Telephones When a ke is pressed on a touch-tone telephone, the kepad generates two pure tones, which combine to produce a sound that uniquel identifies the ke. The figure shows the low frequenc f and the high frequenc f associated with each ke. Pressing a ke produces the sound wave sinpf t sinpf t. (a) Find the function that models the sound produced when the 4 ke is pressed. (b) Use a Sum-to-Product Formula to epress the sound generated b the 4 ke as a product of a sine and a cosine function. (c) Graph the sound wave generated b the 4 ke, from t 0 to t s. Low frequenc f 697 Hz 770 Hz 4 85 Hz 7 94 Hz * High frequenc f Hz # DISCOVERY DISCUSSION WRITING 09. Geometric Proof of a Double-Angle Formula Use the figure to prove that sin u sin u cos u. A O [Hint: Find the area of triangle ABC in two different was. You will need the following facts from geometr: An angle inscribed in a semicircle is a right angle, so ACB is a right angle. The central angle subtended b the chord of a circle is twice the angle subtended b the chord on the circle, so BOC is u.] DISCOVERY PROJECT Where to Sit at the Movies In this project we use trigonometr to find the best location to observe such things as a painting or a movie. You can find the project at the book companion website: C B 7.4 BASIC TRIGONOMETRIC EQUATIONS Basic Trigonometric Equations Solving Trigonometric Equations b Factoring An equation that contains trigonometric functions is called a trigonometric equation. For eample, the following are trigonometric equations: sin u cos u The first equation is an identit that is, it is true for ever value of the variable u. The other two equations are true onl for certain values of u. To solve a trigonometric equation, we find all the values of the variable that make the equation true. Basic Trigonometric Equations sin u 0 tan u 0 Solving an trigonometric equation alwas reduces to solving a basic trigonometric equation an equation of the form Tu c, where T is a trigonometric function and c is a constant. In the net three eamples we solve such basic equations. EXAMPLE Solving a Basic Trigonometric Equation Solve the equation sin u. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

26 58 CHAPTER 7 Analtic Trigonometr = 5π 6 SOLUTION Find the solutions in one period. Because sine has period p, we first find the solutions in an interval of length p. To find these solutions, we look at the unit circle in Figure. We see that sin u in Quadrants I and II, so the solutions in the interval 30, p are _ = π 6 0 u p u 5p 6 6 Find all solutions. Because the sine function repeats its values ever p units, we get all solutions of the equation b adding integer multiples of p to these solutions: _ u p 6 kp u 5p 6 kp FIGURE where k is an integer. Figure gives a graphical representation of the solutions. =ß = 7π _ 6 π 6 5π 6 π 3π 6 7π 6 5π 6 FIGURE _ NOW TRY EXERCISE 5 EXAMPLE Solve the equation Solving a Basic Trigonometric Equation cos u, and list eight specific solutions. SOLUTION Find the solutions in one period. Because cosine has period p, we first find the solutions in an interval of length p. From the unit circle in Figure 3 we see that cos u /in Quadrants II and III, so the solutions in the interval 30, p are u 3p 4 u 5p 4 = 5π 4 = 3π 4 _ FIGURE 3 œ Find all solutions. Because the cosine function repeats its values ever p units, we get all solutions of the equation b adding integer multiples of p to these solutions: u 3p 4 kp u 5p 4 kp Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

27 where k is an integer. You can check that for k, 0,, we get the following specific solutions: u 5p 4, 3p 4, 3p 4, 5p 4, p 4, 3p 4, 9p 4, p 4 k k 0 k k Figure 4 gives a graphical representation of the solutions. _ 5π 4 _ 3π 4 0 3π 4 e π SECTION 7.4 Basic Trigonometric Equations 59 5π 4 π =cos π 4 3π 4 FIGURE 4 NOW TRY EXERCISE 7 _ œ = _ EXAMPLE 3 Solving a Basic Trigonometric Equation Solve the equation cos u SOLUTION Find the solutions in one period. We first find one solution b taking cos of each side of the equation. cos u 0.65 Given equation u cos 0.65 Take cos of each side u 0.86 Calculator (in radian mode) Because cosine has period p, we net find the solutions in an interval of length p. To find these solutions, we look at the unit circle in Figure 5. We see that cos u 0.85 in Quadrants I and IV, so the solutions are u 0.86 u p =0.86 _ = FIGURE 5 _ Find all solutions. To get all solutions of the equation, we add integer multiples of p to these solutions: u 0.86 kp u 5.4 kp where k is an integer. NOW TRY EXERCISE Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

28 50 CHAPTER 7 Analtic Trigonometr EXAMPLE 4 Solving a Basic Trigonometric Equation Solve the equation tan u. SOLUTION Find the solutions in one period. We first find one solution b taking tan of each side of the equation: tan u Given equation u tan Take tan of each side u. Calculator (in radian mode) B the definition of tan, the solution that we obtained is the onl solution in the interval p/, p/ (which is an interval of length p). Find all solutions. Since tangent has period p, we get all solutions of the equation b adding integer multiples of p: u. kp where k is an integer. A graphical representation of the solutions is shown in Figure 6. You can check that the solutions shown in the graph correspond to k, 0,,, 3. =tan = π _ π FIGURE 6 _.0. NOW TRY EXERCISE 3 In the net eample we solve trigonometric equations that are algebraicall equivalent to basic trigonometric equations. EXAMPLE 5 Solving Trigonometric Equations Find all solutions of the equation. (a) sin u 0 (b) tan u 3 0 SOLUTION (a) We start b isolating sin u: sin u 0 sin u Given equation Add sin u Divide b This last equation is the same as that in Eample. The solutions are u p 6 kp u 5p 6 kp where k is an integer. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

29 SECTION 7.4 Basic Trigonometric Equations 5 (b) We start b isolating tan u: tan u 3 0 tan u 3 tan u 3 Given equation Add 3 Take the square root Because tangent has period p, we first find the solutions in an interval of length p. In the interval p/, p/ the solutions are u p/3 and u p/3. To get all solutions, we add integer multiples of p to these solutions: u p 3 kp u p 3 kp where k is an integer. NOW TRY EXERCISES 7 AND 33 Zero-Product Propert If AB 0, then A 0 or B 0. Equation of Quadratic Tpe C 7C 3 0 C C 3 0 Solving Trigonometric Equations b Factoring Factoring is one of the most useful techniques for solving equations, including trigonometric equations. The idea is to move all terms to one side of the equation, factor, and then use the Zero-Product Propert (see Section.5). EXAMPLE 6 A Trigonometric Equation of Quadratic Tpe Solve the equation cos u 7 cos u 3 0. SOLUTION We factor the left-hand side of the equation. cos u 7 cos u 3 0 cos u cos u 3 0 Given equation Factor cos u 0 or cos u 3 0 Set each factor equal to 0 cos u or cos u 3 Solve for cos u Because cosine has period p, we first find the solutions in the interval 30, p. For the first equation the solutions are u p/3 and u 5p/3 (see Figure 7). The second equation has no solution because cos u is never greater than. Thus the solutions are u p 3 kp u 5p 3 kp where k is an integer. = π 3 _ = 5π FIGURE 7 _ NOW TRY EXERCISE 4 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

30 5 CHAPTER 7 Analtic Trigonometr =π+0.93 _ FIGURE 8 _0.8 _ =_0.93 EXAMPLE 7 Solving a Trigonometric Equation b Factoring Solve the equation 5 sin u cos u 4 cos u 0. SOLUTION We factor the left-hand side of the equation: 5 sin u cos u cos u 0 cos u5 sin u 0 Given equation Factor cos u 0 or 5 sin u 4 0 sin u 0.8 Set each factor equal to 0 Solve for sin u Because sine and cosine have period p, we first find the solutions of these equations in an interval of length p. For the first equation the solutions in the interval 30, p are u p/ and u 3p/. To solve the second equation, we take sin of each side: sin u 0.80 u sin 0.80 u 0.93 Second equation Take sin of each side Calculator (in radian mode) So the solutions in an interval of length p are u 0.93 and u p (see Figure 8). We get all the solutions of the equation b adding integer multiples of p to these solutions. u p kp, u 3p kp, u 0.93 kp, u 4.07 kp where k is an integer. NOW TRY EXERCISE EXERCISES CONCEPTS. Because the trigonometric functions are periodic, if a basic trigonometric equation has one solution, it has (several/infinitel man) solutions.. The basic equation sin has (no/one/infinitel man) solutions, whereas the basic equation sin 0.3 has (no/one/infinitel man) solutions. 3. We can find some of the solutions of sin 0.3 graphicall b graphing sin and. Use the graph below to estimate some of the solutions. π 4. We can find the solutions of sin 0.3 algebraicall. (a) First we find the solutions in the interval 3p, p4. We get one such solution b taking sin to get. The other solution in this interval is. (b) We find all solutions b adding multiples of to the solutions in 3p, p4. The solutions are SKILLS 5 6 Solve the given equation. and. 5. sin u cos u 8. sin u cos u 3 9. cos u 4 0. sin u 0.3. sin u cos u tan u 3 4. tan u 5. tan u 5 6. tan u 3 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

31 SECTION 7.4 Basic Trigonometric Equations Solve the given equation, and list si specific solutions. 7. cos u 3 8. cos u 9. sin u 0.. cos u tan u Find all solutions of the given equation. 5. cos u 0 6. sin u 0 7. sin u sin u cos u tan u 0 3. cot u cos u tan u sin u sec u csc u Solve the given equation tan u 4 cos u 0 tan u 6 sin u 0 4 cos u 4 cos u 0 sin u sin u 0 3 sin u 7 sin u 0 tan 4 u 3 tan u 36 0 cos u 7 cos u 3 0 sin u sin u cos u cos u sin u 5 sin u sin u sin u tan 3 u tan u 5. cos u sin u 0 5. sec u cos u cos u sin u cos u tan u sin u sin u tan u sin u tan u cos u sin u 3 cos u 0 APPLICATIONS 57. Refraction of Light It has been observed since ancient times that light refracts or bends as it travels from one medium to another (from air to water, for eample). If is the speed of light in one medium and its speed in another medium, then according to Snell s Law, sin u sin u sin u 3 tan u.5 sin u 0.9 cos u 0 4 sin u 3 0 where u is the angle of incidence and u is the angle of refraction (see the figure). The number / is called the inde of refraction. The inde of refraction for several substances is given in the table. If a ra of light passes through the surface of a lake at an angle of incidence of 70, what is the angle of refraction? Water Air Substance 58. Total Internal Reflection When light passes from a more-dense to a less-dense medium from glass to air, for eample the angle of refraction predicted b Snell s Law (see Eercise 57) can be 90 or larger. In this case the light beam is actuall reflected back into the denser medium. This phenomenon, called total internal reflection, is the principle behind fiber optics. Set u 90 in Snell s Law, and solve for u to determine the critical angle of incidence at which total internal reflection begins to occur when light passes from glass to air. (Note that the inde of refraction from glass to air is the reciprocal of the inde from air to glass.) 59. Phases of the Moon As the moon revolves around the earth, the side that faces the earth is usuall just partiall illuminated b the sun. The phases of the moon describe how much of the surface appears to be in sunlight. An astronomical measure of phase is given b the fraction F of the lunar disc that is lit. When the angle between the sun, earth, and moon is u (0 u 360), then F cos u Refraction from air to substance Water.33 Alcohol.36 Glass.5 Diamond.4 Determine the angles u that correspond to the following phases: (a) F 0 (new moon) (b) F 0.5 (a crescent moon) (c) F 0.5 (first or last quarter) (d) F (full moon) DISCOVERY DISCUSSION WRITING 60. Equations and Identities Which of the following statements is true? A. Ever identit is an equation. B. Ever equation is an identit. Give eamples to illustrate our answer. Write a short paragraph to eplain the difference between an equation and an identit. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

32 54 CHAPTER 7 Analtic Trigonometr 7.5 MORE TRIGONOMETRIC EQUATIONS Solving Trigonometric Equations b Using Identities Equations with Trigonometric Functions of Multiples of Angles In this section we solve trigonometric equations b first using identities to simplif the equation. We also solve trigonometric equations in which the terms contain multiples of angles. Solving Trigonometric Equations b Using Identities In the net two eamples we use trigonometric identities to epress a trigonometric equation in a form in which it can be factored. EXAMPLE Using a Trigonometric Identit Solve the equation sin u cos u. SOLUTION We first need to rewrite this equation so that it contains onl one trigonometric function. To do this, we use a trigonometric identit: sin u cos u sin u sin u sin u sin u 0 sin u sin u 0 Given equation Pthagorean identit Put all terms on one side Factor sin u 0 or sin u 0 Set each factor equal to 0 sin u or sin u Solve for sin u Solve for u in the u p or u 3p 6, 5p 6 interval 30, p Because sine has period p, we get all the solutions of the equation b adding integer multiples of p to these solutions. Thus the solutions are u p 6 kp u 5p 6 kp u 3p kp where k is an integer. NOW TRY EXERCISE 3 EXAMPLE Using a Trigonometric Identit Solve the equation sin u cos u 0. SOLUTION The first term is a function of u, and the second is a function of u, so we begin b using a trigonometric identit to rewrite the first term as a function of u onl: sin u cos u 0 sin u cos u cos u 0 cos u sin u 0 Given equation Double-Angle Formula Factor Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

33 SECTION 7.5 More Trigonometric Equations 55 cos u 0 or sin u 0 Set each factor equal to 0 sin u Solve for sin u u p, 3p or u p 6, 5p 6 Solve for u in 30, p Both sine and cosine have period p, so we get all the solutions of the equation b adding integer multiples of p to these solutions. Thus the solutions are u p kp u 3p kp u p 6 kp u 5p 6 kp where k is an integer. NOW TRY EXERCISES 7 AND EXAMPLE 3 Squaring and Using an Identit Solve the equation cos u sin u in the interval 30, p. SOLUTION To get an equation that involves either sine onl or cosine onl, we square both sides and use a Pthagorean identit: cos u 0 cos u 0 cos u sin u cos u cos u sin u cos u cos u cos u cos u cos u 0 cos ucos u 0 Given equation Square both sides Pthagorean identit Simplif Factor or cos u 0 Set each factor equal to 0 or cos u Solve for cos u u p, 3p or u p Solve for u in 30, p Because we squared both sides, we need to check for etraneous solutions. From Check Your Answers we see that the solutions of the given equation are p/ and p. CHECK YOUR ANSWER u p u 3p u p p cos sin p 0 3p cos sin 3p cos p sin p 0 0 NOW TRY EXERCISE 3 EXAMPLE 4 Finding Intersection Points Find the values of for which the graphs of f sin and g cos intersect. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

34 56 CHAPTER 7 Analtic Trigonometr SOLUTION : Graphical The graphs intersect where f g. In Figure we graph sin and cos on the same screen, for between 0 and p. Using TRACE or the intersect command on the graphing calculator, we see that the two points of intersection in this interval occur where and Since sine and cosine are periodic with period p, the intersection points occur where kp and 3.97 kp where k is an integer FIGURE _.5 Intersection X= Y= (a) _.5 Intersection X= Y= (b) SOLUTION : Algebraic To find the eact solution, we set f g and solve the resulting equation algebraicall: sin cos Equate functions Since the numbers for which cos 0 are not solutions of the equation, we can divide both sides b cos : sin Divide b cos cos tan Reciprocal identit The onl solution of this equation in the interval p/, p/ is p/4. Since tangent has period p, we get all solutions of the equation b adding integer multiples of p: p 4 kp where k is an integer. The graphs intersect for these values of. You should use our calculator to check that, rounded to three decimals, these are the same values that we obtained in Solution. NOW TRY EXERCISE 35 Equations with Trigonometric Functions of Multiples of Angles When solving trigonometric equations that involve functions of multiples of angles, we first solve for the multiple of the angle, then divide to solve for the angle. EXAMPLE 5 A Trigonometric Equation Involving a Multiple of an Angle Consider the equation sin 3u 0. (a) Find all solutions of the equation. (b) Find the solutions in the interval 30, p. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

35 SECTION 7.5 More Trigonometric Equations 57 SOLUTION (a) We first isolate sin 3u and then solve for the angle 3u. sin 3u 0 Given equation sin 3u Add sin 3u Divide b 3u p 6, 5p 6 Solve for 3u in the interval 30, p (see Figure ) To get all solutions, we add integer multiples of p to these solutions. So the solutions are of the form 3u p 6 kp 3u 5p 6 kp To solve for u, we divide b 3 to get the solutions where k is an integer. u p 8 kp 3 u 5p kp 8 3 (b) The solutions from part (a) that are in the interval 30, p correspond to k 0,, and. For all other values of k the corresponding values of u lie outside this interval. So the solutions in the interval 30, p are u p 8, 5p 8, 3p 8, 7p 8, 5p 8, 9p 8 e k 0 k k 3 = 5π 6 _ 3 = π 6 0 FIGURE _ NOW TRY EXERCISE 7 EXAMPLE 6 A Trigonometric Equation Involving a Half Angle Consider the equation 3 tan u 0. (a) Find all solutions of the equation. (b) Find the solutions in the interval 30, 4p. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

36 58 CHAPTER 7 Analtic Trigonometr SOLUTION (a) We start b isolating tan u : 3 tan u 0 Given equation 3 tan u tan u 3 u p 6 Add Divide b 3 Solve for u in the interval a p, p b Since tangent has period p, to get all solutions we add integer multiples of p to this solution. So the solutions are of the form Multipling b, we get the solutions u p kp 6 u p 3 kp where k is an integer. (b) The solutions from part (a) that are in the interval 30, 4p correspond to k 0 and k. For all other values of k the corresponding values of lie outside this interval. Thus the solutions in the interval 30, 4p are p 3, 7p 3 NOW TRY EXERCISE EXERCISES CONCEPTS We can use identities to help us solve trigonometric equations.. Using a Pthagorean identit we see that the equation sin sin cos is equivalent to the basic equation whose solutions are.. Using a Double-Angle Formula we see that the equation sin sin 0 is equivalent to the equation. Factoring we see that solving this equation is equivalent to solving the two basic equations and. SKILLS 3 6 Solve the given equation. 3. cos u sin u 4. sin u 4 cos u 5. tan u sec u 6. csc u cot u 3 7. sin u 3 sin u sin u sin u 0 9. cos u 3 sin u 0. cos u cos u. sin u cos u. tan u 3 cot u 0 3. sin u cos u 4. cos u sin u 5. tan u sec u 6. tan u sec u An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval 30, p. 7. cos 3u 8. 3 csc u 4 9. cos u 0 0. sin 3u 0. 3 tan 3u 0. sec 4u 0 3. cos u 0 4. tan u sin u sec u cos u 3 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

37 SECTION 7.5 More Trigonometric Equations sin u 3 cos u 8. csc 3u 5 sin 3u 9. sec u tan u cos u 30. tan 3u sec 3u 3. 3 tan 3 u 3 tan u tan u sin u cos u sin u cos u sin u tan u tan u sin u 34. sec u tan u cos u cot u sin u (a) Graph f and g in the given viewing rectangle and find the intersection points graphicall, rounded to two decimal places. (b) Find the intersection points of f and g algebraicall. Give eact answers. 35. f 3 cos, g cos ; 3p, p4 b 3.5, f sin, g sin ; 3p, p4 b 3.5, f tan, g 3; c p, p d b 30, f sin, g cos ; 3p, p4 b 3.5, Use an Addition or Subtraction Formula to simplif the equation. Then find all solutions in the interval 30, p. 39. cos u cos 3u sin u sin 3u cos u cos u sin u sin u 4. sin u cos u cos u sin u 3/ 4. sin 3u cos u cos 3u sin u Use a Double- or Half-Angle Formula to solve the equation in the interval 30, p. 43. sin u cos u tan u sin u cos u cos u 46. tan u cot u 4 sin u 47. cos u cos u sin u cos u 49. cos u cos 4u sin 3u sin 6u 0 APPLICATIONS 63. Range of a Projectile If a projectile is fired with velocit 0 at an angle u, then its range, the horizontal distance it travels (in feet), is modeled b the function Ru 0 sin u 3 (See page 576.) If 0 00 ft/s, what angle (in degrees) should be chosen for the projectile to hit a target on the ground 5000 ft awa? 64. Damped Vibrations The displacement of a spring vibrating in damped harmonic motion is given b 4e 3t sin pt Find the times when the spring is at its equilibrium position Hours of Dalight In Philadelphia the number of hours of dalight on da t (where t is the number of das after Januar ) is modeled b the function Lt.83 sin a p t 80 b 365 (a) Which das of the ear have about 0 hours of dalight? (b) How man das of the ear have more than 0 hours of dalight? 66. Belts and Pulles A thin belt of length L surrounds two pulles of radii R and r, as shown in the figure. (a) Show that the angle u (in radians) where the belt crosses itself satisfies the equation u cot u L R r p [Hint: Epress L in terms of R, r, and u b adding up the lengths of the curved and straight parts of the belt.] (b) Suppose that R.4 ft, r. ft, and L 7.78 ft. Find u b solving the equation in part (a) graphicall. Epress our answer both in radians and in degrees. 5. cos u sin u sin u 5. sin u cos u Solve the equation b first using a Sum-to-Product Formula. R R r r 53. sin u sin 3u cos 5u cos 7u cos 4u cos u cos u 56. sin 5u sin 3u cos 4u 57 6 Use a graphing device to find the solutions of the equation, correct to two decimal places. 57. sin 58. cos sin 60. sin 3 cos cos e e DISCOVERY DISCUSSION WRITING 67. A Special Trigonometric Equation What makes the equation sincos 0 different from all the other equations we ve looked at in this section? Find all solutions of this equation. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

38 530 CHAPTER 7 Analtic Trigonometr CHAPTER 7 REVIEW CONCEPT CHECK. (a) State the reciprocal identities. (b) State the Pthagorean identities. (c) State the even-odd identities. (d) State the cofunction identities.. Eplain the difference between an equation and an identit. 3. How do ou prove a trigonometric identit? 4. (a) State the Addition Formulas for Sine, Cosine, and Tangent. (b) State the Subtraction Formulas for Sine, Cosine, and Tangent. 5. (a) State the Double-Angle Formulas for Sine, Cosine, and Tangent. (b) State the formulas for lowering powers. (c) State the Half-Angle Formulas. 6. (a) State the Product-to-Sum Formulas. (b) State the Sum-to-Product Formulas. 7. Eplain how ou solve a trigonometric equation b factoring. 8. What identit would ou use to solve the equation cos sin 0? EXERCISES 4 Verif the identit.. sin u cot u tan u sec u. sec u sec u tan u 3. cos csc csc sin cos 7. sin cos sec tan 8. tan cot sec csc 9. sin cot cos tan 0. tan cot csc sec.. sin tan cos tan sin sec sec sin cos sin tan cos cot sec cos cot tan cos sin 3. tan csc cot sin sin 4. tan cos cos 5. sin sin sin sin tan a p (a) Graph f and g. (b) Do the graphs suggest that the equation f g is an identit? Prove our answer (a) Graph the function(s) and make a conjecture, and (b) prove our conjecture. 9. f sin 3 cos a cos sin b sin cos 3 cos 7 sin 3 sin 7 sin sin cos cos sec sin sec tan tan sec cos cos sin sin cos b tan tan f a cos sin b, g sin f sin cos, g sin cos f tan tan, g cos f 8 sin 8 sin 4, g cos 4 f sin cot, g cos csc tan cot tan tan sec sin 3 cos 3 cos sin sin 3 48 Solve the equation in the interval 30, p sin u cos u cos sin sin sin sin 0 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

39 CHAPTER 7 Review sin 5 sin sin cos tan 37. cos 7 cos sin cos 3 cos 39. cos sin cos 4. tan 3 tan 3 tan cos csc cos 43. tan sin csc 44. cos 3 cos cos tan sec cos 3 tan cos 48. e sin 49. If a projectile is fired with velocit 0 at an angle u, then the maimum height it reaches (in feet) is modeled b the function Mu 0 sin u 64 Suppose ft/s. (a) At what angle u should the projectile be fired so that the maimum height it reaches is 000 ft? (b) Is it possible for the projectile to reach a height of 3000 ft? (c) Find the angle u for which the projectile will travel highest Find the eact value of the epression given that sec 3, csc 3, and and are in Quadrant I. 6. sin 6. cos 63. tan 64. sin 65. cos 66. tan Find the eact value of the epression. 67. tana cos 3 7B 68. sinatan 3 4 cos 3B Write the epression as an algebraic epression in the variable(s). 69. tan tan 70. cossin cos 7. A 0-ft-wide highwa sign is adjacent to a roadwa, as shown in the figure. As a driver approaches the sign, the viewing angle u changes. (a) Epress viewing angle u as a function of the distance between the driver and the sign. (b) The sign is legible when the viewing angle is or greater. At what distance does the sign first become legible? 0 ft M( ) 50. The displacement of an automobile shock absorber is modeled b the function ft 0.t sin 4pt Find the times when the shock absorber is at its equilibrium position (that is, when ft 0. [Hint: 0 for all real.] 5 60 Find the eact value of the epression. 5. cos 5 5. sin 5p 53. tan p 54. sin p cos p 8 7. A 380-ft-tall building supports a 40-ft communications tower (see the figure). As a driver approaches the building, the viewing angle u of the tower changes. (a) Epress the viewing angle u as a function of the distance between the driver and the building. (b) At what distance from the building is the viewing angle u as large as possible? 40 ft 55. sin 5 cos 40 cos 5 sin tan 66 tan 6 tan 66 tan cos p p sin 8 cos p 3 sin p 59. cos 37.5 cos cos 67.5 cos ft Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

40 CHAPTER 7 TEST. Verif each identit. (a) tan u sin u cos u sec u tan (b) csc sec cos tan (c) sin tan. Let sin u, p/ u p/. Simplif the epression 4 3. Find the eact value of each epression. (a) sin 8 cos cos 8 sin (b) sin 75 (c) sin p 4. For the angles a and b in the figures, find cosa b. å 3 5. (a) Write sin 3 cos 5 as a sum of trigonometric functions. (b) Write sin sin 5 as a product of trigonometric functions. 6. If sin u 4 5 and u is in Quadrant III, find tanu/. 7. Solve each trigonometric equation in the interval 30, p, rounded to two decimal places. (a) (b) (c) (d) 3 sin u 0 cos u sin u 0 cos u 5 cos u 0 sin u cos u 0 8. Find all solutions in the interval 30, p, rounded to five decimal places: 5 cos u 9. Find the eact value of cosa tan 40B Rewrite the epression as an algebraic function of and : sincos tan 53 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

41 FOCUS ON MODELING Traveling and Standing Waves We ve learned that the position of a particle in simple harmonic motion is described b a function of the form A sin vt (see Section 5.6). For eample, if a string is moved up and down as in Figure, then the red dot on the string moves up and down in simple harmonic motion. Of course, the same holds true for each point on the string. FIGURE What function describes the shape of the whole string? If we fi an instant in time t 0 and snap a photograph of the string, we get the shape in Figure, which is modeled b A sin k where is the height of the string above the -ais at the point. A _A π k π k FIGURE A sin k Traveling Waves If we snap photographs of the string at other instants, as in Figure 3, it appears that the waves in the string travel or shift to the right. FIGURE 3 The velocit of the wave is the rate at which it moves to the right. If the wave has velocit, then it moves to the right a distance t in time t. So the graph of the shifted wave at time t is, t A sin k t This function models the position of an point on the string at an time t. We use the notation, t to indicate that the function depends on the two variables and t. Here is how this function models the motion of the string. If we fi, then, t is a function of t onl, which gives the position of the fied point at time t. If we fi t, then, t is a function of onl, whose graph is the shape of the string at the fied time t. 533 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

42 534 Focus on Modeling EXAMPLE A Traveling Wave A traveling wave is described b the function, t 3 sin a p t b, 0 (a) Find the function that models the position of the point p/6 at an time t. Observe that the point moves in simple harmonic motion. 3 (b) Sketch the shape of the wave when t 0, 0.5,.0,.5, and.0. Does the wave appear to be traveling to the right? (c) Find the velocit of the wave. SOLUTION (a) Substituting p/6 we get a p 6, t b 3 sin a # p 6 p t b 3 sin a p 3 p t b _3 FIGURE 4 Traveling wave The function 3 sina p 3 p tb describes simple harmonic motion with amplitude 3 and period p/p/ 4. (b) The graphs are shown in Figure 4. As t increases, the wave moves to the right. (c) We epress the given function in the standard form, t A sin k t:, t 3 sin a p t b Given 3 sin a p 4 t b Factor Comparing this to the standard form, we see that the wave is moving with velocit p/4. Standing Waves If two waves are traveling along the same string, then the movement of the string is determined b the sum of the two waves. For eample, if the string is attached to a wall, then the waves bounce back with the same amplitude and speed but in the opposite direction. In this case, one wave is described b A sin k t and the reflected wave b A sin k t. The resulting wave is, t A sin k t A sin k t Add the two waves Sum-to-Product Formula The points where k is a multiple of p are special, because at these points 0 for an time t. In other words, these points never move. Such points are called nodes. Figure 5 shows the graph of the wave for several values of t. We see that the wave does not travel, but simpl vibrates up and down. Such a wave is called a standing wave. A A sin k cos k t FIGURE 5 A standing wave _A Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

43 EXAMPLE A Standing Wave Traveling waves are generated at each end of a wave tank 30 ft long, with equations.5 sin a p 3t b 5 Traveling and Standing Waves 535 and.5 sin a p 3t b 5 (a) Find the equation of the combined wave, and find the nodes. (b) Sketch the graph for t 0, 0.7, 0.34, 0.5, 0.68, 0.85, and.0. Is this a standing wave? SOLUTION (a) The combined wave is obtained b adding the two equations:.5 sin a p 5 3t b.5 sin a p 3t b 5 Add the two waves 3 sin p cos 3t 5 Sum-to-Product Formula The nodes occur at the values of for which sin p p 5 0, that is, where 5 kp (k an integer). Solving for, we get 5k. So the nodes occur at 0, 5, 0, 5, 0, 5, 30 (b) The graphs are shown in Figure 6. From the graphs we see that this is a standing wave. t=0 t=0.7 t=0.34 t=0.5 t=0.68 t=0.85 t= _3 FIGURE 6, t 3 sin p cos 3t 5 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

44 536 Focus on Modeling PROBLEMS. Wave on a Canal A wave on the surface of a long canal is described b the function, t 5 sin a p t b, 0 (a) Find the function that models the position of the point 0 at an time t. (b) Sketch the shape of the wave when t 0, 0.4, 0.8,., and.6. Is this a traveling wave? (c) Find the velocit of the wave.. Wave in a Rope Traveling waves are generated at each end of a tightl stretched rope 4 ft long, with equations 0. sin t and 0. sin t (a) Find the equation of the combined wave, and find the nodes. (b) Sketch the graph for t 0,,, 3, 4, 5, and 6. Is this a standing wave? 3. Traveling Wave A traveling wave is graphed at the instant t 0. If it is moving to the right with velocit 6, find an equation of the form, t A sink k t for this wave _.7 4. Traveling Wave A traveling wave has period p/3, amplitude 5, and velocit 0.5. (a) Find the equation of the wave. (b) Sketch the graph for t 0, 0.5,,.5, and. 5. Standing Wave A standing wave with amplitude 0.6 is graphed at several times t as shown in the figure. If the vibration has a frequenc of 0 Hz, find an equation of the form, t A sin a cos bt that models this wave _0.6 _0.6 _0.6 t=0 s t=0.00 s t=0.05 s Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).

45 Traveling and Standing Waves Standing Wave A standing wave has maimum amplitude 7 and nodes at 0, p/, p, 3p/, p, as shown in the figure. Each point that is not a node moves up and down with period 4p. Find a function of the form, t A sin a cos bt that models this wave. 7 π π 3π π _7 7. Vibrating String When a violin string vibrates, the sound produced results from a combination of standing waves that have evenl placed nodes. The figure illustrates some of the possible standing waves. Let s assume that the string has length p. (a) For fied t, the string has the shape of a sine curve Asin a. Find the appropriate value of a for each of the illustrated standing waves. (b) Do ou notice a pattern in the values of a that ou found in part (a)? What would the net two values of a be? Sketch rough graphs of the standing waves associated with these new values of a. (c) Suppose that for fied t, each point on the string that is not a node vibrates with frequenc 440 Hz. Find the value of b for which an equation of the form A cos bt would model this motion. (d) Combine our answers for parts (a) and (c) to find functions of the form, t A sin a cos bt that model each of the standing waves in the figure. (Assume that A.) Waves in a Tube Standing waves in a violin string must have nodes at the ends of the string because the string is fied at its endpoints. But this need not be the case with sound waves in a tube (such as a flute or an organ pipe). The figure shows some possible standing waves in a tube. Suppose that a standing wave in a tube 37.7 ft long is modeled b the function, t 0.3 cos cos 50pt Here, t represents the variation from normal air pressure at the point feet from the end of the tube, at time t seconds. (a) At what points are the nodes located? Are the endpoints of the tube nodes? (b) At what frequenc does the air vibrate at points that are not nodes? Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s).