Analytic Trigonometry

Transcription

1 0 Analytic Trigonometry In this chapter, you will study analytic trigonometry. Analytic trigonometry is used to simplify trigonometric epressions and solve trigonometric equations. In this chapter, you should learn the following. How to use the fundamental trigonometric identities to evaluate, simplify, and rewrite trigonometric epressions. (0.) How to verify trigonometric identities. (0.) How to use standard algebraic techniques to solve trigonometric equations, solve trigonometric equations of quadratic type, solve trigonometric equations involving multiple angles, and use inverse trigonometric functions to solve trigonometric equations. (0.3) How to use sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations. (0.4) How to use multiple-angle formulas, power-reducing formulas, half-angle formulas, product-to-sum formulas, and sum-to-product formulas to rewrite and evaluate trigonometric functions, and rewrite real-life problems. (0.5) Steve Chenn/Brand X Pictures/Jupiter Images Given a function that models the range of a javelin in terms of the velocity and the angle thrown, how can you determine the angle needed to throw a javelin 30 feet at a velocity of 75 feet per second? (See Section 0.5, Eercise 4.) Many trigonometric equations have an infinite number of solutions. You will learn how to use fundamental trigonometric identities and the rules of algebra to find all possible solutions to trigonometric equations. (See Section 0.3.) 645 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).

2 646 Chapter 0 Analytic Trigonometry 0. Using Fundamental Trigonometric Identities Recognize and write the fundamental trigonometric identities. Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric epressions, and rewrite trigonometric epressions. STUDY TIP Recall that an identity is an equation that is true for every value in the domain of the variable. For instance, csc u sin u is an identity because it is true for all values of u for which csc u is defined. Introduction In Chapter 9, 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 trigonometric identities to do the following.. Evaluate trigonometric functions.. Simplify trigonometric epressions. 3. Develop additional trigonometric identities. 4. Solve trigonometric equations. STUDY TIP 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. 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 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 0. Using Fundamental Trigonometric Identities 647 Using the Fundamental Trigonometric Identities One common application of trigonometric identities is to use given values of trigonometric functions to evaluate other trigonometric functions. EXAMPLE Using Trigonometric Identities to Evaluate a Function Use the values sec u 3 and tan u > 0 to find the values of all si trigonometric functions. Solution Using a reciprocal identity, you have cos u Reciprocal identity sec u Substitute 3 for sec u Using a Pythagorean identity, you have sin u cos u 3 Simplify. Pythagorean identity Substitute 3 for cos u Simplify. TECHNOLOGY You can use a graphing utility to check the result of Eample. To do this, graph and y sin cos sin y sin 3 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. 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 EXAMPLE Simplifying a Trigonometric Epression Simplify sin cos sin. csc u 3 sin u sec u cos u 3 cot u tan u Solution First factor out a common monomial factor and then use a fundamental identity. sin cos sin sin cos sin cos Factor out common monomial factor. Factor out. sin sin Pythagorean identity sin 3 Multiply. 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).

4 648 Chapter 0 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. EXAMPLE 3 Factoring Trigonometric Epressions Factor each epression. a. sec b. 4 tan tan 3 Solution a. This epression has the form u v, which is the difference of two squares. It factors as sec sec sec ). b. This epression has the polynomial form a b c, and it factors as 4 tan tan 3 4 tan 3tan. 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 shown in Eamples 4 and 5, respectively. EXAMPLE 4 Factoring a Trigonometric Epression Factor csc cot 3. Solution Use the identity csc cot to rewrite the epression in terms of the cotangent. csc cot 3 cot cot 3 cot cot cot cot Pythagorean identity Combine like terms. Factor. STUDY TIP Remember that when adding rational epressions, you must first find the least common denominator (LCD). In Eample 5, the LCD is sin t. EXAMPLE 5 Simplifying a Trigonometric Epression Simplify sin t cot t cos t. Solution 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 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 0. Using Fundamental Trigonometric Identities 649 EXAMPLE 6 Adding Trigonometric Epressions Perform the addition and simplify. sin cos cos sin Solution sin cos sin sin cos cos cos sin cos sin sin cos cos cos sin cos cos sin sin csc 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 useful when integrating. In particular, it is often useful to convert a fraction with a binomial denominator into one with a monomial denominator. EXAMPLE 7 Rewriting a Trigonometric Epression Rewrite sin so that it is not in fractional form. Solution From the Pythagorean identity cos sin sin sin you can see that multiplying both the numerator and the denominator by will produce a monomial denominator. sin sin sin sin sin sin sin sin cos sin cos cos sin cos cos cos sec tan sec 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).

6 650 Chapter 0 Analytic Trigonometry EXAMPLE 8 Trigonometric Substitution That Removes a Radical Use the substitution tan, 0 < 4 <, to write as a trigonometric function of. Solution Begin by letting tan. Then, you can obtain 4 4 tan 4 4 tan 4 tan 4 sec sec. Substitute tan Rule of eponents Factor. Pythagorean identity sec > 0 for 0 < for. < 4 + θ = arctan Angle whose tangent is /. Figure 0. Figure 0. shows the right triangle illustration of the trigonometric substitution tan in Eample 8. For 0 < <, you have opp, adj, and hyp 4. With these epressions, you can write the following. 4 sec sec 4 0. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 0, fill in the blank to complete the trigonometric identity.. sin u cos u. 3. tan u csc u tan u sin u sec u cosu tanu csc u cos u In Eercises 4, use the given values to evaluate (if possible) all si trigonometric functions.. sin cos 3, tan 3 3, sec, csc 5 7, tan 8 5, cos 3 sin tan 7 4 sec cot 3, sin sec 3 csc 35, 5 8. cos 3 5, cos sin tan 3, 4 0. sec 4, sin > 0. tan,. csc 5, cos < 0 3. sin, cot 0 4. tan is undefined, sin > 0 sin < 0 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 0. Using Fundamental Trigonometric Identities 65 In Eercises 5 30, 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 sin cos cos In Eercises 3 48, use the fundamental trigonometric identities to simplify the epression. There is more than one correct form of each answer. 3. cot sec 3. cos tan 33. sin csc sin 34. sec sin 35. cot csc 36. csc sec 37. sin 38. csc tan 39. tan sec sin 40. tan sec cot sec cos 43. cos y sin y 44. cos t tan t 45. sin tan cos 46. csc tan sec 47. cot u sin u tan u cos u 48. sin sec cos csc In Eercises 49 60, factor the epression and use the fundamental identities to simplify. There is more than one correct form of each answer. 49. tan tan sin 50. sin csc sin 5. sin sec sin 5. cos cos tan 53. sec cos sec cos 55. tan 4 tan 56. cos cos sin 4 cos sec 4 tan csc 3 csc csc sec 3 sec sec In Eercises 6 64, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer. 6. sin cos 6. cot csc cot csc csc csc 3 3 sin 3 3 sin In Eercises 65 68, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer sec cos cos sec cos sin tan sec sin cos tan In Eercises 69 7, 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 tan sec 3 tan sec tan csc WRITING ABOUT CONCEPTS In Eercises 73 78, determine whether the equation is an identity, and give a reason for your answer. 73. cos sin 74. cot csc 75. sin kcos k tan, k is a constant cos 5 sec 77. sin csc 78. csc 79. Epress each of the other trigonometric functions of in terms of sin. 80. Epress each of the other trigonometric functions of in terms of cos. Numerical and Graphical Analysis In Eercises 8 84, use a graphing utility to complete the table and graph the functions. Make a conjecture about and y y y 8. y cos y sin, 8. y sec cos, y sin tan sin 83. y cos y sin, cos 84. y y tan tan 4 sec 4 sec, 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).

8 65 Chapter 0 Analytic Trigonometry In Eercises 85 88, use a graphing utility to determine which of the si trigonometric functions is equal to the epression. Verify your answer algebraically. 85. cos cot sin 86. sec csc tan sin cos cos sin cos cos sin In Eercises 89 94, use the trigonometric substitution to write the algebraic epression as a trigonometric function of, where 0 < < / , 3 cos , cos , 4, 00, 9 5, 7 sin sec 0 tan 3 5 tan In Eercises 95 98, use the trigonometric substitution to write the algebraic equation as a trigonometric equation of, where / < < /. Then find sin and cos , 3 36, 3 sin 6 sin , 53 00, cos 0 cos In Eercises 99 0, use a graphing utility to solve the equation for, where 0 <. 99. sin cos 00. cos sin 0. sec tan 0. csc cot In Eercises 03 06, rewrite the epression as a single logarithm and simplify the result. 03. lncos lnsin 04. lnsec lnsin 05. ln cot t ln tan t 06. lncos t ln tan t In Eercises 07 0, use a calculator to demonstrate the identity for each value of csc cot (a) (b) tan sec (a) (b) 09. cos sin (a) (b) 0. sin sin (a) (b) 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 W cos W sin where is the coefficient of friction. Solve the equation for and simplify the result. 3. Rate of Change The rate of change of the function f sec cos is given by the epression sec tan sin. Show that this epression can also be written as sin tan. True or False? In Eercises 4 and 5, determine whether the statement is true or false. Justify your answer. 4. The even and odd trigonometric identities are helpful for determining whether the value of a trigonometric function is positive or negative. 5. A cofunction identity can be used to transform a tangent function so that it can be represented by a cosecant function. In Eercises 6 9, fill in the blanks. (Note: The notation c indicates that approaches c from the right and c indicates that approaches c from the left.) 6. As, sin and csc. 7. As and sec. 0, cos CAPSTONE. (a) Use the definitions of sine and cosine to derive the Pythagorean identity sin cos. 8. As, tan and cot. 9. As and csc., sin θ (b) 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. W 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 0. Verifying Trigonometric Identities Verifying Trigonometric Identities Verify trigonometric identities. 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. 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).

10 654 Chapter 0 Analytic Trigonometry EXAMPLE Verifying a Trigonometric Identity STUDY TIP 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 sec sin. sec Solution The left side is more complicated, so start with it. sec tan Pythagorean identity sec sec tan Simplify. sec tan cos Reciprocal identity sin Quotient identity cos cos sin 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. There can be 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 EXAMPLE Verifying a Trigonometric Identity Verify the identity sec. sin sin Algebraic Solution The right side is more complicated, so start with it. sin sin Add fractions. sin sin sin sin Simplify. sin Pythagorean identity cos sec Reciprocal identity Numerical Solution Use the table feature of a graphing utility set in radian mode to create a table that shows the values of y cos and y sin sin for different values of, as shown in Figure 0.. From the table, you can see that the values appear to be identical, so sec sin sin appears to be an identity. Figure 0. 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 0. Verifying Trigonometric Identities 655 EXAMPLE 3 Verifying a Trigonometric Identity Verify the identity tan cos tan. Algebraic Solution 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 Graphical Solution Use a graphing utility set in radian mode to graph the left side of the identity y tan cos and the right side of the identity y tan in the same viewing window, as shown in Figure 0.3. (Select the line style for y and the path style for y.) Because the graphs appear to coincide, tan cos tan appears to be an identity. y = (tan + )(cos ) 3 y = tan Figure 0.3 STUDY TIP Although a graphing utility can be useful in helping to verify an identity, you must use algebraic techniques to produce a valid proof. EXAMPLE 4 Converting to Sines and Cosines Verify the identity tan cot sec csc. Solution Try converting the left side into sines and cosines. tan cot sin cos cos sin sin cos cos sin cos sin cos sin sec csc Quotient identities Add fractions. Pythagorean identity Product of fractions Reciprocal identities STUDY TIP 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 cos cos cos sin csc cos This technique is demonstrated in the net eample. 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).

12 656 Chapter 0 Analytic Trigonometry EXAMPLE 5 Verifying a Trigonometric Identity Verify the identity sec tan cos sin. Algebraic Solution Graphical Solution Begin with the right side because you can create a monomial denominator by multiplying the numerator and denominator by Use a graphing utility set in the radian and dot modes to graph y sec tan and y cos sin in the same sin. viewing window, as shown in Figure 0.4. Because the graphs cos cos sin sin sin sin Multiply numerator appear to coincide, sec tan cos sin appears and denominator by to be an identity. sin. cos cos sin sin cos cos sin cos cos cos sin cos cos sin cos cos sec tan Multiply. Pythagorean identity Write as separate fractions. Simplify. Identities 7 Figure y = sec + tan 5 y = 9 cos sin 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. EXAMPLE 6 Working with Each Side Separately Verify the identity cot sin. csc sin Algebraic Solution Working with the left side, you have cot csc Pythagorean identity csc csc csc csc Factor. csc csc. Simplify. Now, simplifying the right side, you have sin sin Write as separate fractions. sin sin sin csc. Reciprocal identity The identity is verified because both sides are equal to csc. Numerical Solution Use the table feature of a graphing utility set in radian mode to create a table that shows the values of y cot csc and y sin sin for different values of, as shown in Figure 0.5. From the table you can see that the values appear to be identical, so cot csc sin sin appears to be an identity. Figure 0.5 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).

13 0. Verifying Trigonometric Identities 657 In Eample 7, powers of trigonometric functions are rewritten as more complicated sums of products of trigonometric functions. This is a common procedure used to integrate trigonometric power functions. EXAMPLE 7 Verifying Trigonometric Identities Verify each identity. a. tan 4 tan sec tan b. sin 3 cos 4 cos 4 cos 6 sin c. csc 4 cot csc cot cot 3 Solution a. tan 4 tan tan tan sec tan sec tan Write as separate factors. Pythagorean identity Multiply. b. sin 3 cos 4 sin cos 4 sin cos cos 4 sin cos 4 cos 6 sin Write as separate factors. Pythagorean identity Multiply. c. csc 4 cot csc csc cot csc cot cot csc cot cot 3 Write as separate factors. Pythagorean identity Multiply. 0. Eercises See for worked-out solutions to odd-numbered eercises. 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. cos u cot u sin u cos u sin u 7. cscu 8. secu In Eercises 9 46, verify the identity. 9. tan t cot t 0. sec y cos y. cot ysec y cos sin tan sec sin sin cos cos sin cos 5. cos sin sin 6. sin sin 4 cos cos 4 tan 7. sin tan sec cot 3 t 8. csc t cos tcsc t cot t tan sec csc t sin t sin t tan tan. sin cos sin 5 cos cos 3 sin. sec 6 sec tan sec 4 sec tan sec 5 tan 3 cot sec 3. csc sin 4. sec sec cos 5. csc sin cos cot 6. sec cos sin tan 7. tan tan cot cot 8. sin csc sin csc sin 9. cos sec 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).

14 658 Chapter 0 Analytic Trigonometry cos cot 30. csc sin 3. cos csc cot cos 3. cos cos sin cos tan sin cos 33. tan cos 34. tan tan sin tan cot csc 35. sec 36. cos sec cot 37. sin y siny cos y tan tan y cot cot y 38. tan tan y cot cot y tan cot y 39. tan y cot tan cot y cos cos y sin sin y 40. sin sin y cos cos y 0 4. sin sin sin cos 4. cos cos cos sin 43. cos cos 44. sec y cot y 45. sin t csc t tan t 46. sec cot In Eercises 47 54, (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. 47. cot cos cot sin cos 48. csc csc sin cot csc sin 49. cos 3 cos 4 sin 3 cos 50. tan 4 tan 3 sec 4 tan 3 5. csc 4 csc cot 4 5. sin 4 sin cos cos cos sin sin cos 54. cot csc csc cot In Eercises 55 58, verify the identity. 55. tan 5 tan 3 sec tan sec 4 tan tan tan 4 sec 57. cos 3 sin sin sin 4 cos 58. sin 4 cos 4 cos cos 4 WRITING ABOUT CONCEPTS In Eercises 59 64, 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 6. tan tan 6. sin cos sin cos 63. tan sec 64. csc cot n 65. Verify that for all integers, cos Verify that for all integers n n, sin 6. In Eercises 67 70, use the cofunction identities to evaluate the epression without using a calculator. 67. sin 5 sin cos 55 cos cos 0 cos 5 cos 38 cos 70 tan 63 cot 6 sec 74 csc 7 7. 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. CAPSTONE 7. Write a short paper in your own words eplaining to a classmate the difference between a trigonometric identity and a conditional equation. Include suggestions on how to verify a trigonometric identity. True or False? In Eercises 73 and 74, determine whether the statement is true or false. Justify your answer. 73. There can be more than one way to verify a trigonometric identity. 74. The equation sin cos tan is an identity because sin 0 cos 0 and tan 0. 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 0.3 Solving Trigonometric Equations Solving Trigonometric Equations 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. 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 in the equation. For eample, to solve the equation sin, divide each side by to obtain sin To solve for note in Figure 0.6 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 where n is an integer, as shown in Figure 0.6. y 5 n 6 General solution = π π = π = π + π 6 y = 6 6 π π Figure 0.6 Another way to show that the equation sin has infinitely many solutions is indicated in Figure 0.7. Any angles that are coterminal with 6 or 56 will also be solutions of the equation. ( ) = 5π π = 5π = 5π + π y = sin ( ) sin 5π + nπ = sin π + nπ = 6 6 5π 6 π 6 Figure 0.7 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).

16 660 Chapter 0 Analytic Trigonometry EXAMPLE Collecting Like Terms Solve sin sin. Solution Begin by rewriting the equation so that sin is isolated on one side of the equation. sin sin Write original equation. sin sin 0 Add sin to each side. sin sin Subtract from each side. sin Combine like terms. sin 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. and 7 n 4 General solution EXAMPLE Etracting Square Roots STUDY TIP When you etract square roots, make sure you account for both the positive and negative solutions. Solve 3 tan 0. Solution Begin by rewriting the equation so that tan is isolated on one side of the equation. 3 tan 0 Write original equation. 3 tan Add to each side. tan 3 Divide each side by 3. tan ± 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 ± 3 3 and 5 n 6 General solution where n is an integer. 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 0.3 Solving Trigonometric Equations 66 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. π Figure y = cot cos cot y π EXAMPLE 3 Factoring Solve cot cos cot. Solution Begin by rewriting the equation so that all terms are collected on one side of the equation. cot cos cot 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 cos 0 cos cos ±. 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 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 0.8. 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. NOTE In Eample 3, don t make the mistake of dividing each side of the equation by cot. If you do this, you lose the solutions. Can you see why? 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).

18 66 Chapter 0 Analytic Trigonometry EXAMPLE 4 Factoring an Equation of Quadratic Type Find all solutions of sin sin 0 in the interval 0,. Algebraic Solution Begin by treating the equation as a quadratic in sin and factoring. sin sin 0 Write original equation. sin sin 0 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 Solution Use a graphing utility set in radian mode to graph y sin sin for 0 <, as shown in Figure 0.9. 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 , π Figure 0.9 EXAMPLE 5 Rewriting with a Single Trigonometric Function STUDY TIP In Eample 5, conversion to cosine was chosen because the identity relating sine and cosine sin cos involves their squares. When using the Pythagorean identities to convert equations to one function, keep in mind their function pairs and powers. Solve sin 3 cos 3 0. Solution 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. sin 3 cos 3 0 cos 3 cos 3 0 cos 3 cos 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 and cos 0 cos cos 0 3, 5 3 Because cos has a period of, the general form of the solution is obtained by adding multiples of to get n, n, 3 5 n 3 General solution where n is an integer. 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 0.3 Solving Trigonometric Equations 663 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. EXAMPLE 6 Squaring and Converting to Quadratic Type STUDY TIP 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 for the squares of the cosecant and cotangent functions. Find all solutions of cos sin in the interval 0,. Solution 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 sin Write original equation. cos cos sin Square each side. cos cos cos Pythagorean identity cos cos cos 0 Rewrite equation. cos cos 0 Combine like terms. cos cos 0 Factor. Setting each factor equal to zero produces and cos 0 cos. Because you squared the original equation, check for etraneous solutions. Check / cos? sin 0 Check 3/ cos cos 0 cos 0 3, 3? sin 3 0 Check cos? sin 0 Substitute for. Solution checks. Substitute 3 for. Solution does not check. Substitute for. Solution checks. Of the three possible solutions, 3 is etraneous. So, in the interval 0,, the only two solutions are and. NOTE In Eample 6, the general solution is n and n where n is an integer. 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).

20 664 Chapter 0 Analytic Trigonometry Functions Involving Multiple Angles The net two eamples involve trigonometric functions of multiple angles of the forms cos ku and tan ku. To solve equations of these forms, first solve the equation for ku, then divide your result by k. EXAMPLE 7 Functions of Multiple Angles Solve cos 3t 0. Solution cos 3t 0 cos 3t cos 3t Write original equation. Add to each side. Divide each side by. NOTE Two different intervals, 0, and 0,, that correspond to the periods of the cosine and tangent functions are considered in Eamples 7 and 8, respectively. In the interval 0,, you know that 3t 3 and 3t 53 are the only solutions, so, in general, you have 3t n 3 and Dividing these results by 3, you obtain the general solution t 9 n 3 where n is an integer. and 3t 5 n. 3 t 5 n 9 3 General solution EXAMPLE 8 Functions of Multiple Angles Solve 3 tan 3 0. Solution 3 tan tan 3 tan Write original equation. Subtract 3 from each side. 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 General solution where n is an integer. 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 0.3 Solving Trigonometric Equations 665 Using Inverse Functions In the net eample, you will see how inverse trigonometric functions can be used to solve an equation. EXAMPLE 9 Using Inverse Functions Solve sec tan 4. Solution sec tan 4 tan tan 4 0 tan tan 3 0 tan 3tan 0 Write original equation. Pythagorean identity Combine like terms. 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 and tan 0 tan 3 arctan 3 4 Finally, because tan has a period of, you obtain the general solution by adding multiples of arctan 3 n and tan General solution where n is an integer. You can use a calculator to approimate the value of arctan 3. n Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 4, fill in the blanks.. When solving a trigonometric equation, the preliminary goal is to the trigonometric function involved in the equation.. The equation sin 0 has the solutions n and n, which are called 6 6 solutions. 3. The equation tan 3 tan 0 is a trigonometric equation that is of type. 4. A solution of an equation that does not satisfy the original equation is called an solution. In Eercises 5 0, verify that the -values are solutions of the equation cos 0 (a) 3 (b) sec 0 (a) (b) tan 0 (a) (b) 5 cos 4 0 (a) (b) sin sin 0 (a) (b) 7 6 csc 4 4 csc 0 (a) (b) 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).

22 666 Chapter 0 Analytic Trigonometry In Eercises 0, solve the equation.. cos 0. sin csc 0 4. tan sec cot 0 7. sin sin tan tan cos 0 sin 3 cos In Eercises 3, 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 9. cos cos sin 3 sin 0 sec tan 3 0 cos sin tan In Eercises 33 40, solve the multiple-angle equation. 33. cos 34. sin tan sec cos 38. sin sin 40. tan 3 3 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 y 5 y WRITING ABOUT CONCEPTS In Eercises 45 and 46, solve both equations. How do the solutions of the algebraic equation compare with the solutions of the trigonometric equation? 45. 6y 3y cos 3 cos y y 0 0 sin sin 0 0 In Eercises 47 56, use a graphing utility to approimate the solutions (to three decimal places) of the equation in the interval [0,. 47. sin cos sin 3 sin sin sin cos cos sin 4 cos cot 50. sin 3 5. tan 0 5. cos sec 0.5 tan csc 0.5 cot tan 7 tan sin 7 sin 0 In Eercises 57 60, use the Quadratic Formula to solve the equation in the interval [0,. Then use a graphing utility to approimate the angle. 57. sin 3 sin tan 4 tan tan 3 tan cos 4 cos 0 In Eercises 6 64, use inverse functions where needed to find all solutions of the equation in the interval [0,. 6. tan 6 tan sec tan cos 5 cos sin 7 sin In Eercises 65 68, (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. 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 0.3 Solving Trigonometric Equations 667 Function 65. f sin cos 66. f cos sin 67. f sin cos 68. f sin cos 69. Graphical Reasoning Consider the function given by f cos and its graph shown in the figure. π (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. 70. Graphical Reasoning Consider the function given by f sin and its graph shown in the figure. π 3 3 y y Trigonometric Equation sin cos sin 0 sin cos cos 0 cos sin 0 cos 4 sin cos 0 (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 0 have in the interval 8, 8? Find the solutions. π π 7. Harmonic Motion A weight is oscillating on the end of a spring. 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. 7. 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. 73. 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 in which sales eceed 00,000 units. 74. 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. 75. 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. θ t 6 r = 300 ft r = 000 yd θ Not drawn to scale 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).

24 668 Chapter 0 Analytic Trigonometry 76. Data Analysis: Meteorology The table shows the average daily high temperatures in Houston H (in degrees Fahrenheit) for month t, with t corresponding to January. (Source: National Climatic Data Center) t H t H (a) Create a scatter plot of the data. (b) Find a cosine model for the temperatures in Houston. (c) Use a graphing utility to graph the data points and the model for the temperatures in Houston. How well does the model fit the data? (d) What is the overall average daily high temperature in Houston? (e) Use a graphing utility to describe the months during which the average daily high temperature is above 86F and below 86F. 77. Geometry The area of a rectangle (see figure) inscribed in one arc of the graph of y cos is given by A cos, 0 < <. π π y (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. CAPSTONE 78. Consider the equation sin 0. Eplain the similarities and differences between finding all solutions in the interval 0, finding all solutions, in the interval 0,, and finding the general solution. 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 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. 8. Think About It Eplain what would happen if you divided each side of the equation cot cos cot by cot. Is this a correct method to use when solving equations? 8. Graphical Reasoning Use a graphing utility to confirm the solutions found in Eample 6 in two different ways. (a) Graph both sides of the equation and find the -coor- dinates of the points at which the graphs intersect. Left side: y cos Right side: y sin (b) Graph the equation y cos sin and find the -intercepts of the graph. Do both methods produce the same -values? Which method do you prefer? Eplain. SECTION PROJECT Modeling a Sound Wave A particular sound wave is modeled by pt p t 30p t p 3 t p 5 t 30p 6 t 4 where p n t sin54 nt, and t is the time in seconds. n (a) Find the sine components p n t and use a graphing utility to graph each component. Then verify the graph of p that is shown at the right. (b) Find the period of each sine component of p. Is p periodic? If so, what is its period? (c) Use the zero or root feature or the zoom and trace features to find the t-intercepts of the graph of p over one cycle. (d) Use the maimum and minimum features of a graphing utility to approimate the absolute maimum and absolute minimum values of p over one cycle. y y = p(t).4 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).

25 0.4 Sum and Difference Formulas Sum and Difference Formulas Use sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations. Using Sum and Difference Formulas In this and the following section, you will study the uses of several trigonometric identities and formulas. (Proofs of these formulas are given in Appendi A.) 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 tanu v tan u tan v tan u tan v tanu v tan u tan v tan u tan v NOTE Note that sinu v sin u sin v. Similar statements can be made for cosu v and tanu v. 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. EXAMPLE Evaluating a Trigonometric Function Find the eact value of sin. Solution To find the eact value of sin use the fact that, 3 4. Consequently, the formula for sinu v yields sin sin 3 sin 3 cos cos 4 3 sin Try checking this result on your calculator. You will find that 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).

26 670 Chapter 0 Analytic Trigonometry STUDY TIP Another way to solve Eample is to use the fact that together with the formula for cosu v. y 5 u = 3 EXAMPLE Evaluating a Trigonometric Function Find the eact value of cos 75. Solution Using the fact that , together with the formula for cosu v, you obtain cos 75 cos30 45 cos 30 cos 45 sin 30 sin EXAMPLE 3 Evaluating a Trigonometric Epression Figure 0.0 y Find the eact value of sinu v given sin u 4 where and cos v where 5, 0 < u <, 3, < v <. 3 = 5 3 v Figure 0. Solution Because sin u 45 and u is in Quadrant I, cos u 35, as shown in Figure 0.0. Because cos v 3 and v is in Quadrant II, sin v 53, as shown in Figure 0.. You can find sinu v as follows. sinu v sin u cos v cos u sin v u v Figure 0. EXAMPLE 4 An Application of a Sum Formula Write cosarctan arccos as an algebraic epression. Solution This epression fits the formula for cosu v. Angles u arctan and v arccos are shown in Figure 0.. So cosu v cosarctan cosarccos sinarctan sinarccos. NOTE In Eample 4, you can test the reasonableness of your solution by evaluating both epressions for particular values of. Try doing this for 0. 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 0.4 Sum and Difference Formulas 67 Eample 5 shows how to use a difference formula to prove the cofunction identity cos sin. The Granger Collection, New York HIPPARCHUS Hipparchus, considered the most eminent of Greek astronomers, was born about 90 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. EXAMPLE 5 Proving a Cofunction Identity Prove the cofunction identity cos sin. Solution Using the formula for cosu v, you have cos cos cos sin sin 0cos sin sin. Sum and difference formulas can be used to rewrite epressions such as sin n and cos n where n is an integer, as epressions involving only sin or cos. The resulting formulas are called reduction formulas. EXAMPLE 6 Deriving Reduction Formulas Simplify each epression. a. cos 3 b. Solution a. Using the formula for cosu v, you have tan 3 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. 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).