Analytic Trigonometry

Transcription

1 Analytic Trigonometry 7. Using Fundamental Identities 7. Verifying Trigonometric Identities 7.3 Solving Trigonometric Equations 7.4 Sum and Difference Formulas 7.5 Multiple-Angle and Product-to-Sum Formula 7 Concepts of trigonometry can be used to model the height above ground of a seat on a Ferris wheel. Patrick Ward/Corbis SELECTED APPLICATIONS Trigonometric equations and identities have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. Friction, Eercise 99, page 543 Shadow Length, Eercise 56, page 550 Ferris Wheel, Eercise 75, page 560 Data Analysis: Unemployment Rate, Eercise 76, page 560 Harmonic Motion, Eercise 75, page 567 Mach Number, Eercise, page 579 Projectile Motion, Eercise 0, page 583 Ocean Depth, Eercise 0, page Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

2 536 Chapter 7 Analytic Trigonometry 7. Using Fundamental Identities What you should learn Recognize and write the fundamental trigonometric identities. Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric epressions, and rewrite trigonometric epressions. Why you should learn it Fundamental trigonometric identities can be used to simplify trigonometric epressions. For instance, in Eercise 99 on page 543, you can use trigonometric identities to simplify an epression for the coefficient of friction. Introduction In Chapter 6, you studied the basic definitions, properties, graphs, and applications of the individual trigonometric functions. In this chapter, you will learn how to use the fundamental identities to do the following.. Evaluate trigonometric functions.. Simplify trigonometric epressions. 3. Develop additional trigonometric identities. 4. Solve trigonometric equations. Fundamental Trigonometric Identities Reciprocal Identities sin u csc u csc u sin u Quotient Identities tan u sin u cos u Pythagorean Identities sin u cos u Cofunction Identities sin u cos u tan u cot u sec u csc u Even/Odd Identities sinu sin u cscu csc u cos u sec u sec u cos u cot u cos u sin u tan u sec u cos u sin u cot u tan u csc u sec u cosu cos u secu sec u tan u cot u cot u tan u cot u csc u tanu tan u cotu cot u The Online Study Center CD-ROM and Eduspace for this tet contain additional resources related to the concepts discussed in this chapter. Pythagorean identities are sometimes used in radical form such as sin u ± cos u or tan u ±sec u where the sign depends on the choice of u. Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

3 Section 7. Using Fundamental Identities 537 You should learn the fundamental trigonometric identities well, because they are used frequently in trigonometry and they will also appear later in calculus. Note that u can be an angle, a real number, or a variable. You can use a graphing utility to check the result of Eample. To do this, graph and in the same viewing window, as shown below. Because Eample shows the equivalence algebraically and the two graphs appear to coincide, you can conclude that the epressions are equivalent. π Technology y sin cos sin y sin 3 π Using the Fundamental Identities One common use of trigonometric identities is to use given values of trigonometric functions to evaluate other trigonometric functions. Eample Using Identities to Evaluate a Function Use the values sec u 3 and tan u > 0 to find the values of all si trigonometric functions. Using a reciprocal identity, you have cos u sec u 3 3. Using a Pythagorean identity, you have sin u cos u Pythagorean identity Substitute Simplify. for cos u. Because sec u < 0 and tan u > 0, it follows that u lies in Quadrant III. Moreover, because sin u is negative when u is in Quadrant III, you can choose the negative root and obtain sin u 53. Now, knowing the values of the sine and cosine, you can find the values of all si trigonometric functions. sin u 5 3 cos u 3 tan u sin u 53 5 cos u 3 Eample Now try Eercise. Simplify sin cos sin Simplifying a Trigonometric Epression First factor out a common monomial factor and then use a fundamental identity. sin cos sin sin cos sin cos sin sin sin 3 Now try Eercise 45. csc u 3 sin u sec u cos u 3 cot u tan u Factor out common monomial factor. Factor out. Pythagorean identity Multiply. 3 Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

4 538 Chapter 7 Analytic Trigonometry When factoring trigonometric epressions, it is helpful to find a special polynomial factoring form that fits the epression, as shown in Eample 3. Eample 3 Factoring Trigonometric Epressions Factor each epression. a. sec b. 4 tan tan 3 a. Here you have the difference of two squares, which factors as b. This epression has the polynomial form a b c, and it factors as Now try Eercise 47. On occasion, factoring or simplifying can best be done by first rewriting the epression in terms of just one trigonometric function or in terms of sine and cosine only. These strategies are illustrated in Eamples 4 and 5, respectively. Factoring a Trigonometric Epression Eample 4 Factor csc cot 3. sec sec sec ). 4 tan tan 3 4 tan 3tan. Use the identity csc cot to rewrite the epression in terms of the cotangent. csc cot 3 cot cot 3 cot cot cot cot Now try Eercise 5. Pythagorean identity Combine like terms. Factor. Remember that when adding rational epressions, you must first find the least common denominator (LCD). In Eample 5, the LCD is sin t. Simplifying a Trigonometric Epression Eample 5 Simplify sin t cot t cos t. Begin by rewriting cot t in terms of sine and cosine. sin t cot t cos t sin t cos t sin t cos t sin t cos t sin t sin t csc t Now try Eercise 57. Quotient identity Add fractions. Pythagorean identity Reciprocal identity Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

5 Section 7. Using Fundamental Identities 539 Eample 6 Adding Trigonometric Epressions Perform the addition and simplify. sin cos cos sin sin cos sin sin (cos cos cos sin cos sin sin cos cos cos sin cos cos sin sin csc Now try Eercise 6. Multiply. Pythagorean identity: sin cos Divide out common factor. Reciprocal identity The last two eamples in this section involve techniques for rewriting epressions in forms that are used in calculus. Eample 7 Rewriting a Trigonometric Epression Rewrite sin so that it is not in fractional form. From the Pythagorean identity cos sin sin sin, you can see that multiplying both the numerator and the denominator by sin will produce a monomial denominator. sin sin sin sin sin sin sin cos sin cos cos sin cos cos cos sec tan sec Now try Eercise 65. Multiply numerator and denominator by sin. Multiply. Pythagorean identity Write as separate fractions. Product of fractions Reciprocal and quotient identities Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

6 540 Chapter 7 Analytic Trigonometry Eample 8 Trigonometric Substitution 4 + θ = arctan Angle whose tangent is. FIGURE 7. Use the substitution tan, 0 < 4 as a trigonometric function of. <, to write Begin by letting tan. Then, you can obtain 4 4 tan Now try Eercise 77. Substitute tan Rule of eponents Factor. Pythagorean identity for. Figure 7. shows the right triangle illustration of the trigonometric substitution tan in Eample 8. You can use this triangle to check the solution of Eample 8. For 0 < <, you have opp, and With these epressions, you can write the following. sec hyp adj 4 4 tan 4 sec sec. adj, 4 sec sec 4 So, the solution checks. 4 tan hyp 4. sec > 0 for 0 < < Eample 9 Rewriting a Logarithmic Epression Rewrite ln csc ln tan ln csc ln tan ln csc tan Now try Eercise 9. as a single logarithm and simplify the result. ln sin sin cos ln cos ln sec Product Property of Logarithms Reciprocal and quotient identities Simplify. Reciprocal identity Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

7 Section 7. Using Fundamental Identities Eercises The HM mathspace CD-ROM and Eduspace for this tet contain step-by-step solutions to all odd-numbered eercises. They also provide Tutorial Eercises for additional help. VOCABULARY CHECK: sin u.. cos u tan u Fill in the blank to complete the trigonometric identity. 5. csc u 6. tan u sec sin u u PREREQUISITE SKILLS REVIEW: sec u sin u 9. cosu 0. tanu Practice and review algebra skills needed for this section at In Eercises 4, use the given values to evaluate (if possible) all si trigonometric functions cot 3, 7. sec 3, sin 3, 0. sec 4,. sin 3, tan 3 3, sec, cos 3 5, tan,. csc 5, 3. sin, sin > 0 sin < 0 4. tan is undefined, cos cos 3 sin csc 5 tan 3 3, 4 tan 5 sec 3, sin 0 0 csc 35 5 tan 4 cos < 0 cot 0 cos 4 5 sin > 0 In Eercises 5 0, match the trigonometric epression with one of the following. (a) sec (b) (c) cot (d) (e) tan (f) sin 5. sec cos 6. tan csc 7. cot csc 8. cos csc sin cos In Eercises 6, match the trigonometric epression with one of the following. (a) csc (b) tan (c) (d) sin tan (e) sec (f) sec tan. sin sec. cos sec 3. sec 4 tan 4 4. cot sec sec sin In Eercises 7 44, use the fundamental identities to simplify the epression. There is more than one correct form of each answer. 7. cot sec 8. cos tan 9. sin csc sin 30. sec sin cot csc sin csc 35. sec sin 36. tan 37. cos 38. sec cos y cos t tan t sin y 4. sin tan cos 4. csc tan sec 43. cot u sin u tan u cos u 44. sin sec cos csc sin cos cos cos csc sec tan tan sec sin cot cos Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

8 54 Chapter 7 Analytic Trigonometry In Eercises 45 56, factor the epression and use the fundamental identities to simplify. There is more than one correct form of each answer. 45. tan tan sin 46. sin csc sin 47. sin sec sin 48. cos cos tan sec sec 5. tan 4 tan 5. cos cos sin 4 cos sec 4 tan csc 3 csc csc 56. sec 3 sec sec In Eercises 57 60, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer. 57. sin cos 58. cot csc cot csc 59. csc csc sin 3 3 sin In Eercises 6 64, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer cos cos cos sin sin cos In Eercises 65 68, rewrite the epression so that it is not in fractional form.there is more than one correct form of each answer. sin y cos y sec tan cos 4 cos sec sec tan sec tan 5 tan sec tan csc Numerical and Graphical Analysis In Eercises 69 7, use a graphing utility to complete the table and graph the functions. Make a conjecture about and y. y In Eercises 73 76, use a graphing utility to determine which of the si trigonometric functions is equal to the epression. Verify your answer algebraically. 73. cos cot sin y y sec 4 sec, sec csc tan sin cos cos sin cos In Eercises 77 8, use the trigonometric substitution to write the algebraic epression as a trigonometric function of, where 0 < < / , , 79. 9, 80. 4, cos sin, 8. 5, 8. 00, In Eercises 83 86, use the trigonometric substitution to write the algebraic equation as a trigonometric equation of, where / < < /. Then find sin and cos , , y cos sin 3 cos 3 sec sec 5 tan , 0 tan , In Eercises 87 90, use a graphing utility to solve the equation for, where 0 <. 87. sin cos 88. cos sin 89. sec tan 90. csc cot cos 3 sin sin cos y tan tan 4 6 sin cos 0 cos y cos, y y y y sin sec cos, y sin tan In Eercises 9 94, rewrite the epression as a single logarithm and simplify the result ln ln cos ln sin ln sec ln sin cot t ln tan t 94. lncos t ln tan t Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

9 Section 7. Using Fundamental Identities 543 In Eercises 95 98, use a calculator to demonstrate the identity for each value of. 95. csc cot (a) (b) 96. tan sec 97. (a) (a) (b) (b) 98. sin sin (a) 3, 346, cos 80, 50, (b) 99. Friction The forces acting on an object weighing W units on an inclined plane positioned at an angle of with the horizontal (see figure) are modeled by sin W cos W sin where is the coefficient of friction. Solve the equation for and simplify the result. θ W In Eercises 03 06, fill in the blanks. (Note: The notation c indicates that approaches c from the right and c indicates that approaches c from the left.) 03. As, sin and csc. 04. As and sec. 05. As, tan and cot. 06. As and csc. In Eercises 07, determine whether or not the equation is an identity, and give a reason for your answer. 07. cos sin 08. cot csc sin k 09. k is a constant. cos k tan, 0. 0, cos, sin 5 cos 5 sec. sin csc. csc 3. Use the definitions of sine and cosine to derive the Pythagorean identity sin cos. 4. Writing Use the Pythagorean identity sin cos to derive the other Pythagorean identities, tan sec and cot csc. Discuss how to remember these identities and other fundamental identities. Skills Review 00. Rate of Change The rate of change of the function f csc sin is given by the epression csc cot cos. Show that this epression can also be written as cos cot. Synthesis True or False? In Eercises 0 and 0, determine whether the statement is true or false. Justify your answer. 0. The even and odd trigonometric identities are helpful for determining whether the value of a trigonometric function is positive or negative. 0. A cofunction identity can be used to transform a tangent function so that it can be represented by a cosecant function. In Eercises 5 and 6, perform the operation and simplify In Eercises 7 0, perform the addition or subtraction and simplify In Eercises 4, sketch the graph of the function. (Include two full periods.). f. sin 4 z f tan 3. f 4. f 3 cos 3 sec Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

10 544 Chapter 7 Analytic Trigonometry 7. Verifying Trigonometric Identities What you should learn Verify trigonometric identities. Why you should learn it You can use trigonometric identities to rewrite trigonometric equations that model real-life situations. For instance, in Eercise 56 on page 550, you can use trigonometric identities to simplify the equation that models the length of a shadow cast by a gnomon (a device used to tell time). Introduction In this section, you will study techniques for verifying trigonometric identities. In the net section, you will study techniques for solving trigonometric equations. The key to verifying identities and solving equations is the ability to use the fundamental identities and the rules of algebra to rewrite trigonometric epressions. Remember that a conditional equation is an equation that is true for only some of the values in its domain. For eample, the conditional equation sin 0 Conditional equation is true only for n, where n is an integer. When you find these values, you are solving the equation. On the other hand, an equation that is true for all real values in the domain of the variable is an identity. For eample, the familiar equation sin cos Identity is true for all real numbers. So, it is an identity. Verifying Trigonometric Identities Although there are similarities, verifying that a trigonometric equation is an identity is quite different from solving an equation. There is no well-defined set of rules to follow in verifying trigonometric identities, and the process is best learned by practice. Robert Ginn/PhotoEdit Guidelines for Verifying Trigonometric Identities. Work with one side of the equation at a time. It is often better to work with the more complicated side first.. Look for opportunities to factor an epression, add fractions, square a binomial, or create a monomial denominator. 3. Look for opportunities to use the fundamental identities. Note which functions are in the final epression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. 4. If the preceding guidelines do not help, try converting all terms to sines and cosines. 5. Always try something. Even paths that lead to dead ends provide insights. Verifying trigonometric identities is a useful process if you need to convert a trigonometric epression into a form that is more useful algebraically. When you verify an identity, you cannot assume that the two sides of the equation are equal because you are trying to verify that they are equal. As a result, when verifying identities, you cannot use operations such as adding the same quantity to each side of the equation or cross multiplication. Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

11 Section 7. Verifying Trigonometric Identities 545 Eample Verifying a Trigonometric Identity Remember that an identity is only true for all real values in the domain of the variable. For instance, in Eample the identity is not true when because sec is not defined when. Verify the identity Because the left side is more complicated, start with it. sec sec sec sec tan sec tan cos sin sin. tan sec sin cos cos Pythagorean identity Simplify. Reciprocal identity Quotient identity Simplify. Notice how the identity is verified. You start with the left side of the equation (the more complicated side) and use the fundamental trigonometric identities to simplify it until you obtain the right side. Now try Eercise 5. There is more than one way to verify an identity. Here is another way to verify the identity in Eample. sec sec sec sec sec cos sin Rewrite as the difference of fractions. Reciprocal identity Pythagorean identity Eample Combining Fractions Before Using Identities Verify the identity sec. sin sin sin sin sin sin sin sin sin cos sec Now try Eercise 9. Add fractions. Simplify. Pythagorean identity Reciprocal identity Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

12 546 Chapter 7 Analytic Trigonometry Eample 3 Verifying Trigonometric Identity Verify the identity tan cos tan. Algebraic By applying identities before multiplying, you obtain the following. tan cos sec sin sin cos sin cos tan Pythagorean identities Reciprocal identity Rule of eponents Quotient identity Numerical Use the table feature of a graphing utility set in radian mode to create a table that shows the values of y tan cos and y tan for different values of, as shown in Figure 7.. From the table you can see that the values of y and y appear to be identical, so tan cos tan appears to be an identity. Now try Eercise 39. FIGURE 7. Eample 4 Converting to Sines and Cosines Although a graphing utility can be useful in helping to verify an identity, you must use algebraic techniques to produce a valid proof. Verify the identity tan cot sec csc. Try converting the left side into sines and cosines. tan cot sin cos cos sin sin cos cos sin cos sin cos sin Now try Eercise 9. sec csc Quotient identities Add fractions. Pythagorean identity Reciprocal identities As shown at the right, csc cos is considered a simplified form of cos because the epression does not contain any fractions. Recall from algebra that rationalizing the denominator using conjugates is, on occasion, a powerful simplification technique. A related form of this technique, shown below, works for simplifying trigonometric epressions as well. cos cos cos cos csc cos This technique is demonstrated in the net eample. cos cos cos sin Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

13 Section 7. Verifying Trigonometric Identities 547 Eample 5 Verifying Trigonometric Identities Verify the identity sec y tan y cos y sin y. Begin with the right side, because you can create a monomial denominator by multiplying the numerator and denominator by sin y. cos y cos y sin y sin y sin y sin y cos y cos y sin y sin y cos y cos y sin y cos y cos y cos y sin y cos y cos y sin y cos y cos y sec y tan y Now try Eercise 33. Multiply numerator and denominator by sin y. Multiply. Pythagorean identity Write as separate fractions. Simplify. Identities In Eamples through 5, you have been verifying trigonometric identities by working with one side of the equation and converting to the form given on the other side. On occasion, it is practical to work with each side separately, to obtain one common form equivalent to both sides. This is illustrated in Eample 6. Eample 6 Working with Each Side Separately Verify the identity Working with the left side, you have cot csc csc csc cot sin. csc sin csc csc csc csc. Now, simplifying the right side, you have sin sin sin sin sin csc. Pythagorean identity Factor. Simplify. Write as separate fractions. Reciprocal identity The identity is verified because both sides are equal to csc. Now try Eercise 47. Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

14 548 Chapter 7 Analytic Trigonometry In Eample 7, powers of trigonometric functions are rewritten as more complicated sums of products of trigonometric functions. This is a common procedure used in calculus. Eample 7 Three Eamples from Calculus Verify each identity. a. b. c. tan 4 tan sec tan sin 3 cos 4 cos 4 cos 6 sin csc 4 cot csc cot cot 3 a. tan 4 tan tan Write as separate factors. tan sec tan sec tan Pythagorean identity Multiply. b. sin 3 cos 4 sin cos 4 sin Write as separate factors. cos cos 4 sin cos 4 cos 6 sin Pythagorean identity Multiply. c. csc 4 cot csc csc cot Write as separate factors. csc cot cot csc cot cot 3 Now try Eercise 49. Pythagorean identity Multiply. W RITING ABOUT MATHEMATICS Error Analysis You are tutoring a student in trigonometry. One of the homework problems your student encounters asks whether the following statement is an identity. tan sin? 5 6 tan Your student does not attempt to verify the equivalence algebraically, but mistakenly uses only a graphical approach. Using range settings of Xmin 3 Xma 3 Xscl Ymin 0 Yma 0 Yscl your student graphs both sides of the epression on a graphing utility and concludes that the statement is an identity. What is wrong with your student s reasoning? Eplain. Discuss the limitations of verifying identities graphically. Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

15 Section 7. Verifying Trigonometric Identities Eercises VOCABULARY CHECK: In Eercises and, fill in the blanks.. An equation that is true for all real values in its domain is called an.. An equation that is true for only some values in its domain is called a. In Eercises 3 8, fill in the blank to complete the trigonometric identity cot u 5. sin u 6. PREREQUISITE SKILLS REVIEW: cos u sin u cos u 7. cscu 8. secu Practice and review algebra skills needed for this section at In Eercises 38, verify the identity.. sin t csc t. sec y cos y 3. sin sin cos 4. cot ysec y 5. cos sin sin 6. cos sin cos cos sin tan sec csc 9. csc sec 0. cot cot t. csc t sin t. csc t sin sin 4 cos cos 4 sin cos sin 5 cos cos 3 sin sec 6 sec tan sec 4 sec tan sec 5 tan 3 csc sin sec tan sec sec cos csc sin cos cot sec cos sin tan tan tan cot cot sin csc sin csc cos cot csc sin sin cos sec cos sin cot 3 t csc t cos t csc t tan sec tan tan cos cos sin cos tan sin cos 5. tan cos 6. tan tan sin sin y siny cos y sin csc csc sec cot tan cot cos tan tan y cot cot y tan tan y cot cot y tan cot y tan cot y cos cos y sin sin y sin sin y cos cos y 0 sin sin sin cos cos cos cos sin cos cos sec y cot y sin t csc t tan t sec tan y cot 38. sec cot Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

16 550 Chapter 7 Analytic Trigonometry In Eercises 39 46, (a) use a graphing utility to graph each side of the equation to determine whether the equation is an identity, (b) use the table feature of a graphing utility to determine whether the equation is an identity, and (c) confirm the results of parts (a) and (b) algebraically sin 4 sin cos cos cos sin cot 46. csc csc sin cos cot In Eercises 47 50, verify the identity In Eercises 5 54, use the cofunction identities to evaluate the epression without the aid of a calculator. 5. sin 5 sin cos 55 cos sec sec sin sin cos csc csc sin csc 4 csc cot 4 tan 5 tan 3 sec tan 3 sin cos cot csc sin cos 3 cos 4 sin 3 cos tan 4 tan 3 sec 4 tan 3 sec 4 tan tan tan 4 sec cos 3 sin sin sin 4 cos sin 4 cos 4 cos cos 4 cos 0 cos 5 cos 38 cos 70 sin sin 40 sin 50 sin Rate of Change The rate of change of the function f sin csc with respect to change in the variable is given by the epression cos csc cot. Show that the epression for the rate of change can also be cos cot. 56. Shadow Length The length s of a shadow cast by a vertical gnomon (a device used to tell time) of height h when the angle of the sun above the horizon is (see figure) can be modeled by the equation h sin90 s. sin Model It Synthesis True or False? In Eercises 57 and 58, determine whether the statement is true or false. Justify your answer. 57. The equation sin cos tan is an identity, because sin 0 cos 0 and tan The equation tan cot is not an identity, because it is true that tan 6 3, and cot 6 4. Think About It In Eercises 59 and 60, eplain why the equation is not an identity and find one value of the variable for which the equation is not true. 59. sin cos 60. tan sec Skills Review Model It (continued) (a) Verify that the equation for s is equal to h cot. (b) Use a graphing utility to complete the table. Let h 5 feet. s s (c) Use your table from part (b) to determine the angles of the sun for which the length of the shadow is the greatest and the least. (d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is 90? In Eercises 6 64, perform the operation and simplify. 6. 3i i i 3 50 h ft s θ In Eercises 65 68, use the Quadratic Formula to solve the quadratic equation Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

17 Section 7.3 Solving Trigonometric Equations Solving Trigonometric Equations What you should learn Use standard algebraic techniques to solve trigonometric equations. Solve trigonometric equations of quadratic type. Solve trigonometric equations involving multiple angles. Use inverse trigonometric functions to solve trigonometric equations. Why you should learn it You can use trigonometric equations to solve a variety of real-life problems. For instance, in Eercise 7 on page 560, you can solve a trigonometric equation to help answer questions about monthly sales of skiing equipment. Introduction To solve a trigonometric equation, use standard algebraic techniques such as collecting like terms and factoring. Your preliminary goal in solving a trigonometric equation is to isolate the trigonometric function involved in the equation. For eample, to solve the equation sin, divide each side by to obtain sin. To solve for, note in Figure 7.3 that the equation sin has solutions 6 and 56 in the interval 0,. Moreover, because sin has a period of, there are infinitely many other solutions, which can be written as n 6 and 5 n 6 where n is an integer, as shown in Figure 7.3. = π π 6 y = y = π 6 General solution = π + π 6 π π FIGURE 7.3 = 5π π = 5π 6 6 y = sin = 5π + π 6 Tom Stillo/Inde Stock Imagery Another way to show that the equation sin has infinitely many solutions is indicated in Figure 7.4. Any angles that are coterminal with 6 or 56 will also be solutions of the equation. ( ) sin π ( + nπ) = sin 5π + nπ = 6 5π 6 π 6 6 FIGURE 7.4 When solving trigonometric equations, you should write your answer(s) using eact values rather than decimal approimations. Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

18 55 Chapter 7 Analytic Trigonometry Eample Collecting Like Terms Solve sin sin. Begin by rewriting the equation so that sin equation. is isolated on one side of the Write original equation. Add sin to each side. Subtract from each side. Combine like terms. Divide each side by. Because sin has a period of, first find all solutions in the interval 0,. These solutions are 54 and 74. Finally, add multiples of to each of these solutions to get the general form 5 n 4 where n is an integer. sin sin sin sin 0 sin sin sin sin and Now try Eercise 7. 7 n 4 General solution Eample Etracting Square Roots Solve 3 tan 0. Begin by rewriting the equation so that equation. 3 tan 0 3 tan tan is isolated on one side of the Write original equation. Add to each side. tan 3 Divide each side by 3. Etract square roots. Because tan has a period of, first find all solutions in the interval 0,. These solutions are 6 and 56. Finally, add multiples of to each of these solutions to get the general form n 6 tan ± 3 where n is an integer. and ±3 3 Now try Eercise. 5 n 6 General solution Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

19 Section 7.3 Solving Trigonometric Equations 553 The equations in Eamples and involved only one trigonometric function. When two or more functions occur in the same equation, collect all terms on one side and try to separate the functions by factoring or by using appropriate identities. This may produce factors that yield no solutions, as illustrated in Eample 3. Eploration Using the equation from Eample 3, eplain what would happen if you divided each side of the equation by cot. Is this a correct method to use when solving equations? π FIGURE y y = cot cos cot π Eample 3 Factoring Solve cot cos cot. Begin by rewriting the equation so that all terms are collected on one side of the equation. cot cos cot 0 cot cos 0 Write original equation. Subtract cot from each side. Factor. By setting each of these factors equal to zero, you obtain cot 0 and The equation cot 0 has the solution [in the interval 0, ]. No solution is obtained for cos ± because ± are outside the range of the cosine function. Because cot has a period of, the general form of the solution is obtained by adding multiples of to, to get n cot cos cot cos 0 cos General solution where n is an integer. You can confirm this graphically by sketching the graph of y cot cos cot, as shown in Figure 7.5. From the graph you can see that the -intercepts occur at 3,,, 3, and so on. These -intercepts correspond to the solutions of cot cos cot 0. Now try Eercise 5. cos ±. Equations of Quadratic Type Many trigonometric equations are of quadratic type a b c 0. Here are a couple of eamples. Quadratic in sin sin sin 0 sin sin 0 Quadratic in sec sec 3 sec 0 sec 3sec 0 To solve equations of this type, factor the quadratic or, if this is not possible, use the Quadratic Formula. Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

20 554 Chapter 7 Analytic Trigonometry Eample 4 Factoring an Equation of Quadratic Type Find all solutions of sin sin 0 in the interval 0,. Algebraic Begin by treating the equation as a quadratic in sin and factoring. sin sin 0 sin sin 0 Write original equation. Factor. Setting each factor equal to zero, you obtain the following solutions in the interval 0,. sin 0 sin 7 6, 6 and sin 0 sin Graphical Use a graphing utility set in radian mode to graph y sin sin for 0 <, as shown in Figure 7.6. Use the zero or root feature or the zoom and trace features to approimate the -intercepts to be.57, , and These values are the approimate solutions of sin sin 0 in the interval 0,. 3 y = sin sin Now try Eercise 9. FIGURE 7.6 Eample 5 Rewriting with a Single Trigonometric Function Solve sin 3 cos 3 0. This equation contains both sine and cosine functions. You can rewrite the equation so that it has only cosine functions by using the identity sin cos. cos 3 cos 3 0 cos cos 0 Write original equation. Pythagorean identity Multiply each side by. Factor. Set each factor equal to zero to find the solutions in the interval 0,. cos 0 cos 0 Because cos has a period of, the general form of the solution is obtained by adding multiples of to get n, cos 3 cos 0 where n is an integer. sin 3 cos 3 0 n, 3 cos cos Now try Eercise 3. 5 n 3 3, General solution Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

21 Section 7.3 Solving Trigonometric Equations 555 Sometimes you must square each side of an equation to obtain a quadratic, as demonstrated in the net eample. Because this procedure can introduce etraneous solutions, you should check any solutions in the original equation to see whether they are valid or etraneous. Eample 6 Squaring and Converting to Quadratic Type You square each side of the equation in Eample 6 because the squares of the sine and cosine functions are related by a Pythagorean identity. The same is true for the squares of the secant and tangent functions and the cosecant and cotangent functions. Eploration Use a graphing utility to confirm the solutions found in Eample 6 in two different ways. Do both methods produce the same -values? Which method do you prefer? Why?. Graph both sides of the equation and find the -coordinates of the points at which the graphs intersect. Left side: Right side: y sin. Graph the equation y cos y cos sin and find the -intercepts of the graph. Find all solutions of cos sin in the interval 0,. It is not clear how to rewrite this equation in terms of a single trigonometric function. Notice what happens when you square each side of the equation. cos cos cos 0 Setting each factor equal to zero produces cos 0 cos 0 and Write original equation. Square each side. Pythagorean identity Rewrite equation. Combine like terms. Factor. Because you squared the original equation, check for etraneous solutions. Check / cos? sin 0 Check 3/ cos 3? sin 3 Check, 3 0 cos? sin 0 cos cos sin cos cos cos cos cos 0 cos cos 0 Substitute for. checks. Substitute 3 for. does not check. Substitute for. checks. Of the three possible solutions, 3 is etraneous. So, in the interval 0,, the only two solutions are and. Now try Eercise 33. cos sin cos 0 cos. Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

22 556 Chapter 7 Analytic Trigonometry Functions Involving Multiple Angles The net two eamples involve trigonometric functions of multiple angles of the forms sin ku and cos ku. To solve equations of these forms, first solve the equation for ku, then divide your result by k. Eample 7 Functions of Multiple Angles Solve cos 3t 0. cos 3t 0 cos 3t cos 3t Write original equation. Add to each side. Divide each side by. In the interval 0,, you know that 3t 3 and 3t 53 are the only solutions, so, in general, you have and Dividing these results by 3, you obtain the general solution t 9 3t n 3 n 3 where n is an integer. and Now try Eercise 35. 3t 5 n. 3 t 5 n 9 3 General solution Eample 8 Functions of Multiple Angles Solve 3 tan tan tan 3 Write original equation. Subtract 3 from each side. tan Divide each side by 3. In the interval 0,, you know that 34 is the only solution, so, in general, you have 3 n. 4 Multiplying this result by, you obtain the general solution 3 n where n is an integer. Now try Eercise 39. General solution Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

23 Using Inverse Functions Section 7.3 Solving Trigonometric Equations 557 In the net eample, you will see how inverse trigonometric functions can be used to solve an equation. Eample 9 Using Inverse Functions Solve sec tan 4. sec tan 4 Write original equation. tan tan 4 0 Pythagorean identity tan tan 3 0 Combine like terms. tan 3tan 0 Factor. Setting each factor equal to zero, you obtain two solutions in the interval,. [Recall that the range of the inverse tangent function is,.] tan 3 0 tan 3 arctan 3 and tan 0 tan Finally, because tan has a period of, you obtain the general solution by adding multiples of arctan 3 n and n 4 General solution where n is an integer. You can use a calculator to approimate the value of arctan 3. Now try Eercise W RITING ABOUT MATHEMATICS Equations with No s One of the following equations has solutions and the other two do not. Which two equations do not have solutions? a. b. c. sin 5 sin 6 0 sin 4 sin 6 0 sin 5 sin 6 0 Find conditions involving the constants b and c that will guarantee that the equation sin b sin c 0 has at least one solution on some interval of length. Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

24 558 Chapter 7 Analytic Trigonometry 7.3 Eercises VOCABULARY CHECK: Fill in the blanks. 7. The equation sin 0 has the solutions n and n, which are called solutions The equation tan 3 tan 0 is a trigonometric equation that is of type. 3. A solution to an equation that does not satisfy the original equation is called an solution. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Eercises 6, verify that the -values are solutions of the equation... (a) (a) 3. 3 tan 0 (a) 4. cos (a) (a) (a) (b) (b) (b) (b) (b) (b) In Eercises 7 0, solve the equation. 3. cos 0 3 sec sin sin 0 csc 4 4 csc 0 6 sin sin tan tan cos 0 8. sin csc 0 0. tan sec cot cos 0 6. sin 3 cos 7. sin 8. tan tan 3tan 0 0. cos cos 0 In Eercises 34, find all solutions of the equation in the interval [0,.. cos 3 cos. sec tan 3 tan 4. sin cos 5. sec sec 6. sec csc csc 7. sin csc 0 8. sec tan cos cos 0 sin 3 sin 0 sec tan 3 0 cos sin tan csc cot sin cos In Eercises 35 40, solve the multiple-angle equation. 35. cos tan sec cos 40. In Eercises 4 44, find the -intercepts of the graph. 4. y sin 4. y sin cos y 43. y 44. y sec 4 8 tan y sin 3 sin 3 y y Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

25 Section 7.3 Solving Trigonometric Equations 559 In Eercises 45 54, use a graphing utility to approimate the solutions (to three decimal places) of the equation in the interval [0, In Eercises 55 58, use the Quadratic Formula to solve the equation in the interval [0,. Then use a graphing utility to approimate the angle In Eercises 59 6, use inverse functions where needed to find all solutions of the equation in the interval [0, In Eercises 63 and 64, (a) use a graphing utility to graph the function and approimate the maimum and minimum points on the graph in the interval [0,, and (b) solve the trigonometric equation and demonstrate that its solutions are the -coordinates of the maimum and minimum points of f. (Calculus is required to find the trigonometric equation.) sin cos 0 4 sin 3 sin sin 0 sin cos cos cot sin 3 Function f sin cos cos sin tan cos 0 sec 0.5 tan 0 csc 0.5 cot 5 0 tan 7 tan sin 7 sin 0 sin 3 sin tan 4 tan 4 0 tan 3 tan 0 4 cos 4 cos 0 tan 6 tan 5 0 sec tan 3 0 cos 5 cos 0 sin 7 sin 3 0 f sin cos Trigonometric Equation cos sin 0 cos 4 sin cos 0 Fied Point In Eercises 65 and 66, find the smallest positive fied point of the function f. [ A fied point of a function f is a real number c such that fc c.] 65. f tan 66. f cos Graphical Reasoning Consider the function given by f cos and its graph shown in the figure. (a) What is the domain of the function? (b) Identify any symmetry and any asymptotes of the graph. (c) Describe the behavior of the function as 0. (d) How many solutions does the equation cos 0 have in the interval,? Find the solutions. (e) Does the equation cos 0 have a greatest solution? If so, approimate the solution. If not, eplain why. 68. Graphical Reasoning Consider the function given by f sin and its graph shown in the figure. (a) What is the domain of the function? (b) Identify any symmetry and any asymptotes of the graph. (c) Describe the behavior of the function as 0. (d) How many solutions does the equation sin π π y y have in the interval 8, 8? Find the solutions. π π Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

26 560 Chapter 7 Analytic Trigonometry 69. Harmonic Motion A weight is oscillating on the end of a spring (see figure). The position of the weight relative to the point of equilibrium is given by y cos 8t 3 sin 8t, where y is the displacement (in meters) and t is the time (in seconds). Find the times when the weight is at the point of equilibrium y 0 for 0 t. 74. Projectile Motion A sharpshooter intends to hit a target at a distance of 000 yards with a gun that has a muzzle velocity of 00 feet per second (see figure). Neglecting air resistance, determine the gun s minimum angle of elevation if the range r is given by r 3 v 0 sin. Equilibrium θ r = 000 yd y Not drawn to scale 70. Damped Harmonic Motion The displacement from equilibrium of a weight oscillating on the end of a spring is given by y.56e 0.t cos 4.9t, where y is the displacement (in feet) and t is the time (in seconds). Use a graphing utility to graph the displacement function for 0 t 0. Find the time beyond which the displacement does not eceed foot from equilibrium. 7. Sales The monthly sales S (in thousands of units) of a seasonal product are approimated by S sin where t is the time (in months), with t corresponding to January. Determine the months when sales eceed 00,000 units. 7. Sales The monthly sales S (in hundreds of units) of skiing equipment at a sports store are approimated by t S cos 6 where t is the time (in months), with t corresponding to January. Determine the months when sales eceed 7500 units. 73. Projectile Motion A batted baseball leaves the bat at an angle of with the horizontal and an initial velocity of v 0 00 feet per second. The ball is caught by an outfielder 300 feet from home plate (see figure). Find if the range r of a projectile is given by r 3 v 0 sin. t 6 θ 75. Ferris Wheel A Ferris wheel is built such that the height h (in feet) above ground of a seat on the wheel at time t (in minutes) can be modeled by ht sin 6 t. The wheel makes one revolution every 3 seconds. The ride begins when t 0. (a) During the first 3 seconds of the ride, when will a person on the Ferris wheel be 53 feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts 60 seconds, how many times will a person be at the top of the ride, and at what times? 76. Data Analysis: Unemployment Rate The table shows the unemployment rates r in the United States for selected years from 990 through 004. The time t is measured in years, with t 0 corresponding to 990. (Source: U.S. Bureau of Labor Statistics) Time, t Model It Rate, r Time, t (a) Create a scatter plot of the data. Rate, r r = 300 ft Not drawn to scale Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

27 Section 7.3 Solving Trigonometric Equations Geometry The area of a rectangle (see figure) inscribed in one arc of the graph of y cos is given by A cos, (a) Use a graphing utility to graph the area function, and approimate the area of the largest inscribed rectangle. (b) Determine the values of for which A. 78. Quadratic Approimation Consider the function given by f 3 sin0.6. (a) Approimate the zero of the function in the interval 0, 6. (b) A quadratic approimation agreeing with f at 5 is g Use a graphing utility to graph f and g in the same viewing window. Describe the result. (c) Use the Quadratic Formula to find the zeros of g. Compare the zero in the interval 0, 6 with the result of part (a). Synthesis Model It (continued) (b) Which of the following models best represents the data? Eplain your reasoning. () () (3) r.4 sin0.47t r.4 sin0.47t r sin0.0t (4) r 896 sin0.57t (c) What term in the model gives the average unemployment rate? What is the rate? (d) Economists study the lengths of business cycles such as unemployment rates. Based on this short span of time, use the model to find the length of this cycle. (e) Use the model to estimate the net time the unemployment rate will be 5% or less. π 0 < <. True or False? In Eercises 79 and 80, determine whether the statement is true or false. Justify your answer. 79. The equation sin 4t 0 has four times the number of solutions in the interval 0, as the equation sin t 0. y π 80. If you correctly solve a trigonometric equation to the statement sin 3.4, then you can finish solving the equation by using an inverse function. In Eercises 8 and 8, use the graph to approimate the number of points of intersection of the graphs of and y. 8. y sin 8. sin y Skills Review In Eercises 83 and 84, solve triangle ABC by finding all missing angle measures and side lengths. 83. B B A 3 y A y y y π 4.6 In Eercises 85 88, use reference angles to find the eact values of the sine, cosine, and tangent of the angle with the given measure C C y Angle of Depression Find the angle of depression from the top of a lighthouse 50 feet above water level to the water line of a ship miles offshore. 90. Height From a point 00 feet in front of a public library, the angles of elevation to the base of the flagpole and the top of the pole are 8 and 39 45, respectively. The flagpole is mounted on the front of the library s roof. Find the height of the flagpole. 9. Make a Decision To work an etended application analyzing the normal daily high temperatures in Phoeni and in Seattle, visit this tet s website at college.hmco.com. (Data Source: NOAA) y π y y y. Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

28 56 Chapter 7 Analytic Trigonometry 7.4 Sum and Difference Formulas What you should learn Use sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations. Why you should learn it You can use identities to rewrite trigonometric epressions. For instance, in Eercise 75 on page 567, you can use an identity to rewrite a trigonometric epression in a form that helps you analyze a harmonic motion equation. Using Sum and Difference Formulas In this and the following section, you will study the uses of several trigonometric identities and formulas. Sum and Difference Formulas sinu v sin u cos v cos u sin v sinu v sin u cos v cos u sin v cosu v cos u cos v sin u sin v cosu v cos u cos v sin u sin v For a proof of the sum and difference formulas, see Proofs in Mathematics on page 586. Eploration tanu v tanu v tan u tan v tan u tan v tan u tan v tan u tan v Use a graphing utility to graph y cos and y cos cos in the same viewing window. What can you conclude about the graphs? Is it true that cos cos cos? Use a graphing utility to graph y sin 4 and y sin sin 4 in the same viewing window. What can you conclude about the graphs? Is it true that sin 4 sin sin 4? Richard Megna/Fundamental Photographs Eamples and show how sum and difference formulas can be used to find eact values of trigonometric functions involving sums or differences of special angles. Eample Evaluating a Trigonometric Function Find the eact value of cos 75. To find the eact value of cos 75, use the fact that Consequently, the formula for cosu v yields cos 75 cos30 45 cos 30 cos 45 sin 30 sin 45 3 Try checking this result on your calculator. You will find that cos Now try Eercise Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

29 Section 7.4 Sum and Difference Formulas 563 Eample Evaluating a Trigonometric Epression The Granger Collection, New York Historical Note Hipparchus, considered the most eminent of Greek astronomers, was born about 60 B.C. in Nicaea.He was credited with the invention of trigonometry. He also derived the sum and difference formulas for sina ± B and cosa ± B. Find the eact value of sin. Using the fact that together with the formula for sinu v, you obtain sin sin 3 sin 3 cos cos 4 3 sin Now try Eercise 3. Eample 3 Evaluating a Trigonometric Epression Find the eact value of sin 4 cos cos 4 sin. Recognizing that this epression fits the formula for sinu v, you can write sin 4 cos cos 4 sin sin4 sin 30. Eample 4 Now try Eercise 3. An Application of a Sum Formula u v FIGURE 7.7 Write cosarctan arccos as an algebraic epression. This epression fits the formula for cosu v. Angles u arctan and v arccos are shown in Figure 7.7. So cosu v cosarctan cosarccos sinarctan sinarccos. Now try Eercise 5. Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

30 564 Chapter 7 Analytic Trigonometry Eample 5 shows how to use a difference formula to prove the cofunction identity cos sin. Eample 5 Proving a Cofunction Identity Prove the cofunction identity Using the formula for cosu v, you have cos cos Now try Eercise 55. Sum and difference formulas can be used to rewrite epressions such as sin n cos sin. cos sin sin 0cos sin sin. and cos n where n is an integer, as epressions involving only sin or cos. The resulting formulas are called reduction formulas. Eample 6 Deriving Reduction Formulas Simplify each epression. a. cos 3 b. a. Using the formula for cosu v, you have cos 3 cos cos 3 sin sin 3 cos 0 sin sin. b. Using the formula for tanu v, you have tan 3 tan tan 3 tan tan 3 tan 0 tan 0 tan. tan 3 Now try Eercise 65. Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

31 Section 7.4 Sum and Difference Formulas 565 Eample 7 Solving a Trigonometric Equation y 3 3 π π π π ( ( ( π y = sin + + sin FIGURE 7.8 ( Find all solutions of sin Using sum and difference formulas, rewrite the equation as So, the only solutions in the interval 0, are 5 4 and You can confirm this graphically by sketching the graph of 4 sin 7 4. in the interval 0,. 4 sin cos cos sin sin cos cos sin y sin for 0 <, 4 sin 4 as shown in Figure 7.8. From the graph you can see that the -intercepts are 54 and 74. Now try Eercise 69. sin cos 4 sin sin sin. The net eample was taken from calculus. It is used to derive the derivative of the sine function. Eample 8 An Application from Calculus Verify that sin h sin h where h 0. sin h cos h sin Using the formula for sinu v, you have cos h h sin h sin sin cos h cos sin h sin h h cos sin h sin cos h h sin h cos h sin cos h h. Now try Eercise 9. Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

32 566 Chapter 7 Analytic Trigonometry 7.4 Eercises VOCABULARY CHECK: Fill in the blank to complete the trigonometric identity.. sinu v. cosu v 3. tanu v 4. sinu v 5. cosu v 6. tanu v PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Eercises 6, find the eact value of each epression.. (a) cos0 45 (b) cos 0 cos 45. (a) sin35 30 (b) sin 35 cos (a) cos (b) cos cos (a) (b) sin 3 5 sin sin (a) sin (b) 6 6. (a) sin35 60 (b) sin 35 sin 60 In Eercises 7, find the eact values of the sine, cosine, and tangent of the angle by using a sum or difference formula In Eercises 3 30, write the epression as the sine, cosine, or tangent of an angle sin 40 cos 50 cos 40 sin cos 5 cos 5 sin 5 sin 5 tan 35 tan 86 tan 35 tan 86 tan 40 tan 60 tan 40 tan sin 7 sin sin 3 cos. cos 3 sin. 8. cos 7 cos sin 5 7 sin In Eercises 3 36, find the eact value of the epression cos 5 cos 60 sin 5 sin sin cos cos 4 sin cos tan tan tan tan cos 3 cos y sin 3 sin y sin 330 cos 30 cos 330 sin 30 cos 3 6 sin 6 tan 5 tan 0 tan 5 tan 0 tan54 tan tan54 tan In Eercises 37 44, find the eact value of the trigonometric function given that sin u 5 and cos v (Both u and v are in Quadrant II.) 37. sinu v 38. cosu v 39. cosu v 40. sinv u 4. tanu v 4. cscu v 43. secv u 44. cotu v In Eercises 45 50, find the eact value of the trigonometric 7 function given that sin u and cos v (Both u and v are in Quadrant III.) 45. cosu v 46. sinu v 47. tanu v 48. cotv u 49. secu v 50. cosu v sin 3 6 Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

33 In Eercises 5 54, write the trigonometric epression as an algebraic epression. Section 7.4 Sum and Difference Formulas sinarcsin arccos 5. sinarctan arccos 75. Harmonic Motion A weight is attached to a spring 53. cosarccos arcsin suspended vertically from a ceiling. When a driving 54. cosarccos arctan force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by In Eercises 55 64, verify the identity cos y cos y cos cos y In Eercises 65 68, simplify the epression algebraically and use a graphing utility to confirm your answer graphically cos sin 68. In Eercises 69 7, find all solutions of the equation in the interval [0, sin3 sin sin 6 cos 3 sin cos 5 4 cos tan 4 sin sin 6 sin cos sin sin 0 3 sin 7. cos 4 cos 4 7. tan sin 0 tan tan cos y cos y cos sin y sin y sin y) sin sin y sin y sin y sin cos y 3 6 sin cos cos tan 76. Standing Waves The equation of a standing wave is obtained by adding the displacements of two waves traveling in opposite directions (see figure). Assume that each of the waves has amplitude A, period T, and wavelength. If the models for these waves are y A cos t T show that y y A cos t T t = 0 t = y 3 sin t cos t 4 where y is the distance from equilibrium (in feet) and t is the time (in seconds). (a) Use the identity T 8 y Model It a sin B b cos B a b sinb C where C arctanba, a > 0, to write the model in the form y a b sinbt C. (b) Find the amplitude of the oscillations of the weight. (c) Find the frequency of the oscillations of the weight. and cos. y y + y y y + y y A cos t T y In Eercises 73 and 74, use a graphing utility to approimate the solutions in the interval [0,. 73. cos 4 cos tan cos 0 t = T 8 y y + y y Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

34 568 Chapter 7 Analytic Trigonometry Synthesis True or False? In Eercises 77 80, determine whether the statement is true or false. Justify your answer. 77. sinu ± v sin u ± sin v 78. cosu ± v cos u ± cos v 79. cos 80. sin sin cos In Eercises 8 84, verify the identity. 8. n cos, n is an integer 8. n sin, n is an integer 83. a sin B b cos B a b sinb C, where C arctanba and a > a sin B b cos B a b cosb C, where C arctanab and b > 0 In Eercises 85 88, use the formulas given in Eercises 83 and 84 to write the trigonometric epression in the following forms. (a) cosn sinn a b sinb C 85. sin cos sin 4 cos In Eercises 89 and 90, use the formulas given in Eercises 83 and 84 to write the trigonometric epression in the form a sin B b cos B. (b) 87. sin 3 5 cos sin cos 89. sin cos 9. Verify the following identity used in calculus. cos h cos h cos cos h h 9. Eploration Let 6 in the identity in Eercise 9 and define the functions f and g as follows. f h cos6 h cos6 h cos h gh cos 6 h sin 6 sin h h (a) What are the domains of the functions f and g? (b) Use a graphing utility to complete the table. h f h gh a b cosb C sin sin h h (c) Use a graphing utility to graph the functions f and g. (d) Use the table and the graphs to make a conjecture about the values of the functions f and g as h 0. In Eercises 93 and 94, use the figure, which shows two lines whose equations are y m b and Assume that both lines have positive slopes. Derive a formula for the angle between the two lines.then use your formula to find the angle between the given pair of lines. 93. y and y y and y 95. Conjecture Consider the function given by f sin 4 sin Use a graphing utility to graph the function and use the graph to create an identity. Prove your conjecture. 96. Proof (a) Write a proof of the formula for sinu v. (b) Write a proof of the formula for sinu v. Skills Review y = m + b 4 3 In Eercises 97 00, find the inverse function of f. Verify that f f and f f. 97. f f f f 6 In Eercises 0 04, apply the inverse properties of ln and e to simplify the epression. 0. log 0. log ln 03. e ln e 6 4 y m b. y θ y = m + b 4. Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).

35 Section 7.5 Multiple-Angle and Product-to-Sum Formulas Multiple Angle and Product-to-Sum Formulas What you should learn Use multiple-angle formulas to rewrite and evaluate trigonometric functions. Use power-reducing formulas to rewrite and evaluate trigonometric functions. Use half-angle formulas to rewrite and evaluate trigonometric functions. Use product-to-sum and sum-to-product formulas to rewrite and evaluate trigonometric functions. Use trigonometric formulas to rewrite real-life models. Why you should learn it You can use a variety of trigonometric formulas to rewrite trigonometric functions in more convenient forms. For instance, in Eercise 9 on page 579, you can use a double-angle formula to determine at what angle an athlete must throw a javelin. Mark Dadswell/Getty Images Multiple-Angle Formulas In this section, you will study four other categories of trigonometric identities.. The first category involves functions of multiple angles such as sin ku and cos ku.. The second category involves squares of trigonometric functions such as sin u. 3. The third category involves functions of half-angles such as sinu. 4. The fourth category involves products of trigonometric functions such as sin u cos v. You should learn the double-angle formulas because they are used often in trigonometry and calculus. For proofs of the formulas, see Proofs in Mathematics on page 587. Double-Angle Formulas sin u sin u cos u tan u Eample Solve cos sin 0. Solving a Multiple-Angle Equation Begin by rewriting the equation so that it involves functions of rather than. Then factor and solve as usual. cos 0 and sin 0, 3 So, the general solution is n tan u tan u cos sin 0 cos sin cos 0 cos sin 0 and Write original equation. Double-angle formula Factor. Set factors equal to zero. s in 0, where n is an integer. Try verifying these solutions graphically. Now try Eercise 9. 3 cos u cos u sin u 3 n cos u sin u Copyright 0 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the ebook and/or echapter(s).