Trigonometric Identities and Equations

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1 LIALMC07_0768.QXP /6/0 0:7 AM Page Trigonometric Identities and Equations In 8 Michael Faraday discovered that when a wire passes by a magnet, a small electric current is produced in the wire. Now we generate massive amounts of electricity by simultaneously rotating thousands of wires near large electromagnets. Because electric current alternates its direction on electrical wires, it is modeled accurately by either the sine or the cosine function. We give many eamples of applications of the trigonometric functions to electricity and other phenomena in the eamples and eercises in this chapter, including a model of the wattage consumption of a toaster in Section 7., Eample Fundamental Identities 7. Verifying Trigonometric Identities 7. Sum and Difference Identities 7. Double-Angle Identities and Half- Angle Identities Summary Eercises on Verifying Trigonometric Identities 7.5 Inverse Circular Functions 7.6 Trigonometric Equations 7.7 Equations Involving Inverse Trigonometric Functions 605

2 LIALMC07_0768.QXP /6/0 0:7 AM Page CHAPTER 7 Trigonometric Identities and Equations 7. Fundamental Identities Negative-Angle Identities Fundamental Identities Using the Fundamental Identities y (, y) y r y O r (, y) sin( ) = y r Figure = sin Negative-Angle Identities As suggested by the circle shown in Figure, an angle having the point, y on its terminal side has a corresponding angle with the point, y on its terminal side. From the definition of sine, sin y r and sin y r, so sin and sin are negatives of each other, or sin sin. (Section 5.) Figure shows an angle in quadrant II, but the same result holds for in any quadrant. Also, by definition, cos r and cos r, (Section 5.) TEACHING TIP Point out that in trigonometric identities, can be an angle in degrees, a real number, or a variable. so cos cos. We can use these identities for sin and cos to find tan in terms of tan : tan sin cos sin sin cos cos tan tan. Similar reasoning gives the following identities. csc csc, sec sec, cot cot This group of identities is known as the negative-angle or negative-number identities. Fundamental Identities In Chapter 5 we used the definitions of the trigonometric functions to derive the reciprocal, quotient, and Pythagorean identities. Together with the negative-angle identities, these are called the fundamental identities. Fundamental Identities Reciprocal Identities cot tan Quotient Identities tan sin cos sec cos cot cos sin csc sin (continued)

3 LIALMC07_0768.QXP /6/0 0:7 AM Page Fundamental Identities 607 TEACHING TIP Encourage students to memorize the identities presented in this section as well as subsequent sections. Point out that numerical values can be used to help check whether or not an identity was recalled correctly. Pythagorean Identities sin cos tan sec Negative-Angle Identities sin() sin cos() cos csc() csc sec() sec cot csc tan() tan cot() cot NOTE The most commonly recognized forms of the fundamental identities are given above. Throughout this chapter you must also recognize alternative forms of these identities. For eample, two other forms of sin cos are sin cos and cos sin. Using the Fundamental Identities One way we use these identities is to find the values of other trigonometric functions from the value of a given trigonometric function. Although we could find such values using a right triangle, this is a good way to practice using the fundamental identities. EXAMPLE Finding Trigonometric Function Values Given One Value and the Quadrant If tan 5 and is in quadrant II, find each function value. (a) sec (b) sin (c) cot TEACHING TIP Warn students that the given information in Eample, tan 5 5, does not mean that sin 5 and cos. Ask them why these values cannot be correct. Solution (a) Look for an identity that relates tangent and secant. tan sec 5 sec 5 9 sec 9 sec 9 sec sec Pythagorean identity tan 5 Combine terms. Take the negative square root. (Section.) Simplify the radical. (Section R.7) We chose the negative square root since sec is negative in quadrant II.

4 LIALMC07_0768.QXP /6/0 0:7 AM Page CHAPTER 7 Trigonometric Identities and Equations (b) (c) tan cot tan sin cot tan cos cos tan sin sin sec tan 5 sin sin 5 cot 5 5 Quotient identity Multiply by cos. Reciprocal identity From part (a), sec ; tan 5 Reciprocal identity Negative-angle identity tan 5 ; Simplify. (Section R.5) Now try Eercises 5, 7, and 9. CAUTION To avoid a common error, when taking the square root, be sure to choose the sign based on the quadrant of and the function being evaluated. Any trigonometric function of a number or angle can be epressed in terms of any other function. EXAMPLE Epressing One Function in Terms of Another Epress cos in terms of tan. Solution Since sec is related to both cos and tan by identities, start with tan sec. tan sec tan cos tan cos cos tan cos tan tan Take reciprocals. Reciprocal identity Take square roots. Quotient rule (Section R.7); rewrite. Rationalize the denominator. (Section R.7) Choose the sign or the sign, depending on the quadrant of. Now try Eercise.

5 LIALMC07_0768.QXP /6/0 0:7 AM Page Fundamental Identities 609 We can use a graphing calculator to decide whether two functions are identical. See Figure, which supports the identity sin cos. With an identity, you should see no difference in the two graphs. All other trigonometric functions can easily be epressed in terms of sin and/or cos. We often make such substitutions in an epression to simplify it. Y = Y Figure y = tan + cot y = cos sin EXAMPLE Rewriting an Epression in Terms of Sine and Cosine Write tan cot in terms of sin and cos, and then simplify the epression. Solution tan tan cot sin cot cos cos sin sin cos cos sin cos sin sin cos cos sin cos sin Quotient identities Write each fraction with the LCD. (Section R.5) Add fractions. Pythagorean identity Now try Eercise 55. The graph supports the result in Eample. The graphs of y and y appear to be identical. CAUTION When working with trigonometric epressions and identities, be sure to write the argument of the function. For eample, we would not write sin cos ; an argument such as is necessary in this identity. 7. Eercises Concept Check Fill in the blanks.. If tan.6, then tan.. If cos.65, then cos.. If tan.6, then cot.. If cos.8 and sin.6, then tan. Find sin s. See Eample cos s, s in quadrant I 6. cot s, s in quadrant IV 7. coss 5, tan s 0 8. tan s 7, sec s sec s, tan s 0 0. csc s 8 5. Why is it unnecessary to give the quadrant of s in Eercise 0?

6 LIALMC07_0768.QXP /6/0 0:7 AM Page CHAPTER 7 Trigonometric Identities and Equations. sin. odd. cos 5. even 6. tan 7. odd f f f f f f. cos 5 ; tan 5 ; cot 5 ; 5 sec 5 ; csc 5. sin 6 ; tan 6; 5 cot 6 ; sec 5; csc 56. sin 7 ; 7 cos 7 ; cot ; 7 sec 7 ; csc 7. sin ; 5 cos ; tan ; 5 cot ; sec 5 5. sin ; cos ; 5 5 tan ; sec 5 ; csc 5 Relating Concepts For individual or collaborative investigation (Eercises 7) A function is called an even function if f f for all in the domain of f. A function is called an odd function if f f for all in the domain of f. Work Eercises 7 in order, to see the connection between the negative-angle identities and even and odd functions.. Complete the statement: sin.. Is the function defined by f sin even or odd?. Complete the statement: cos. 5. Is the function defined by f cos even or odd? 6. Complete the statement: tan. 7. Is the function defined by f tan even or odd? Concept Check For each graph, determine whether f f or f f is true cos ; tan ; 5 cot ; sec 5 ; csc 5 7. sin 7 ; cos ; tan 7 ; cot 7 ; 7 csc 7 7 Find the remaining five trigonometric functions of. See Eample.. sin, in quadrant II. cos, in quadrant I 5. tan, in quadrant IV. csc 5, in quadrant III 5. cot, sin 0 6. sin, cos sec, sin 0 8. cos, sin 0

7 LIALMC07_0768.QXP /6/0 0:7 AM Page 6 7. Fundamental Identities 6 8. sin 5 ; tan 5; cot 5 ; 5 sec ; csc B 0. D. E. C. A. C 5. A 6. E 7. D 8. B sin tan p p sin cos cot sin sin tan sec cot csc csc cos cos sec sin sin 9. cos cot 5. csc 5. cos 5. tan 55. sec cos 56. cot 57. cot tan 58. sin cos 59. sin cos 60. cot tan 6. cos 6. tan 9 9 Concept Check For each epression in Column I, choose the epression from Column II that completes an identity. I II cos 9. sin A. sin cos 0. tan B. cot. cos C. sec. tan D.. E. cos Concept Check For each epression in Column I, choose the epression from Column II that completes an identity. You may have to rewrite one or both epressions. I II. tan cos A. sin cos sin cos 5. sec B. sec 6. sec csc C. sin 7. sin D. csc cot sin 8. cos E. tan 9. A student writes cot csc. Comment on this student s work. 0. Another student makes the following claim: Since sin cos, I should be able to also say that sin cos if I take the square root of both sides. Comment on this student s statement.. Concept Check Suppose that cos. Find sin.. Concept Check Find tan if sec p. Write the first trigonometric function in terms of the second trigonometric function. See Eample.. sin ; cos. cot ; sin 5. tan ; sec 6. cot ; csc 7. csc ; cos 8. sec ; sin p Write each epression in terms of sine and cosine, and simplify it. See Eample. 9. cot sin 50. sec cot sin 5. cos csc 5. cot tan 5. sin csc 5. sec sec 55. cos sec 56. cos sin sin sin cos cot 59. sec cos 60. sec csc cos sin 6. sin csc sin 6. cos sin sin tan cot

8 LIALMC07_0768.QXP /6/0 0:7 AM Page 6 6 CHAPTER 7 Trigonometric Identities and Equations 6. sec 6. sin ; ; sin 68. It is the negative of sin. 69. cos 70. It is the same function. 7. (a) sin (b) cos (c) 5 sin 7. identity 7. not an identity 7. not an identity 75. identity 76. not an identity 6. sin tan cos Concept Check Let cos 5. Find all possible values for sin. sin cos 66. Concept Check Let csc. Find all possible values for. Relating Concepts For individual or collaborative investigation (Eercises 67 7) In Chapter 6 we graphed functions defined by tan sec y c a f b d sec tan with the assumption that b 0. To see what happens when b 0, work Eercises 67 7 in order. 67. Use a negative-angle identity to write y sin as a function of. 68. How does your answer to Eercise 67 relate to y sin? 69. Use a negative-angle identity to write y cos as a function of. 70. How does your answer to Eercise 69 relate to y cos? 7. Use your results from Eercises to rewrite the following with a positive value of b. (a) sin (b) cos (c) 5 sin sec Use a graphing calculator to decide whether each equation is an identity. (Hint: In Eercise 76, graph the function of for a few different values of y (in radians).) 7. cos sin sin cos cos y cos cos y sin s sin s cos cos sin 7. Verifying Trigonometric Identities Verifying Identities by Working with One Side Verifying Identities by Working with Both Sides Recall that an identity is an equation that is satisfied for all meaningful replacements of the variable. One of the skills required for more advanced work in mathematics, especially in calculus, is the ability to use identities to write epressions in alternative forms. We develop this skill by using the fundamental identities to verify that a trigonometric equation is an identity (for those values of the variable for which it is defined). Here are some hints to help you get started.

9 LIALMC07_0768.QXP /6/0 0:7 AM Page 6 7. Verifying Trigonometric Identities 6 Looking Ahead to Calculus Trigonometric identities are used in calculus to simplify trigonometric epressions, determine derivatives of trigonometric functions, and change the form of some integrals. TEACHING TIP There is no substitute for eperience when it comes to verifying identities. Guide students through several eamples, giving hints such as Apply a reciprocal identity, or Use a different form of the Pythagorean identity sin cos. Hints for Verifying Identities. Learn the fundamental identities given in the last section. Whenever you see either side of a fundamental identity, the other side should come to mind. Also, be aware of equivalent forms of the fundamental identities. For eample, sin is an alternative form of the identity sin cos. Try to rewrite the more complicated side of the equation so that it is identical to the simpler side.. It is sometimes helpful to epress all trigonometric functions in the equation in terms of sine and cosine and then simplify the result.. Usually, any factoring or indicated algebraic operations should be performed. For eample, the epression sin sin can be factored as sin. The sum or difference of two trigonometric epressions, such as sin cos, can be added or subtracted in the same way as any other rational epression. cos. cos sin sin cos sin cos sin cos cos sin sin cos 5. As you select substitutions, keep in mind the side you are not changing, because it represents your goal. For eample, to verify the identity tan cos, try to think of an identity that relates tan to cos. In this case, since sec cos and sec tan, the secant function is the best link between the two sides. 6. If an epression contains sin, multiplying both numerator and denominator by sin would give sin, which could be replaced with cos. Similar results for sin, cos, and cos may be useful. CAUTION Verifying identities is not the same as solving equations. Techniques used in solving equations, such as adding the same terms to both sides, or multiplying both sides by the same term, should not be used when working with identities since you are starting with a statement (to be verified) that may not be true. Verifying Identities by Working with One Side To avoid the temptation to use algebraic properties of equations to verify identities, work with only one side and rewrite it to match the other side, as shown in Eamples.

10 LIALMC07_0768.QXP /6/0 0:7 AM Page 6 6 CHAPTER 7 Trigonometric Identities and Equations For s =, cot + = csc (cos + sin ) The graphs coincide, supporting the conclusion in Eample. EXAMPLE Verifying an Identity (Working with One Side) Verify that the following equation is an identity. Solution We use the fundamental identities from Section 7. to rewrite one side of the equation so that it is identical to the other side. Since the right side is more complicated, we work with it, using the third hint to change all functions to sine or cosine. Steps Right side of given equation cot s csc scos s sin s csc scos s sin s cos s sin s sin s cos s sin s sin s sin s cot s Left side of given equation Reasons csc s sin s Distributive property (Section R.) cos s sin s sin s cot s; sin s The given equation is an identity since the right side equals the left side. Now try Eercise. tan ( + cot ) = sin The screen supports the conclusion in Eample. EXAMPLE Verifying an Identity (Working with One Side) Verify that the following equation is an identity. tan cot Solution We work with the more complicated left side, as suggested in the second hint. Again, we use the fundamental identities from Section 7.. tan cot tan tan cot tan tan tan sec cos sin sin tan Distributive property cot tan tan tan tan sec sec cos cos sin Since the left side is identical to the right side, the given equation is an identity. Now try Eercise 7.

11 LIALMC07_0768.QXP /6/0 0:7 AM Page Verifying Trigonometric Identities 65 EXAMPLE Verifying an Identity (Working with One Side) Verify that the following equation is an identity. Solution tan t cot t sin t cos t We transform the more complicated left side to match the right side. tan t cos t sin t sec t csc t tan t cot t sin t cos t tan t sin t cos t cot t sin t cos t sec t csc t sin t cos t cot t sin t cos t sin t cos t sin t cos t cos t sin t sin t cos t a b c a b a b a c b c (Section R.5) tan t cos sin t cos t t ; cot t sin t The third hint about writing all trigonometric functions in terms of sine and cosine was used in the third line of the solution. cos t sec t; sin t csc t Now try Eercise. TEACHING TIP Show that the identity in Eample can also be verified by multiplying the numerator and denominator of the left side by sin. EXAMPLE Verifying an Identity (Working with One Side) Verify that the following equation is an identity. Solution We work on the right side, using the last hint in the list given earlier to multiply numerator and denominator on the right by sin. sin cos sin sin cos sin sin cos sin cos cos sin cos sin cos sin sin cos Multiply by. (Section R.) y y y sin cos Lowest terms (Section R.5) (Section R.) Now try Eercise 7. Verifying Identities by Working with Both Sides If both sides of an identity appear to be equally comple, the identity can be verified by working independently on the left side and on the right side, until each side is changed into some common third result. Each step, on each side, must be reversible.

12 LIALMC07_0768.QXP /6/0 0:7 AM Page CHAPTER 7 Trigonometric Identities and Equations left right common third epression With all steps reversible, the procedure is as shown in the margin. The left side leads to a common third epression, which leads back to the right side. This procedure is just a shortcut for the procedure used in Eamples : one side is changed into the other side, but by going through an intermediate step. EXAMPLE 5 Verifying an Identity (Working with Both Sides) Verify that the following equation is an identity. Solution Both sides appear equally comple, so we verify the identity by changing each side into a common third epression. We work first on the left, multiplying numerator and denominator by cos. Multiply by. On the right side of the original equation, begin by factoring. We have shown that sec tan sin sin sec tan cos sec tan sec tan cos sec tan sec tan cos sec cos tan cos sec cos tan cos tan cos tan cos sin cos cos sin cos cos sin sin sin sin sin cos cos sin sin sin sin Distributive property y y y (Section R.) Factor sin. Lowest terms Left side of Common third Right side of given equation epression given equation verifying that the given equation is an identity. sin sin sin sec cos tan sin cos cos sin sec tan sin sin sin, sec tan sin cos Now try Eercise 5.

13 LIALMC07_0768.QXP /6/0 0:7 AM Page Verifying Trigonometric Identities 67 CAUTION Use the method of Eample 5 only if the steps are reversible. There are usually several ways to verify a given identity. For instance, another way to begin verifying the identity in Eample 5 is to work on the left as follows. sin sec tan cos cos sec tan sin cos cos sin cos sin cos sin sin Fundamental identities (Section 7.) Add and subtract fractions. (Section R.5) Simplify the comple fraction. (Section R.5) Compare this with the result shown in Eample 5 for the right side to see that the two sides indeed agree. L C An Inductor and a Capacitor Figure EXAMPLE 6 Applying a Pythagorean Identity to Radios Tuners in radios select a radio station by adjusting the frequency. A tuner may contain an inductor L and a capacitor C, as illustrated in Figure. The energy stored in the inductor at time t is given by and the energy stored in the capacitor is given by where F is the frequency of the radio station and k is a constant. The total energy E in the circuit is given by Show that E is a constant function. (Source: Weidner, R. and R. Sells, Elementary Classical Physics, Vol., Allyn & Bacon, 97.) Solution Lt k sin Ft Ct k cos Ft, Et Lt Ct. Et Lt Ct Given equation k sin Ft k cos Ft Substitute. ksin Ft cos Ft Factor. (Section R.) k sin cos (Here Ft). k Since k is constant, Et is a constant function. Now try Eercise 85.

14 LIALMC07_0768.QXP /6/0 0:7 AM Page CHAPTER 7 Trigonometric Identities and Equations 7. Eercises. csc sec or sin cos. csc sec or sin cos. sec s. cot or sec cos or csc sin 9. sin t 0. sec s. cos or cot csc sin... sin cos sin sin sec sec 5. sin 6. tan cot or 7. sin sin 8. tan tan 9. cos 0. cot cot or csc cot sin cos sin cos sin cos sin cos sin cos tan 8. sec 9. tan 0. cot t. sec. csc Perform each indicated operation and simplify the result. sec. cot. csc csc cot sec. tan scot s csc s. cos sec csc sin csc sec sin 7. cos cos 8. sin sec sin csc sin cos 9. sin t cos t 0. tan s tan s. cos cos. sin cos Factor each trigonometric epression.. sin. sec 5. sin sin 6. tan cot tan cot 7. sin sin 8. tan tan 9. cos cos 0. cot cot. sin cos. sin cos Each epression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each epression.. tan cos. cot sin 5. sec r cos r sin tan csc sec 6. cot t tan t cos cot 9. sec 0. csc t.. cot tan tan sin sin csc cos In Eercises 68, verify that each trigonometric equation is an identity. See Eamples 5. cot tan. cos. sin csc sec sin tan 5. cos 6. sec cos sec 7. cos tan 8. sin cot 9. cot s tan s sec s csc s 0. sin tan cos sec cos sin. sin sec tan. sec cos sec csc cos. sin cos sin. 5. cos cos sin sin 6. tan sin tan cos cos sin cot

15 LIALMC07_0768.QXP /6/0 0:7 AM Page Verifying Trigonometric Identities cos 7. cos 8. tan sec sin 60. sin cos csc cos cos cos sec sin sin sec tan sec tan tan s cos s sin s cot s sec s csc s cos s cos cot csc cos cot cot tan tan 5. tan tan sec tan sec 55. sin sec sin csc sec csc cot 56. cot csc tan sin 57. sec sec tan tan sin sec sec tan tan sin sin cos sin sec s tan s sec s tan s sec s tan s cot t cot t sin t tan t sec t sin cos sin cos cos cos cos cot csc cos 66. sec tan sin sin 67. sec csc cos sin cot tan sin cos 68. sin cos 69. A student claims that the equation cos sin is an identity, since by letting or radians we get 0, a true statement. Comment on this student s reasoning. 70. An equation that is an identity has an infinite number of solutions. If an equation has an infinite number of solutions, is it necessarily an identity? Eplain. 90 tan t cot t tan t cot t cos sin cos sin cos sec tan tan sec csc tan csc

16 LIALMC07_0768.QXP /6/0 0:7 AM Page CHAPTER 7 Trigonometric Identities and Equations 7. sec tan sin cos 7. csc cot sec tan cos 7. cot sin tan 7. tan sin cos sec 75. identity 76. identity 77. not an identity 78. not an identity 8. It is true when sin (a) I k sin (b) For for all integers n, cos, its maimum value, and I attains a maimum value of k. 85. (a) P 6k cos t (b) P 6k sin t n Concept Check Graph each epression and conjecture an identity. Then verify your conjecture algebraically. 7. sec tan sin 7. csc cot sec cos tan sin cos sin tan Graph the epressions on each side of the equals sign to determine whether the equation might be an identity. (Note: Use a domain whose length is at least.) If the equation looks like an identity, prove it algebraically. See Eample. 5 cos s cot s sec s csc s 5 cot s sin s sec s tan s cot s sin s sin s sec s tan s cot s sin s By substituting a number for s or t, show that the equation is not an identity. 79. sincsc s 80. cos s cos s 8. csc t cot t 8. cos t sin t 8. When is sin cos a true statement? (Modeling) Work each problem. 8. Intensity of a Lamp According to Lambert s law, the intensity of light from a single source on a flat surface at point P is given by I k cos, where k is a constant. (Source: Winter, C., Solar Power Plants, Springer-Verlag, 99.) (a) Write I in terms of the sine function. 9(b) Eplain why the maimum value of I occurs when P. y 85. Oscillating Spring The distance or displacement y of a weight attached to an oscillating spring from its natural position is modeled by y cost, where t is time in seconds. Potential energy is the energy of position and is given by P ky, where k is a constant. The weight has the greatest potential energy when the spring is stretched the most. (Source: Weidner, R. and R. Sells, Elementary Classical Physics, Vol., Allyn & Bacon, 97.) (a) Write an epression for P that involves the cosine function. (b) Use a fundamental identity to write P in terms of sint. 86. Radio Tuners Refer to Eample 6. Let the energy stored in the inductor be given by 0 Lt cos 6,000,000t and the energy in the capacitor be given by Ct sin 6,000,000t,

17 LIALMC07_0768.QXP /6/0 0:7 AM Page 6 7. Sum and Difference Identities (a) The sum of L and C equals where t is time in seconds. The total energy E in the circuit is given by Et Lt Ct. (a) Graph L, C, and E in the window 0, 0 6 by,, with Xscl 0 7 and Yscl. Interpret the graph. (b) Make a table of values for L, C, and E starting at t 0, incrementing by 0 7. Interpret your results. (c) Use a fundamental identity to derive a simplified epression for Et. (b) Let Y Lt, Y Ct, and Y Et. Y for all inputs. (c) Et 7. Sum and Difference Identities Cosine Sum and Difference Identities Cofunction Identities Sine and Tangent Sum and Difference Identities Cosine Sum and Difference Identities Several eamples presented earlier should have convinced you by now that cosa B does not equal cos A cos B. For eample, if A and B 0, then cosa B cos 0 cos while cos A cos B cos cos 0 0. We can now derive a formula for cosa B. We start by locating angles A and B in standard position on a unit circle, with B A. Let S and Q be the points where the terminal sides of angles A and B, respectively, intersect the circle. Locate point R on the unit circle so that angle POR equals the difference A B. See Figure. 0, (cos(a B), sin(a B)) R (cos A, sin A) S y Q (cos B, sin B O A A B B P (, 0) Figure Point Q is on the unit circle, so by the work with circular functions in Chapter 6, the -coordinate of Q is the cosine of angle B, while the y-coordinate of Q is the sine of angle B. Q has coordinates cos B,sin B. In the same way, S has coordinates cos A,sin A, and R has coordinates cosa B,sinA B.

18 LIALMC07_0768.QXP /6/0 0:7 AM Page 6 6 CHAPTER 7 Trigonometric Identities and Equations Angle SOQ also equals A B. Since the central angles SOQ and POR are equal, chords PR and SQ are equal. By the distance formula, since PR SQ, cosa B sina B 0 Squaring both sides and clearing parentheses gives (Section.) Since sin cos for any value of, we can rewrite the equation as Subtract ; divide by. Although Figure shows angles A and B in the second and first quadrants, respectively, this result is the same for any values of these angles. To find a similar epression for cosa B, rewrite A B as A B and use the identity for cosa B. cosa B cosa B cos A cos B sin A sin B. cos A B cosa B sin A B cos A cos A cos B cos B sin A sin A sin B sin B. cosa B cos A cos B sin A sin B cosa B cos A cos B sin A sin B. cos A cosb sin A sinb cos A cos B sin Asin B cosa B cos A cos B sin A sin B Cosine difference identity Negative-angle identities (Section 7.) Cosine of a Sum or Difference cos(a B) cos A cos B sin A sin B cos(a B) cos A cos B sin A sin B These identities are important in calculus and useful in certain applications. Although a calculator can be used to find an approimation for cos 5, for eample, the method shown below can be applied to get an eact value, as well as to practice using the sum and difference identities. EXAMPLE Finding Eact Cosine Function Values Find the eact value of each epression. (a) cos 5 (b) cos 5 (c) cos 87 cos 9 sin 87 sin 9 Solution (a) To find cos 5, we write 5 as the sum or difference of two angles with known function values. Since we know the eact trigonometric function values of 5 and 0, we write 5 as 5 0. (We could also use ) Then we use the identity for the cosine of the difference of two angles.

19 LIALMC07_0768.QXP /6/0 0:7 AM Page 6 7. Sum and Difference Identities 6 TEACHING TIP In Eample (b), students may benefit from converting radians to 75 in order to realize that can be used in place of 5. The screen supports the solution in Eample (b) by showing that cos 5 = 6. (b) (c) cos 5 cos5 0 cos 5 cos 5 cos 0 sin 5 sin 0 cos 6 cos 6 6 cos 6 sin Cosine difference identity (Section 5.) Cosine sum identity (Section 6.) cos 87 cos 9 sin 87 sin 9 cos87 9 Cosine sum identity 6 sin cos 80 6 ; (Section 5.) Now try Eercises 7, 9, and. Cofunction Identities We can use the identity for the cosine of the difference of two angles and the fundamental identities to derive cofunction identities. TEACHING TIP Mention that these identities state that the trigonometric function of an acute angle is the same as the cofunction of its complement. Verify the cofunction identities for acute angles using complementary angles in a right triangle along with the righttriangle-based definitions of the trigonometric functions. Emphasize that these identities apply to any angle, not just acute angles. Cofunction Identities cos(90 ) sin cot(90 ) tan sin(90 ) cos sec(90 ) csc tan(90 ) cot csc(90 ) sec Similar identities can be obtained for a real number domain by replacing 90 with. Substituting 90 for A and for B in the identity for cosa B gives cos90 cos 90 cos sin 90 sin 0 cos sin sin. This result is true for any value of since the identity for cosa B is true for any values of A and B.

20 LIALMC07_0768.QXP /6/0 0:7 AM Page 6 6 CHAPTER 7 Trigonometric Identities and Equations TEACHING TIP Verify results from Eample using a calculator. EXAMPLE Using Cofunction Identities to Find Find an angle that satisfies each of the following. (a) cot tan 5 (b) sin cos0 (c) csc sec Solution (a) Since tangent and cotangent are cofunctions, tan90 cot. (b) (c) 90 sec 0 90 sin cos0 cos90 cos0 0 csc sec sec sec sec cot tan 5 tan90 tan Cofunction identity Cofunction identity Combine terms. Cofunction identity Set angle measures equal. Now try Eercises 5 and 7. NOTE Because trigonometric (circular) functions are periodic, the solutions in Eample are not unique. We give only one of infinitely many possibilities. If one of the angles A or B in the identities for cosa B and cosa B is a quadrantal angle, then the identity allows us to write the epression in terms of a single function of A or B. EXAMPLE Reducing cosa B to a Function of a Single Variable Write cos80 as a trigonometric function of. Solution Use the difference identity. Replace A with 80 and B with. cos80 cos 80 cos sin 80 sin cos 0 sin cos (Section 5.) Now try Eercise 9.

21 LIALMC07_0768.QXP /6/0 0:7 AM Page Sum and Difference Identities 65 Sine and Tangent Sum and Difference Identities We can use the cosine sum and difference identities to derive similar identities for sine and tangent. Since sin cos90, we replace with A B to get sina B cos90 A B Cofunction identity cos90 A B cos90 A cos B sin90 A sin B Cosine difference identity sina B sin A cos B cos A sin B. Cofunction identities Now we write sina B as sina B and use the identity for sina B. sina B sina B sin A cosb cos A sinb sina B sin A cos B cos A sin B Sine sum identity Negative-angle identities Sine of a Sum or Difference sin(a B) sin A cos B cos A sin B sin(a B) sin A cos B cos A sin B To derive the identity for tana B, we start with tana B We epress this result in terms of the tangent function by multiplying both numerator and denominator by tana B tana B sina B cosa B sin A cos B cos A sin B cos A cos B sin A sin B. sin A cos B cos A sin B cos A cos B cos A cos B sin A sin B cos A cos B sin A cos B cos A sin B cos A cos B cos A cos B cos A cos B sin A sin B cos A cos B cos A cos B sin A cos A sin B cos B sin A cos A sin B cos B tan A tan B tan A tan B cos A cos B. Fundamental identity (Section 7.) Sum identities Simplify the comple fraction. (Section R.5) Multiply numerators; multiply denominators. tan sin cos Replacing B with B and using the fact that tanb tan B gives the identity for the tangent of the difference of two angles.

22 LIALMC07_0768.QXP /6/0 0:7 AM Page CHAPTER 7 Trigonometric Identities and Equations Tangent of a Sum or Difference tan(a B) tan A tan B tan A tan B tan(a B) tan A tan B tan A tan B EXAMPLE Finding Eact Sine and Tangent Function Values Find the eact value of each epression. (a) sin 75 (b) tan 7 (c) sin 0 cos 60 cos 0 sin 60 Solution (a) (b) (c) sin 75 sin5 0 7 sin 5 cos 0 cos 5 sin 0 6 tan tan tan tan tan tan Tangent sum identity (Section 6.) Sine sum identity (Section 5.) Rationalize the denominator. (Section R.7) Multiply. (Section R.7) Combine terms. Factor out. (Section R.5) Lowest terms sin 0 cos 60 cos 0 sin 60 sin0 60 Sine difference identity sin0 sin 0 Negative-angle identity (Section 5.) Now try Eercises 9,, and 5.

23 LIALMC07_0768.QXP /6/0 0:7 AM Page Sum and Difference Identities 67 EXAMPLE 5 Writing Functions as Epressions Involving Functions of Write each function as an epression involving functions of. (a) sin0 (b) tan5 (c) sin80 Solution (a) Using the identity for sina B, (b) (c) tan5 sin0 sin 0 cos cos 0 sin sin cos sin. tan 5 tan tan tan 5 tan tan sin80 sin 80 cos cos 80 sin 0 cos sin Now try Eercises and 7. EXAMPLE 6 Finding Function Values and the Quadrant of A B Suppose that A and B are angles in standard position, with sin A and cos B 5 5, A,,. Find each of the following. (a) sina B (b) tana B (c) the quadrant of A B Solution (a) The identity for sina B requires sin A, cos A, sin B, and cos B. We are given values of sin A and cos B. We must find values of cos A and sin B. sin A cos A 6 5 cos A cos A 9 5 B Fundamental identity sin A 5 Subtract 6 5. cos A 5 Since A is in quadrant II, cos A 0. In the same way, sin B. Now use the formula for sina B sina B (b) To find tana B, first use the values of sine and cosine from part (a) to get tan A and tan B 5. tana B

24 LIALMC07_0768.QXP /6/0 0:7 AM Page CHAPTER 7 Trigonometric Identities and Equations (c) From parts (a) and (b), sina B 6 65 and tana B 6 6, both positive. Therefore, A B must be in quadrant I, since it is the only quadrant in which both sine and tangent are positive. Now try Eercise 5. EXAMPLE 7 Applying the Cosine Difference Identity to Voltage Common household electric current is called alternating current because the current alternates direction within the wires. The voltage V in a typical 5-volt outlet can be epressed by the function Vt 6 sin t, where is the angular speed (in radians per second) of the rotating generator at the electrical plant and t is time measured in seconds. (Source: Bell, D., Fundamentals of Electric Circuits, Fourth Edition, Prentice-Hall, 988.) (a) It is essential for electric generators to rotate at precisely 60 cycles per sec so household appliances and computers will function properly. Determine for these electric generators. (b) Graph V in the window 0,.05 by 00, 00. (c) Determine a value of so that the graph of Vt 6 cost is the same as the graph of Vt 6 sin t. Solution (a) Each cycle is radians at 60 cycles per sec, so the angular speed is per sec. (b) Vt 6 sin t 6 sin 0t. Because the amplitude of the function is 6 (from Section 6.), 00, 00 is an appropriate interval for the range, as shown in Figure radians 00 For = t, V(t) = 6 sin 0t Figure 5 (c) Using the negative-angle identity for cosine and a cofunction identity, cos cos cos sin. Therefore, if, then Vt 6 cost 6 sin t. Now try Eercise 8.

25 LIALMC07_0768.QXP /6/0 0:7 AM Page Sum and Difference Identities Eercises. F. A. C. D cot. cos sin 5 6. cos 7. cos 0 8. tan 9. csc cot8. tan. cos. cos. tan Concept Check Match each epression in Column I with the correct epression in Column II to form an identity. I II. cos y A. cos cos y sin sin y. cos y B. sin sin y cos cos y. sin y C. sin cos y cos sin y. sin y D. sin cos y cos sin y E. F. cos sin y sin cos y cos cos y sin sin y Use identities to find each eact value. (Do not use a calculator.) See Eample. 5. cos cos5 7. cos cos05 (Hint: ) (Hint: ) 9. cos 0. cos. cos 0 cos 50 sin 0 sin 50. cos 7 9 Write each function value in terms of the cofunction of a complementary angle. See Eample.. tan 87. sin 5 5. cos 6. sin 5 7. sin 5 8. cot 9 9. sec 6 0. tan cos 9 7 sin sin 9 9 Use the cofunction identities to fill in each blank with the appropriate trigonometric function name. See Eample.. cot. 6. sin 57. Find an angle that makes each statement true. See Eample. 5. tan cot5 6. sin cos sin 5 cos 5 Use identities to find the eact value of each of the following. See Eample. 9. sin 5 0. tan 5. tan. sin. sin. tan sin 76 cos cos 76 sin 6. sin 0 cos 50 cos 0 sin 50 tan 80 tan 55 tan 80 tan tan 80 tan 55 tan 80 tan55 sin 7 cot 8 cot 0 tan 0 6

26 LIALMC07_0768.QXP /6/0 0:7 AM Page CHAPTER 7 Trigonometric Identities and Equations 9. sin 0. cos. sin. sin. cos sin. sin 5. cos sin 6. cos sin 7. tan tan tan 8. tan tan tan tan tan (a) (b) (c) (d) I (a) (b) (c) 66 (d) IV 0 5. (a) 9 5 (b) (c) 0 0 (d) II 6 5. (a) (b) (c) (d) IV 55. (a) 6 (b) (c) (d) II (a) 77 (b) (c) (d) III (a) 6 (b) Use identities to write each epression as a function of or. See Eamples and cos90 0. cos80. cos. cos. sin5. sin80 5. sin 6. sin 7. tan tan 50. tan tan 0 Use the given information to find (a) coss t, (b) sins t, (c) tans t, and (d) the quadrant of s t. See Eample cos s and sin t 5, s and t in quadrant I 5 5. cos s and sin t, s and t in quadrant II sin s and sin t, s in quadrant II and t in quadrant IV 5. sin s and sin t, s in quadrant I and t in quadrant III cos s 8 and cos t, s and t in quadrant III cos s 5 and sin t, s in quadrant II and t in quadrant I 7 5 Relating Concepts For individual or collaborative investigation (Eercises 57 60) The identities for cosa B and cosa B can be used to find eact values of epressions like cos 95 and cos 55, where the angle is not in the first quadrant. Work Eercises in order, to see how this is done. 57. By writing 95 as 80 5, use the identity for cosa B to epress cos 95 as cos Use the identity for cosa B to find cos By the results of Eercises 57 and 58, cos Find each eact value using the method shown in Eercises (a) cos 55 (b) cos Find each eact value. Use the technique developed in Relating Concepts Eercises sin tan sin tan tan 66. sin 6

27 LIALMC07_0768.QXP /6/0 0:7 AM Page 6 7. Sum and Difference Identities 6 7. sin cos tan 7. tan tan and 6; no 8. (a) 5 lb (c) Use the identity cos90 sin, and replace with 90 A, to derive the identity cos A sin90 A. 68. Eplain how the identities for seca B, csca B, and cota B can be found by using the sum identities given in this section. 69. Why is it not possible to use a method similar to that of Eample 5(c) to find a formula for tan70? 70. Concept Check Show that if A, B, and C are angles of a triangle, then sina B C 0. Graph each epression and use the graph to conjecture an identity. Then verify your conjecture algebraically. tan 7. sin 7. tan Verify that each equation is an identity sin y sin y sin cos y tan y tan y cos tan cot cos sin sins t tan s tan t cos s cos t sin y tan tan y sin y tan tan y sins t sin t Eercises 79 and 80 refer to Eample How many times does the current oscillate in.05 sec? 80. What are the maimum and minimum voltages in this outlet? Is the voltage always equal to 5 volts? (Modeling) Solve each problem. 8. Back Stress If a person bends at the waist with a straight back making an angle of degrees with the horizontal, then the force F eerted on the back muscles can be modeled by the equation coss t cos t F.6W sin 90, sin where W is the weight of the person. (Source: Metcalf, H., Topics in Classical Biophysics, Prentice-Hall, 980.) (a) Calculate F when W 70 lb and. (b) Use an identity to show that F is approimately equal to.9w cos. (c) For what value of is F maimum? tan tan y tan tan y sin s sin t cos t 0

28 LIALMC07_0768.QXP /6/0 0:7 AM Page 6 6 CHAPTER 7 Trigonometric Identities and Equations 8. (a) 08 lb (b) (a) The pressure P is oscillating. For = t, cos[ P(t) =. 0 06t ] (b) The pressure oscillates and amplitude decreases as r increases. For = r, P(r) = r r cos[.9 ] 0, (c) P a cos ct n 8. (a) For = t, V = V + V = 0 sin 0t + 0 cos 0t Back Stress Refer to Eercise 8. (a) Suppose a 00-lb person bends at the waist so that. Estimate the force eerted on the person s back muscles. (b) Approimate graphically the value of that results in the back muscles eerting a force of 00 lb. 8. Sound Waves Sound is a result of waves applying pressure to a person s eardrum. For a pure sound wave radiating outward in a spherical shape, the trigonometric function defined by P a r cos r ct can be used to model the sound pressure at a radius of r feet from the source, where t is time in seconds, is length of the sound wave in feet, c is speed of sound in feet per second, and a is maimum sound pressure at the source measured in pounds per square foot. (Source: Beranek, L., Noise and Vibration Control, Institute of Noise Control Engineering, Washington, D.C., 988.) Let ft and c 06 ft per sec. (a) Let a. lb per ft. Graph the sound pressure at distance r 0 ft from its source in the window 0,.05 by.05,.05. Describe P at this distance. (b) Now let a and t 0. Graph the sound pressure in the window 0, 0] by,. What happens to pressure P as radius r increases? (c) Suppose a person stands at a radius r so that r n, where n is a positive integer. Use the difference identity for cosine to simplify P in this situation. 8. Voltage of a Circuit When the two voltages V 0 sin 0t and V 0 cos 0 t are applied to the same circuit, the resulting voltage V will be equal to their sum. (Source: Bell, D., Fundamentals of Electric Circuits, Second Edition, Reston Publishing Company, 98.) (a) Graph the sum in the window 0,.05 by 60, 60. (b) Use the graph to estimate values for a and so that V a sin0t. (c) Use identities to verify that your epression for V is valid (b) a 50; Double-Angle Identities and Half-Angle Identities Double-Angle Identities Product-to-Sum and Sum-to-Product Identities Half-Angle Identities TEACHING TIP A common error is to write cos A as cos A. Double-Angle Identities When A B in the identities for the sum of two angles, these identities are called the double-angle identities. For eample, to derive an epression for cos A, we let B A in the identity cosa B cos A cos B sin A sin B. cos A cosa A cos A cos A sin A sin A Cosine sum identity (Section 7.) cos A cos A sin A

29 LIALMC07_0768.QXP /6/0 0:7 AM Page 6 7. Double-Angle Identities and Half-Angle Identities 6 TEACHING TIP Students might find it helpful to see each formula illustrated with a concrete eample that they can check. For instance, you might show that cos 60 cos 0 cos 0 sin 0. Two other useful forms of this identity can be obtained by substituting either cos A sin A or sin A cos A. Replace cos A with the epression sin A to get cos A cos A sin A sin A sin A cos A sin A, or replace sin A with cos A to get cos A cos A sin A cos A cos A cos A cos A cos A cos A. Fundamental identity (Section 7.) Fundamental identity We find sin A with the identity sina B sin A cos B cos A sin B, letting B A. sin A sina A sin A cos A cos A sin A sin A sin A cos A Using the identity for tana B, we find tan A. tan A tana A tan A tan A tan A tan A tan A tan A tan A Sine sum identity Tangent sum identity Looking Ahead to Calculus The identities cos A sin A and cos A cos A can be rewritten as Double-Angle Identities cos A cos A sin A cos A cos A sin A sin A cos A tan A cos A sin A tan A tan A sin A cos A and cos A cos A. These identities are used to integrate the functions fa sin A and ga cos A. EXAMPLE Finding Function Values of Given Information about Given cos 5 and sin 0, find sin, cos, and tan. Solution To find sin, we must first find the value of sin. sin 5 sin 6 5 sin cos ; Simplify. cos 5 sin 5 Choose the negative square root since sin 0.

30 LIALMC07_0768.QXP /6/0 0:7 AM Page 6 6 CHAPTER 7 Trigonometric Identities and Equations Using the double-angle identity for sine, sin sin cos Now we find cos, using the first of the double-angle identities for cosine. (Any of the three forms may be used.) The value of tan can be found in either of two ways. We can use the doubleangle identity and the fact that tan tan tan Simplify. (Section R.5) Alternatively, we can find tan by finding the quotient of sin and cos. tan sin cos 5 sin 5 ; cos 5 cos cos sin tan sin cos Now try Eercise 9. EXAMPLE Verifying a Double-Angle Identity Verify that the following equation is an identity. cot sin cos Solution We start by working on the left side, using the hint from Section 7. about writing all functions in terms of sine and cosine. cot sin cos sin sin cos sin cos sin cos cos Quotient identity Double-angle identity cos cos, so cos cos The final step illustrates the importance of being able to recognize alternative forms of identities. Now try Eercise 7.

31 LIALMC07_0768.QXP /6/0 0:7 AM Page Double-Angle Identities and Half-Angle Identities 65 EXAMPLE Simplify each epression. (a) cos 7 sin 7 Solution Simplifying Epressions Using Double-Angle Identities (a) This epression suggests one of the double-angle identities for cosine: cos A cos A sin A. Substituting 7 for A gives (b) If this epression were sin 5 cos 5, we could apply the identity for sin A directly since sin A sin A cos A. We can still apply the identity with A 5 by writing the multiplicative identity element as. (b) sin 5 cos 5 cos 7 sin 7 cos 7 cos. sin 5 cos 5 sin 5 cos 5 Multiply by in the form. sin 5 cos 5 Associative property (Section R.) sin 5 sin A cos A sin A, with A 5 sin 0 sin 0 (Section 5.) Now try Eercises and 5. Identities involving larger multiples of the variable can be derived by repeated use of the double-angle identities and other identities. EXAMPLE Deriving a Multiple-Angle Identity Write sin in terms of sin. Solution sin sin sin cos cos sin sin cos cos cos sin sin sin cos cos sin sin sin sin sin sin sin sin sin sin sin sin sin sin Sine sum identity (Section 7.) Double-angle identities Multiply. cos sin Distributive property (Section R.) Combine terms. Now try Eercise.

32 LIALMC07_0768.QXP /6/0 0:7 AM Page CHAPTER 7 Trigonometric Identities and Equations The net eample applies a multiple-angle identity to answer a question about electric current. 000 For = t, (6 sin 0t) W(t) = 5 EXAMPLE 5 Determining Wattage Consumption If a toaster is plugged into a common household outlet, the wattage consumed is not constant. Instead, it varies at a high frequency according to the model where V is the voltage and R is a constant that measures the resistance of the toaster in ohms. (Source: Bell, D., Fundamentals of Electric Circuits, Fourth Edition, Prentice-Hall, 998.) Graph the wattage W consumed by a typical toaster with R 5 and V 6 sin 0t in the window 0,.05 by 500, 000. How many oscillations are there? Solution Substituting the given values into the wattage equation gives W V R W V R, 6 sin 0t. 5 To determine the range of W, we note that sin 0t has maimum value, so 6 the epression for W has maimum value The minimum value is 0. The graph in Figure 6 shows that there are si oscillations Figure 6 Now try Eercise 8. Product-to-Sum and Sum-to-Product Identities Because they make it possible to rewrite a product as a sum, the identities for cosa B and cosa B are used to derive a group of identities useful in calculus. Adding the identities for cosa B and cosa B gives cosa B cos A cos B sin A sin B cosa B cos A cos B sin A sin B cosa B cosa B cos A cos B or cos A cos B cosa B cosa B. Similarly, subtracting cosa B from cosa B gives Looking Ahead to Calculus The product-to-sum identities are used in calculus to find integrals of functions that are products of trigonometric functions. One classic calculus tet includes the following eample: Evaluate cos 5 cos d. The first solution line reads: We may write cos 5 cos cos 8 cos. sin A sin B cosa B cosa B. Using the identities for sina B and sina B in the same way, we get two more identities. Those and the previous ones are now summarized. Product-to-Sum Identities cos A cos B [cos(a B) cos(a B)] sin A sin B [cos(a B) cos(a B)] (continued)

33 LIALMC07_0768.QXP /6/0 0:7 AM Page Double-Angle Identities and Half-Angle Identities 67 sin A cos B [sin(a B) sin(a B)] cos A sin B [sin(a B) sin(a B)] EXAMPLE 6 Using a Product-to-Sum Identity Write cos sin as the sum or difference of two functions. A Solution Use the identity for cos A sin B, with and cos sin sin sin B. sin sin Now try Eercise. From these new identities we can derive another group of identities that are used to write sums of trigonometric functions as products. Sum-to-Product Identities sin A sin B sin A B cos A B sin A sin B cos A B sin A B cos A cos B cos A B cos A B cos A cos B sin A B sin A B EXAMPLE 7 Using a Sum-to-Product Identity Write sin sin as a product of two functions. Solution Use the identity for sin A sin B, with and sin sin cos sin cos 6 sin cos sin cos sin A B. sin sin (Section 7.) Now try Eercise 5.

34 LIALMC07_0768.QXP /6/0 0:7 AM Page CHAPTER 7 Trigonometric Identities and Equations Half-Angle Identities From the alternative forms of the identity for cos A, we derive three additional identities for sin A, cos A, and tan A. These are known as half-angle identities. To derive the identity for sin A, start with the following double-angle identity for cosine and solve for sin. cos sin sin cos cos sin Add sin ; subtract cos. Divide by ; take square roots. (Section.) sin A cos A Let A, so A ; substitute. The sign in this identity indicates that the appropriate sign is chosen depending on the quadrant of. For eample, if A A is a quadrant III angle, we choose the negative sign since the sine function is negative in quadrant III. We derive the identity for cos A using the double-angle identity cos cos. cos cos cos cos cos cos Add. Rewrite; divide by. Take square roots. cos A cos A Replace with A. An identity for tan A comes from the identities for sin A and cos A. tan A sin A cos A cos A cos A cos A cos A We derive an alternative identity for tan A using double-angle identities. tan A sin A cos A sin A cos A cos A tan A sin A cos A sin A cos A Multiply by cos A in numerator and denominator. Double-angle identities From this identity for tan A, we can also derive tan A cos A sin A.

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