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333353_0500.qxp 1/15/07 2:35 PM Page 351 Analytic Trigonometry Chapter 5 y 5.1 Using Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Solving Trigonometric Equations 5.4 Sum and Difference Formulas 5.5 Multiple-Angle and Product-to-Sum Formulas Selected Applications Trigonometric equations and identities have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. ■ Friction, Exercise 71, page 367 ■ Shadow Length, Exercise 72, page 367 ■ Projectile Motion, Exercise 103, page 378 ■ Data Analysis: Unemployment Rate, Exercise 105, page 379 ■ Standing Waves, Exercise 79, page 385 ■ Harmonic Motion, Exercise 80, page 386 ■ Railroad Track, Exercise 129, page 397 ■ Mach Number, Exercise 130, page 398 y y 1 1 1 x π −1 2π x x −π π −1 π 2π 2π −1 You can use multiple approaches–algebraic, numerical, and graphical–to solve trigonometric equations. In Chapter 5, you will use all three approaches to solve trigonometric equations. You will also use trigonometric identities to evaluate trigonometric functions and simplify trigonometric expressions. Dan Donovan/MLB Photos/Getty Images Trigonometry can be used to model projectile motion, such as the flight of a baseball. Given the angle at which the ball leaves the bat and the initial velocity, you can determine the distance the ball will travel. 351 333353_0501.qxp 352 1/15/07 2:37 PM Chapter 5 Page 352 Analytic Trigonometry 5.1 Using Fundamental Identities What you should learn Introduction In Chapter 4, you studied the basic definitions, properties, graphs, and applications of the individual trigonometric functions. In this chapter, you will learn how to use the fundamental identities to do the following. 1. 2. 3. 4. Evaluate trigonometric functions. Simplify trigonometric expressions. Develop additional trigonometric identities. Solve trigonometric equations. The fundamental trigonometric identities can be used to simplify trigonometric expressions. For instance, Exercise 111 on page 359 shows you how trigonometric identities can be used to simplify an expression for the rate of change of a function, a concept used in calculus. Reciprocal Identities 1 csc u 1 csc u sin u 1 sec u 1 sec u cos u cos u 䊏 Recognize and write the fundamental trigonometric identities. Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions. Why you should learn it Fundamental Trigonometric Identities sin u 䊏 1 cot u 1 cot u tan u tan u Quotient Identities tan u sin u cos u cot u cos u sin u Pythagorean Identities sin2 u cos 2 u 1 1 tan2 u sec 2 u 1 cot 2 u csc 2 u Cofunction Identities 冢 2 u冣 cos u tan冢 u冣 cot u 2 sec冢 u冣 csc u 2 sin 冢 2 u冣 sin u cot冢 u冣 tan u 2 csc冢 u冣 sec u 2 cos STUDY TIP Even/Odd Identities sin共u兲 sin u cos共u兲 cos u tan共u兲 tan u csc共u兲 csc u sec共u兲 sec u cot共u兲 cot u Pythagorean identities are sometimes used in radical form such as sin u ± 冪1 cos2 u Using the Fundamental Identities One common use of trigonometric identities is to use given values of trigonometric functions to evaluate other trigonometric functions. or tan u ± 冪sec2 u 1 where the sign depends on the choice of u. 333353_0501.qxp 1/15/07 2:37 PM Page 353 Section 5.1 353 Using Fundamental Identities Example 1 Using Identities to Evaluate a Function 3 Use the values sec u 2 and tan u > 0 to find the values of all six trigonometric functions. Solution Using a reciprocal identity, you have cos u 1 1 2 . sec u 3兾2 3 Using a Pythagorean identity, you have sin2 u 1 cos 2 u 冢 3冣 1 2 Pythagorean identity 2 2 Substitute 3 for cos u. 4 5 1 . 9 9 Simplify. TECHNOLOGY TIP You can use a graphing utility to check the result of Example 2. To do this, enter y1 and y2 as shown below. 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 冪5兾3. Now, knowing the values of the sine and cosine, you can find the values of all six trigonometric functions. sin u 冪5 csc u 3 2 cos u 3 tan u 1 3 3冪5 冪 sin u 5 5 1 3 sec u cos u 2 sin u 冪5兾3 冪5 cos u 2兾3 2 cot u 1 2 2冪5 冪 tan u 5 5 Now try Exercise 7. Example 2 Simplifying a Trigonometric Expression Simplify sin x cos 2 x sin x. Solution Select the line style for y1 and the path style for y2 (see figure above). The path style, denoted by , traces the leading edge of the graph and draws a path. Now, graph both equations in the same viewing window, as shown below. The two graphs appear to coincide, so the expressions appear to be equivalent. Remember that in order to be certain that two expressions are equivalent, you need to show their equivalence algebraically, as in Example 2. First factor out a common monomial factor and then use a fundamental identity. sin x cos 2 x sin x sin x共cos2 x 1兲 Factor out monomial factor. sin x共1 cos 2 x兲 Distributive Property sin x共sin2 x兲 Pythagorean identity sin3 x Multiply. Now try Exercise 29. 2 − −2 333353_0501.qxp 354 1/15/07 Chapter 5 2:37 PM Page 354 Analytic Trigonometry Example 3 Verifying a Trigonometric Identity Determine whether the equation appears to be an identity. ? cos 3x 4 cos3 x 3 cos x Numerical Solution Graphical Solution Use the table feature of a graphing utility set in radian mode to create a table that shows the values of y1 cos 3x and y2 4 cos3 x 3 cos x for different values of x, as shown in Figure 5.1. The values of y1 and y2 appear to be identical, so cos 3x 4 cos3 x 3 cos x appears to be an identity. Use a graphing utility set in radian mode to graph y1 cos 3x and y2 4 cos3 x 3 cos x in the same viewing window, as shown in Figure 5.2. (Select the line style for y1 and the path style for y2.) Because the graphs appear to coincide, cos 3x 4 cos3 x 3 cos x appears to be an identity. 2 y1 = cos 3x − Figure 5.1 −2 y2 = 4 cos3 x − 3 cos x Note that if the values of y1 and y2 were not identical, then the equation would not be an identity. Figure 5.2 Note that if the graphs of y1 and y2 did not coincide, then the equation would not be an identity. Now try Exercise 39. Example 4 Verifying a Trigonometric Identity Verify the identity cos sin csc . 1 cos sin Algebraic Solution Graphical Solution sin cos 共sin 兲共sin 兲 (cos 兲共1 cos 兲 1 cos sin 共1 cos 兲共sin 兲 sin2 cos2 cos 共1 cos 兲共sin 兲 Multiply. 1 cos 共1 cos 兲共sin 兲 Pythagorean identity 1 sin Divide out common factor. csc 4 −2 y1 = sin θ cos θ + 1 + cos θ sin θ 2 Use reciprocal identity. Notice how the identity is verified. You start with the left side of the equation (the more complicated side) and use the fundamental trigonometric identities to simplify it until you obtain the right side. Now try Exercise 45. Use a graphing utility set in radian and dot modes to graph y1 and y2 in the same viewing window, as shown in Figure 5.3. Because the graphs appear to coincide, this equation appears to be an identity. y2 = csc θ −4 Figure 5.3 333353_0501.qxp 1/15/07 2:37 PM Page 355 Section 5.1 Using Fundamental Identities When factoring trigonometric expressions, it is helpful to find a polynomial form that fits the expression, as shown in Example 5. Example 5 Factoring Trigonometric Expressions Factor (a) sec 2 1 and (b) 4 tan2 tan 3. Solution a. Here the expression is a difference of two squares, which factors as sec2 1 共sec 1兲共sec 1). b. This expression has the polynomial form ax 2 bx c and it factors as 4 tan2 tan 3 共4 tan 3兲共tan 1兲. Now try Exercise 51. On occasion, factoring or simplifying can best be done by first rewriting the expression in terms of just one trigonometric function or in terms of sine or cosine alone. These strategies are illustrated in Examples 6 and 7. Example 6 Factoring a Trigonometric Expression Factor csc 2 x cot x 3. Solution Use the identity csc 2 x 1 cot 2 x to rewrite the expression in terms of the cotangent. csc 2 x cot x 3 共1 cot 2 x兲 cot x 3 Pythagorean identity cot 2 x cot x 2 Combine like terms. 共cot x 2兲共cot x 1兲 Factor. Now try Exercise 57. Example 7 Simplifying a Trigonometric Expression 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 冢 sin t 冣cos t cos t Quotient identity sin2 t cos 2 t sin t Add fractions. 1 csc t sin t Pythagorean identity and reciprocal identity Now try Exercise 67. 355 333353_0501.qxp 356 1/15/07 2:37 PM Chapter 5 Page 356 Analytic Trigonometry The last two examples in this section involve techniques for rewriting expressions into forms that are used in calculus. Example 8 Rewriting a Trigonometric Expression Rewrite 1 so that it is not in fractional form. 1 sin x Solution From the Pythagorean identity cos 2 x 1 sin2 x 共1 sin x兲共1 sin x兲, you can see that multiplying both the numerator and the denominator by 共1 sin x兲 will produce a monomial denominator. 1 1 1 sin x 1 sin x 1 sin x 1 sin x Multiply numerator and denominator by 共1 sin x兲. 1 sin x 1 sin2 x Multiply. 1 sin x cos 2 x Pythagorean identity 1 sin x 2 cos x cos 2 x Write as separate fractions. 1 sin x 2 cos x cos x 1 cos x sec2 x tan x sec x Write as separate fractions. Reciprocal and quotient identities Now try Exercise 69. Example 9 Trigonometric Substitution Use the substitution x 2 tan , 0 < < 兾2, to write 冪4 x 2 as a trigonometric function of . Solution Begin by letting x 2 tan . Then you can obtain 冪4 x 2 冪4 共2 tan 兲2 冪4共1 tan2 兲 冪4 sec 2 2 sec . Substitute 2 tan for x. Distributive Property Pythagorean identity sec > 0 for 0 < < 4+ x2 x 2 Now try Exercise 81. θ = arctan 2 Figure 5.4 shows the right triangle illustration of the substitution in Example 9. For 0 < < 兾2, you have opp x, adj 2, and hyp 冪4 x2. Try using these expressions to obtain the result shown in Example 9. Figure 5.4 x 2 333353_0501.qxp 1/15/07 2:37 PM Page 357 Section 5.1 5.1 Exercises Using Fundamental Identities 357 See www.CalcChat.com for worked-out solutions to odd-numbered exercises. Vocabulary Check Fill in the blank to complete the trigonometric identity. 1. 1 _______ cos u 2. 1 _______ cot u 3. cos u _______ sin u 4. 1 _______ sin u 5. 1 _______ sec2 u 7. cos 6. 1 cot2 u _______ 冢2 u冣 _______ 8. csc 9. tan共u兲 _______ 冢2 u冣 _______ 10. cos共u兲 _______ In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions. 1 1. sin x , 2 cos x 2. csc 2, tan 冪3 2 冪3 冪3 3 , cos x 7 5. tan x , 24 17 7. sec , 15 (b) ⴚ1 (c) cot x (e) ⴚtan x (f) sin x 2 19. sin共x兲 cos共x兲 冪3 2 x 17. cot2 冪2 25 sec x 24 6. cot 5, sin (a) sec x (d) 1 15. sec x cos x 3 3. sec 冪2, sin 4. tan x In Exercises 15–20, match the trigonometric expression with one of the following. csc 2 16. tan x csc x 20. sin关共兾2兲 x兴 cos关共兾2兲 x兴 In Exercises 21–26, match the trigonometric expression with one of the following. (a) csc x (b) tan x 冪26 (d) sin x tan x 26 21. sin x sec x 8 sin 17 18. 共1 cos 2 x兲共csc x兲 x 23. sec x 4 tan4 (e) sec2 x (c) sin2 x (f) sec2 x 1 tan2 x 22. cos2 x共sec2 x 1兲 x sec x 1 sin2 x 24. cot x sec x 2 cos2关共兾2兲 x兴 cos x 4 3 x , cos x 8. cos 2 5 5 25. 2冪5 2 9. sin共x兲 , tan x 3 5 In Exercises 27– 38, use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. 冢 冣 10. csc共x兲 5, cos x 冪24 5 11. tan 2, sin < 0 12. sec 3, tan < 0 13. csc is undefined, cos < 0 14. tan is undefined, sin > 0 26. 27. cot x sin x 28. cos tan 29. sin 共csc sin 兲 30. sec 2 x共1 sin2 x兲 31. csc x cot x 33. sec sin tan 32. sec csc 34. tan2 sec2 333353_0501.qxp 358 2:38 PM Chapter 5 35. sin 37. 1/15/07 Page 358 Analytic Trigonometry 冢 2 x冣csc x cos2 y 1 sin y 36. cot 38. 冢 2 x冣cos x 1 cot x 1 2 In Exercises 39– 44, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. 39. sin cos cot csc 40. 共sec tan 兲共csc 1兲 cot 41. cos sec tan 1 sin 42. 1 csc sec cot cos 43. In Exercises 45–50, verify the identity algebraically. Use a graphing utility to check your result graphically. 45. csc tan sec 46. sin csc sin2 cos2 sin2 cos 47. 1 1 cos 1 sec tan 2 csc 1 sec tan cot共 兲 cos 49. csc 50. 冢2 冣 tan共 兲 63. 共csc x 1兲共csc x 1兲 64. 共5 5 sin x兲共5 5 sin x兲 65. 1 1 1 cos x 1 cos x 66. 1 1 sec x 1 sec x 1 sec2 x tan x cos x 1 sin x 1 sin x cos x In Exercises 69–72, rewrite the expression so that it is not in fractional form. 69. sin2 y 1 cos y 70. 5 tan x sec x 71. 3 sec x tan x 72. tan2 x csc x 1 Numerical and Graphical Analysis In Exercises 73 –76, use a graphing utility to complete the table and graph the functions in the same viewing window. Make a conjecture about y1 and y2. 0.2 x 0.4 0.6 0.8 1.0 1.2 1.4 y1 y2 csc 73. y1 cos In Exercises 51– 60, factor the expression and use the fundamental identities to simplify. Use a graphing utility to check your result graphically. 51. cot2 x cot2 x cos2 x 53. 62. 共tan x sec x兲共tan x sec x兲 68. sin 1 cos 2 csc sin 1 cos csc 61. 共sin x cos x兲2 67. tan x sin cos cos sin sec csc 44. sin cos 48. In Exercises 61–68, perform the indicated operation and use the fundamental identities to simplify. cos2 x 4 cos x 2 52. sec2 x tan2 x sec2 x 54. csc2 x 1 csc x 1 冢 2 x冣, y2 sin x 74. y1 cos x sin x tan x, y2 sec x 75. y1 cos x 1 sin x , y2 1 sin x cos x 76. y1 sec4 x sec2 x, y2 tan2 x tan4 x 56. 1 2 sin2 x sin4 x In Exercises 77–80, use a graphing utility to determine which of the six trigonometric functions is equal to the expression. 57. sin4 x cos4 x 77. cos x cot x sin x 55. tan4 x 2 tan2 x 1 58. sec4 x tan4 x 59. csc3 x csc2 x csc x 1 60. sec x sec x sec x 1 3 2 79. sec x cos x 1 sin x 78. sin x共cot x tan x兲 80. cos 1 1 sin 2 cos 1 sin 冢 冣 333353_0501.qxp 1/15/07 2:38 PM Page 359 Section 5.1 In Exercises 81– 92, use the trigonometric substitution to write the algebraic expression as a trigonometric function of , where 0 < < /2. x 5 sin 81. 冪25 x 2, 82. 冪64 x 3 sec 83. 冪x 2 9, x 10 tan 84. 冪x 2 100, 85. 冪9 x2, x 3 sin 86. 冪4 x2, x 2 cos 87. 冪4x2 9, 2x 3 tan 88. 冪9x2 4, 3x 2 tan True or False? In Exercises 113 and 114, determine whether the statement is true or false. Justify your answer. 4x 3 sec 90. 冪9x2 25, 3x 5 sec x 冪2 sin 92. 冪5 x2, x 冪5 cos 91. 冪2 111. Rate of Change The rate of change of the function f 共x兲 csc x sin x is given by the expression csc x cot x cos x. Show that this expression can also be written as cos x cot 2 x. Synthesis 89. 冪16x2 9, x2, 114. cos sec 1 113. sin csc 1 In Exercises 115–118, fill in the blanks. (Note: x → c1 indicates that x approaches c from the right, and x → cⴚ indicates that x approaches c from the left.) In Exercises 93–96, use a graphing utility to solve the equation for , where 0 } < 2. 93. sin 冪1 cos2 115. As x → , sin x → 䊏 and csc x → 䊏. 2 116. As x → 0 , cos x → 䊏 and sec x → 䊏. , tan x → 䊏 and cot x → 䊏. 2 94. cos 冪1 sin2 117. As x → 95. sec 冪1 tan2 118. As x → , sin x → 䊏 and csc x → 䊏. 96. tan 冪sec2 1 In Exercises 97–100, rewrite the expression as a single logarithm and simplify the result. ⱍ ⱍ ⱍ ⱍ lnⱍcsc ⱍ lnⱍtan ⱍ ln共1 sin x兲 lnⱍsec xⱍ lnⱍcot tⱍ ln共1 tan2 t兲 97. ln cos ln sin 98. 99. 359 112. Rate of Change The rate of change of the function f 共x兲 sec x cos x is given by the expression sec x tan x sin x. Show that this expression can also be written as sin x tan2 x. x 2 cos 16x 2, Using Fundamental Identities 119. Write each of the other trigonometric functions of in terms of sin . 120. Write each of the other trigonometric functions of in terms of cos . 121. Use the definitions of sine and cosine to derive the Pythagorean identity sin2 cos2 1. In Exercises 101–106, show that the identity is not true for all values of . (There are many correct answers.) 122. Writing Use the Pythagorean identity sin2 cos2 1 to derive the other Pythagorean identities 1 tan2 sec2 and 1 cot2 csc2 . Discuss how to remember these identities and other fundamental identities. 101. cos 冪1 sin2 102. tan 冪sec2 1 103. sin 冪1 cos2 104. sec 冪1 tan2 Skills Review 105. csc 冪1 cot2 106. cot 冪csc2 1 100. In Exercises 107–110, use the table feature of a graphing utility to demonstrate the identity for each value of . 107. csc2 cot2 1, (a) 132 (b) 2 7 108. tan2 1 sec2 , (a) 346 (b) 3.1 109. cos 冢 sin , (a) 80 (b) 0.8 2 冣 110. sin共 兲 sin , (a) 250 (b) 1 2 In Exercises 123–126, sketch the graph of the function. (Include two full periods.) 123. f 共x兲 1 sin x 2 124. f 共x兲 2 tan x 2 125. f 共x兲 1 cot x 2 4 126. f 共x兲 3 cos共x 兲 3 2 冢 冣 333353_0502.qxp 360 1/15/07 2:44 PM Chapter 5 Page 360 Analytic Trigonometry 5.2 Verifying Trigonometric Identities What you should learn Introduction In this section, you will study techniques for verifying trigonometric identities. In the next section, you will study techniques for solving trigonometric equations. The key to both verifying identities and solving equations is your ability to use the fundamental identities and the rules of algebra to rewrite trigonometric expressions. Remember that a conditional equation is an equation that is true for only some of the values in its domain. For example, the conditional equation sin x 0 Conditional equation 䊏 Verify trigonometric identities. Why you should learn it You can use trigonometric identities to rewrite trigonometric expressions. For instance, Exercise 72 on page 367 shows you how trigonometric identities can be used to simplify an equation that models the length of a shadow cast by a gnomon (a device used to tell time). is true only for x 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 example, the familiar equation sin2 x 1 cos 2 x Identity is true for all real numbers x. So, it is an identity. Verifying Trigonometric Identities BSCHMID/Getty Images 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 1. Work with one side of the equation at a time. It is often better to work with the more complicated side first. 2. Look for opportunities to factor an expression, 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 expression 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 making an attempt that leads to a dead end provides insight. Verifying trigonometric identities is a useful process if you need to convert a trigonometric expression 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. Prerequisite Skills To review the differences among an identity, an expression, and an equation, see Appendix B.3. 333353_0502.qxp 1/15/07 2:44 PM Page 361 Section 5.2 Verifying Trigonometric Identities 361 Example 1 Verifying a Trigonometric Identity Verify the identity sec2 1 sin2 . sec2 Solution Because the left side is more complicated, start with it. sec2 1 共tan2 1兲 1 sec2 sec2 tan2 sec2 Pythagorean identity STUDY TIP Simplify. tan2 共cos 2 兲 Reciprocal identity sin2 共cos2兲 cos2 Quotient identity sin2 Simplify. Remember that an identity is true only for all real values in the domain of the variable. For instance, in Example 1 the identity is not true when 兾2 because sec2 is not defined when 兾2. Now try Exercise 5. There can be more than one way to verify an identity. Here is another way to verify the identity in Example 1. sec2 1 sec2 1 2 2 sec sec sec2 Rewrite as the difference of fractions. 1 cos 2 Reciprocal identity sin2 Pythagorean identity Example 2 Combining Fractions Before Using Identities Verify the identity 1 1 2 sec2 . 1 sin 1 sin Algebraic Solution Numerical Solution 1 1 1 sin 1 sin 1 sin 1 sin 共1 sin 兲共1 sin 兲 Add fractions. 2 1 sin2 Simplify. 2 cos2 Pythagorean identity 2 sec2 Now try Exercise 31. Use the table feature of a graphing utility set in radian mode to create a table that shows the values of y1 1兾共1 sin x兲 1兾共1 sin x兲 and y2 2兾cos2 x for different values of x, as shown in Figure 5.5. From the table, you can see that the values appear to be identical, so 1兾共1 sin x兲 1兾共1sin x兲 2 sec 2 x appears to be an identity. Reciprocal identity Figure 5.5 333353_0502.qxp 362 1/15/07 2:44 PM Chapter 5 Page 362 Analytic Trigonometry Example 3 Verifying a Trigonometric Identity Verify the identity 共tan2 x 1兲共cos 2 x 1兲 tan2 x. Algebraic Solution Graphical Solution By applying identities before multiplying, you obtain the following. 共 tan2 x 1兲共 cos 2 x 1兲 共 兲共 sec2 x 兲 sin2 x sin2 x cos 2 x 冢cos x冣 sin x Pythagorean identities Reciprocal identity 2 tan2 x Rule of exponents Quotient identity Use a graphing utility set in radian mode to graph the left side of the identity y1 共tan2 x 1兲共cos2 x 1兲 and the right side of the identity y2 tan2 x in the same viewing window, as shown in Figure 5.6. (Select the line style for y1 and the path style for y2.) Because the graphs appear to coincide, 共tan2 x 1兲 共cos2 x 1兲 tan2 x appears to be an identity. 2 y1 = (tan2 x + 1)(cos2 x − 1) −2 2 −3 Now try Exercise 39. y2 = −tan2 x Figure 5.6 Example 4 Converting to Sines and Cosines Verify the identity tan x cot x sec x csc x. Solution In this case there appear to be no fractions to add, no products to find, and no opportunities to use the Pythagorean identities. So, try converting the left side to sines and cosines. TECHNOLOGY TIP sin x cos x cos x sin x Quotient identities sin2 x cos 2 x cos x sin x Add fractions. 1 cos x sin x Pythagorean identity tan x cot x 1 cos x 1 sin x sec x csc x Now try Exercise 41. Although a graphing utility can be useful in helping to verify an identity, you must use algebraic techniques to produce a valid proof. For example, graph the two functions y1 sin 50x Product of fractions Reciprocal identities y2 sin 2x in a trigonometric viewing window. Although their graphs seem identical, sin 50x sin 2x. 333353_0502.qxp 1/15/07 2:44 PM Page 363 Section 5.2 Verifying Trigonometric Identities 363 Recall from algebra that rationalizing the denominator using conjugates is, on occasion, a powerful simplification technique. A related form of this technique works for simplifying trigonometric expressions as well. For instance, to simplify 1兾共1 cos x兲, multiply the numerator and the denominator by 1 cos x. 1 1 1 cos x 1 cos x 1 cos x 1 cos x 冢 1 cos x 1 cos2 x 1 cos x sin2 x 冣 csc2 x共1 cos x兲 As shown above, csc2 x共1 cos x兲 is considered a simplified form of 1兾共1 cos x兲 because the expression does not contain any fractions. Example 5 Verifying a Trigonometric Identity Verify the identity sec x tan x TECHNOLOGY SUPPORT For instructions on how to use the radian and dot modes, see Appendix A; for specific keystrokes, go to this textbook’s Online Study Center. cos x . 1 sin x Algebraic Solution Graphical Solution Begin with the right side because you can create a monomial denominator by multiplying the numerator and denominator by 共1 sin x兲. Use a graphing utility set in the radian and dot modes to graph y1 sec x tan x and y2 cos x兾共1 sin x兲 in the same viewing window, as shown in Figure 5.7. Because the graphs appear to coincide, sec x tan x cos x兾共1 sin x兲 appears to be an identity. cos x 1 sin x cos x 1 sin x 1 sin x 1 sin x 冢 冣 Multiply numerator and denominator by 共1 sin x兲. cos x cos x sin x 1 sin2 x Multiply. cos x cos x sin x cos 2 x Pythagorean identity cos x sin x cos x 2 cos x cos2 x Write as separate fractions. 5 − 7 2 9 2 −5 1 sin x cos x cos x Simplify. sec x tan x Identities y1 = sec x + tan x Figure 5.7 Now try Exercise 47. In Examples 1 through 5, you have been verifying trigonometric identities by working with one side of the equation and converting it 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 Example 6. y2 = cos x 1 − sin x 333353_0502.qxp 364 1/15/07 2:44 PM Chapter 5 Page 364 Analytic Trigonometry Example 6 Working with Each Side Separately Verify the identity 1 sin cot 2 . 1 csc sin Algebraic Solution Numerical Solution Working with the left side, you have Use the table feature of a graphing utility set in radian mode to create a table that shows the values of y1 cot2 x兾共1 csc x兲 and y2 共1 sin x兲/sin x for different values of x, as shown in Figure 5.8. From the table you can see that the values appear to be identical, so cot2 x兾共1 csc x兲 共1 sin x兲兾sin x appears to be an identity. cot 2 csc2 1 1 csc 1 csc Pythagorean identity 共csc 1兲共csc 1兲 1 csc csc 1. Factor. Simplify. Now, simplifying the right side, you have 1 sin 1 sin sin sin sin Write as separate fractions. csc 1. Reciprocal identity The identity is verified because both sides are equal to csc 1. Now try Exercise 49. In Example 7, powers of trigonometric functions are rewritten as more complicated sums of products of trigonometric functions. This is a common procedure used in calculus. Example 7 Examples from Calculus Verify each identity. b. sin3 x cos4 x 共cos4 x cos 6 x兲sin x a. tan4 x tan2 x sec2 x tan2 x Solution a. tan4 x 共tan2 x兲共tan2 x兲 Write as separate factors. tan2 x共sec2 x 1兲 tan2 x sec2 x Pythagorean identity tan2 x Multiply. b. sin3 x cos4 x sin2 x cos4 x sin x 共1 cos2 x兲 cos4 x sin x 共cos4 x cos6 x兲sin x Write as separate factors. Pythagorean identity Multiply. Now try Exercise 63. TECHNOLOGY TIP Remember that you can use a graphing utility to assist in verifying an identity by creating a table or by graphing. Figure 5.8 333353_0502.qxp 1/15/07 2:44 PM Page 365 Section 5.2 5.2 Exercises Verifying Trigonometric Identities 365 See www.CalcChat.com for worked-out solutions to odd-numbered exercises. Vocabulary Check In Exercises 1 and 2, fill in the blanks. 1. An equation that is true for only some values in its domain is called a _______ equation. 2. An equation that is true for all real values in its domain is called an _______ . In Exercises 3–10, fill in the blank to complete the trigonometric identity. 3. 1 _______ tan u 4. 1 _______ csc u 5. sin u _______ cos u 6. 1 _______ sec u 7. sin2 u _______ 1 8. tan 9. sin共u兲 _______ 冢2 u冣 _______ 10. sec共u兲 _______ 13. y1 csc x sin x, y2 cos x cot x 1. sin t csc t 1 14. y1 sec x cos x, y2 sin x tan x 2. sec y cos y 1 15. y1 sin x cos x cot x, In Exercises 1–10, verify the identity. 3. 4. 16. y1 cos x sin x tan x, y2 sec x csc2 x csc x sec x cot x sin2 t 17. y1 1 1 , tan x cot x 18. y1 1 1 , y2 csc x sin x sin x csc x cos2 t tan2 t 5. cos 2 sin2 1 2 sin2 6. cos 2 sin2 2 cos 2 1 Error Analysis 7. tan2 6 sec2 5 y2 tan x cot x In Exercises 19 and 20, describe the error. 19. 共1 tan x兲关1 cot共x兲兴 8. 2 csc 2 z 1 cot2 z 共1 tan x兲共1 cot x兲 9. 共1 sin x兲共1 sin x兲 cos2 x 1 cot x tan x tan x cot x 10. tan y共csc y 1兲 1 2 2 Numerical, Graphical, and Algebraic Analysis In Exercises 11–18, use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that y1 ⴝ y2 . Then verify the identity algebraically. 1 cot x tan x 1 2 cot x tan x 20. 1 sec共 兲 1 sec sin共 兲 tan共 兲 sin tan x 0.2 0.4 0.6 0.8 1.0 y1 y2 1 , y2 csc x sin x 11. y1 sec x tan x 12. y1 csc x 1 , 1 sin x y2 csc x y2 csc x 1.2 1.4 1 sec 冤 共sin 兲 1 冢cos1 冣冥 1 sec sin 共1 sec 兲 1 csc sin 333353_0502.qxp 1/15/07 366 2:44 PM Chapter 5 Page 366 Analytic Trigonometry In Exercises 21–30, verify the identity. 21. sin1兾2 22. sec6 23. cot x cos x sin 5兾2 x cos x cos x冪sin x 40. csc x共csc x sin x兲 3 x共sec x tan x兲 sec4 x共sec x tan x兲 sec5 x tan3 x 冢 2 x冣csc x sec x sin x cos x cot x csc2 x sin x 41. cot x tan x csc x sin x 42. 1 csc cot cos sec 24. sec关共兾2兲 x兴 sec x tan关共兾2兲 x兴 43. csc4 x 2 csc2 x 1 cot4 x 25. csc共x兲 cot x sec共x兲 45. sec4 tan4 1 2 tan2 26. 共1 sin y兲关1 sin共y兲兴 cos2 y 44. sin x共1 2 cos2 x cos4 x兲 sin5 x 46. csc4 cot4 2 csc2 1 47. sin 1 cos 1 cos sin 48. 1 csc共兲 sec cos共兲 cot共兲 csc 1 cot csc 1 cot 49. 29. tan x tan y sin x cos y cos x sin y cos x cos y sin x sin y 1 tan x tan y tan3 1 tan2 tan 1 tan 1 50. 30. tan x tan y cot x cot y 1 tan x tan y cot x cot y 1 sin3 cos3 1 sin cos sin cos 27. 28. cos共兲 sec tan 1 sin共兲 In Exercises 31–38, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. 31. 32. sin x sin y cos x cos y 0 sin x sin y cos x cos y tan x cot y tan y cot x tan x cot y 33. 1 sin 冪11 sin sin ⱍcos ⱍ 34. 1 cos 冪11 cos cos ⱍsin ⱍ 35. sin2 冢 2 x冣 sin 2 x1 Conjecture In Exercises 51–54, use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about a simplification of the expression. Verify the resulting identity algebraically. 51. y 1 1 cot x 1 tan x 1 52. y cos x sin x cos x 1 tan x sin x cos x 53. y 1 cos2 x sin x sin x 54. y sin t cot2 t csc t In Exercises 55–58, use the properties of logarithms and trigonometric identities to verify the identity. ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ 55. ln cot ln cos ln sin ⱍ ⱍ ⱍ 36. sec2 y cot 2 y 1 2 56. ln sec ln cos 37. sin x csc x tan x 2 58. ln csc cot ln csc cot 冢 冢 38. sec2 冢 冣 冣 x 1 cot2 x 2 冣 ⱍ 57. ln共1 cos 兲 ln共1 cos 兲 2 ln sin ⱍ ⱍ ⱍ ⱍ In Exercises 59–62, use the cofunction identities to evaluate the expression without using a calculator. 59. sin2 35 sin2 55 In Exercises 39–50, verify the identity algebraically. Use a graphing utility to check your result graphically. 39. 2 sec2 x 2 sec2 x sin2 x sin2 x cos 2 x 1 ⱍ 60. cos 2 14 cos 2 76 61. cos2 20 cos2 52 cos2 38 cos2 70 62. sin2 18 sin2 40 sin2 50 sin2 72 333353_0502.qxp 1/15/07 2:45 PM Page 367 Section 5.2 Verifying Trigonometric Identities 367 In Exercises 63–66, powers of trigonometric functions are rewritten to be useful in calculus. Verify the identity. 76. sin共x2兲 sin2共x兲 63. tan5 x tan3 x sec2 x tan3 x In Exercises 77–80, (a) verify the identity and (b) determine if the identity is true for the given value of x. Explain. tan2 x共 tan2 x x 65. cos 3 x sin x 共sin x sin x兲cos x 66. sin 4 x 2 2 cos 4 x12 tan4 x兲 64. sec4 sec2 x 4 cos 2 x2 In Exercises 67–70, verify the identity. 67. tan共sin1 x兲 x 冪1 x2 68. cos共sin1 x兲 冪1 x2 冢 x1 x1 4 冪16 共x 1兲2 冢 冪4 共x 1兲2 x1 2 x1 69. tan sin1 70. tan cos1 77. sin x 1 cos x , 1 cos x sin x 78. sec x tan x , x tan x sec x cos x cos 4 x 冣 79. csc x cot x 80. 冣 71. Friction The forces acting on an object weighing W units on an inclined plane positioned at an angle of with the horizontal (see figure) are modeled by sin x , 1 cos x 2 In Exercises 81–84, use the trigonometric substitution to write the algebraic expression as a trigonometric function of , where 0 < < /2. Assume a > 0. 81. 冪a2 u2, u a sin 82. 冪a2 u2, u a cos W cos W sin 83. u a tan where is the coefficient of friction. Solve the equation for and simplify the result. 84. 冪u2 a2, u a sec θ x cot x 1 1 tan x , x cot x 1 1 tan x 4 u2, W x0 冪a2 Think About It In Exercises 85 and 86, explain why the equation is not an identity and find one value of the variable for which the equation is not true. 85. 冪tan2 x tan x 86. sin 冪1 cos2 冤 87. Verify that for all integers n, cos 冤 88. Verify that for all integers n, sin 共2n 1兲 0. 2 冥 共12n 1兲 1 . 6 2 冥 Skills Review 72. Shadow Length The length s of the shadow cast by a vertical gnomon (a device used to tell time) of height h when the angle of the sun above the horizon is can be modeled by the equation s h sin共90 兲 . sin Show that the equation is equivalent to s h cot . Synthesis True or False? In Exercises 73–76, determine whether the statement is true or false. Justify your answer. In Exercises 89–92, find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) 89. 1, 8i, 8i 90. i, i, 4i, 4i 91. 4, 6 i, 6 i 92. 0, 0, 2, 1 i In Exercises 93–96, sketch the graph of the function by hand. 93. f 共x兲 2x 3 94. f 共x兲 2x3 95. f 共x兲 2x 1 96. f 共x兲 2x1 3 73. There can be more than one way to verify a trigonometric identity. In Exercises 97–100, state the quadrant in which lies. 74. Of the six trigonometric functions, two are even. 97. csc > 0 and tan < 0 98. cot > 0 and cos < 0 99. sec > 0 and sin < 0 100. cot > 0 and sec < 0 75. The equation sin2 cos2 1 tan2 is an identity, because sin2共0兲 cos2共0兲 1 and 1 tan2共0兲 1. 333353_0503.qxp 1/15/07 368 2:46 PM Chapter 5 Page 368 Analytic Trigonometry 5.3 Solving Trigonometric Equations What you should learn Introduction To solve a trigonometric equation, use standard algebraic techniques such as collecting like terms and factoring. Your preliminary goal is to isolate the trigonometric function involved in the equation. 䊏 䊏 䊏 Example 1 Solving a Trigonometric Equation 2 sin x 1 0 䊏 Original equation 2 sin x 1 Why you should learn it Add 1 to each side. sin x 12 Divide each side by 2. 1 To solve for x, note in Figure 5.9 that the equation sin x 2 has solutions x 兾6 and x 5兾6 in the interval 关0, 2兲. Moreover, because sin x has a period of 2, there are infinitely many other solutions, which can be written as x 2n 6 x and 5 2n 6 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. You can use trigonometric equations to solve a variety of real-life problems.For instance, Exercise 100 on page 378 shows you how solving a trigonometric equation can help answer questions about the position of the sun in Cheyenne,Wyoming. General solution where n is an integer, as shown in Figure 5.9. y π x = − 2π 6 y= x= 1 2 1 π 6 −π x= π + 2π 6 x π x = 5π − 2 π 6 SuperStock x = 5π 6 −1 x = 5π + 2π 6 y = sin x Figure 5.9 Now try Exercise 25. 1 Figure 5.10 verifies that the equation sin x 2 has infinitely many solutions. Any angles that are coterminal with 兾6 or 5兾6 are also solutions of the equation. sin 5π + 2nπ = 1 2 6 ( Figure 5.10 ( 5π 6 π 6 sin π + 2nπ = 1 2 6 ( ( Prerequisite Skills If you have trouble finding coterminal angles, review Section 4.1. 333353_0503.qxp 1/15/07 2:46 PM Page 369 Section 5.3 369 Solving Trigonometric Equations Example 2 Collecting Like Terms Find all solutions of sin x 冪2 sin x in the interval 关0, 2兲. Algebraic Solution Numerical Solution Rewrite the equation so that sin x is isolated on one side of the equation. Use the table feature of a graphing utility set in radian mode to create a table that shows the values of y1 sin x 冪2 and y2 sin x for different values of x. Your table should go from x 0 to x 2 using increments of 兾8, as shown in Figure 5.11. From the table, you can see that the values of y1 and y2 appear to be identical when x ⬇ 3.927 ⬇ 5兾4 and x ⬇ 5.4978 ⬇ 7兾4. These values are the approximate solutions of sin x 冪2 sin x. sin x 冪2 sin x sin x sin x 冪2 2 sin x 冪2 sin x 冪2 2 Write original equation. Add sin x to and subtract 冪2 from each side. Combine like terms. Divide each side by 2. The solutions in the interval 关0, 2兲 are x 5 4 x and 7 . 4 Now try Exercise 35. Figure 5.11 Example 3 Extracting Square Roots Solve 3 tan2 TECHNOLOGY SUPPORT For instructions on how to use the table feature, see Appendix A; for specific keystrokes, go to this textbook’s Online Study Center. x 1 0. Solution Rewrite the equation so that tan x is isolated on one side of the equation. 3 tan2 x 1 tan2 x Add 1 to each side. 1 3 tan x ± Divide each side by 3. 1 Extract square roots. 冪3 6 Because tan x has a period of , first find all solutions in the interval 关0, 兲. These are x 兾6 and x 5兾6. Finally, add multiples of to each of these solutions to get the general form x n 6 and 5 x n 6 − General solution where n is an integer. The graph of y 3 tan2 x 1, shown in Figure 5.12, confirms this result. Now try Exercise 37. Recall that the solutions of an equation correspond to the x-intercepts of the graph of the equation. For instance, the graph in Figure 5.12 has x-intercepts at 兾6, 5兾6, 7兾6, and so on. y = 3 tan2 x − 1 5 2 2 −2 Figure 5.12 333353_0503.qxp 370 1/15/07 2:46 PM Chapter 5 Page 370 Analytic Trigonometry The equations in Examples 1, 2, and 3 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 Example 4. Example 4 Factoring Solve cot x cos2 x 2 cot x. Solution Begin by rewriting the equation so that all terms are collected on one side of the equation. cot x cos 2 x 2 cot x Write original equation. cot x cos 2 x 2 cot x 0 Subtract 2 cot x from each side. cot x共cos2 x 2兲 0 cot x 0 and − 3 x20 cos2 x 2 −3 cos x ± 冪2 The equation cot x 0 has the solution x 兾2 关in the interval 共0, 兲兴. No solution is obtained for cos x ± 冪2 because ± 冪2 are outside the range of the cosine function. Because cot x has a period of , the general form of the solution is obtained by adding multiples of to x 兾2, to get x y = cot x cos2 x − 2 cot x Factor. By setting each of these factors equal to zero, you obtain the following. cos2 3 n 2 General solution where n is an integer. The graph of y cot x cos 2 x 2 cot x (in dot mode), shown in Figure 5.13, confirms this result. Now try Exercise 39. Equations of Quadratic Type Many trigonometric equations are of quadratic type ax2 bx c 0. Here are a few examples. Quadratic in sin x 2 sin2 x sin x 1 0 Quadratic in sec x sec2 x 3 sec x 2 0 2共sin x兲2 sin x 1 0 共sec x兲2 3 共sec x兲 2 0 To solve equations of this type, factor the quadratic or, if factoring is not possible, use the Quadratic Formula. Figure 5.13 Exploration Using the equation in Example 4, explain what would happen if you divided each side of the equation by cot x. Why is this an incorrect method to use when solving an equation? 333353_0503.qxp 1/15/07 2:46 PM Page 371 Section 5.3 371 Solving Trigonometric Equations Example 5 Factoring an Equation of Quadratic Type Find all solutions of 2 sin2 x sin x 1 0 in the interval 关0, 2兲. Algebraic Solution Graphical Solution Treating the equation as a quadratic in sin x and factoring produces the following. Use a graphing utility set in radian mode to graph y 2 sin2 x sin x 1 for 0 ≤ x < 2, as shown in Figure 5.14. Use the zero or root feature or the zoom and trace features to approximate the x-intercepts to be 2 sin2 x sin x 1 0 共2 sin x 1兲共sin x 1兲 0 Write original equation. Factor. Setting each factor equal to zero, you obtain the following solutions in the interval 关0, 2兲. 2 sin x 1 0 sin x x and sin x 1 0 1 2 x ⬇ 1.571 ⬇ 7 11 , x ⬇ 3.665 ⬇ , and x ⬇ 5.760 ⬇ . 2 6 6 These values are the approximate solutions of 2 sin2 x sin x 1 0 in the interval [0, 2兲. sin x 1 7 11 , 6 6 x 3 2 y = 2 sin2 x − sin x − 1 2 0 −2 Now try Exercise 49. Figure 5.14 When working with an equation of quadratic type, be sure that the equation involves a single trigonometric function, as shown in the next example. Example 6 Rewriting with a Single Trigonometric Function Solve 2 sin2 x 3 cos x 3 0. Solution Begin by rewriting the equation so that it has only cosine functions. 2 sin2 x 3 cos x 3 0 Write original equation. 2共1 cos 2 x兲 3 cos x 3 0 2 cos 2 Pythagorean identity Combine like terms and multiply each side by 1. x 3 cos x 1 0 共2 cos x 1兲共cos x 1兲 0 Factor. By setting each factor equal to zero, you can find the solutions in the interval 关0, 2兲 to be x 0, x 兾3, and x 5兾3. Because cos x has a period of 2, the general solution is x 2n, x 2n, 3 x 5 2n 3 − y = 2 sin2 x + 3 cos x − 3 3 2 General solution where n is an integer. The graph of y 2 sin2 x 3 cos x 3, shown in Figure 5.15, confirms this result. Now try Exercise 51. 1 −6 Figure 5.15 333353_0503.qxp 1/15/07 372 2:46 PM Chapter 5 Page 372 Analytic Trigonometry Sometimes you must square each side of an equation to obtain a quadratic. Because this procedure can introduce extraneous solutions, you should check any solutions in the original equation to see whether they are valid or extraneous. Example 7 Squaring and Converting to Quadratic Type Find all solutions of cos x 1 sin x in the interval 关0, 2兲. 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 x 1 sin x x 2 cos x 1 1 Use a graphing utility to confirm the solutions found in Example 7 in two different ways. Do both methods produce the same x-values? Which method do you prefer? Why? Write original equation. cos 2 x 2 cos x 1 sin2 x cos 2 Exploration cos 2 Square each side. x Pythagorean identity 2 cos 2 x 2 cos x 0 Combine like terms. 2 cos x共cos x 1兲 0 Factor. 1. Graph both sides of the equation and find the x-coordinates of the points at which the graphs intersect. Setting each factor equal to zero produces the following. 2 cos x 0 and cos x 1 0 cos x 0 x cos x 1 3 , 2 2 Left side: y cos x 1 Right side: y sin x x 2. Graph the equation y cos x 1 sin x and find the x-intercepts of the graph. Because you squared the original equation, check for extraneous solutions. Check cos cos ? 1 sin 2 2 Substitute 兾2 for x. 011 Solution checks. 3 3 ? 1 sin 2 2 0 1 1 ? cos 1 sin 1 1 0 ✓ Substitute 3兾2 for x. Solution does not check. y = cos x + 1 − sin x Substitute for x. Solution checks. ✓ Of the three possible solutions, x 3兾2 is extraneous. So, in the interval 关0, 2兲, the only solutions are x 兾2 and x . The graph of y cos x 1 sin x, shown in Figure 5.16, confirms this result because the graph has two x-intercepts 共at x 兾2 and x 兲 in the interval 关0, 2兲. Now try Exercise 53. 3 0 −1 Figure 5.16 2 333353_0503.qxp 1/15/07 2:46 PM Page 373 Section 5.3 373 Solving Trigonometric Equations Functions Involving Multiple Angles The next two examples involve trigonometric functions of multiple angles of the forms sin ku and cos ku. To solve equations of these forms, first solve the equation for ku, then divide your result by k. Example 8 Functions of Multiple Angles Solve 2 cos 3t 1 0. Solution 2 cos 3t 1 0 Write original equation. 2 cos 3t 1 cos 3t Add 1 to each side. 1 2 Divide each side by 2. In the interval 关0, 2兲, you know that 3t 兾3 and 3t 5兾3 are the only solutions. So in general, you have 3t 兾3 2n and 3t 5兾3 2n. Dividing this result by 3, you obtain the general solution t 2n 9 3 and t 5 2n 9 3 2 y = 2 cos 3t − 1 2 0 General solution −4 where n is an integer. This solution is confirmed graphically in Figure 5.17. Figure 5.17 Now try Exercise 65. Example 9 Functions of Multiple Angles Solve 3 tan x 3 0. 2 Solution 3 tan x 30 2 Write original equation. 3 tan x 3 2 Subtract 3 from each side. tan x 1 2 Divide each side by 3. In the interval 关0, 兲, you know that x兾2 3兾4 is the only solution. So in general, you have x兾2 3兾4 n. Multiplying this result by 2, you obtain the general solution x 3 2n 2 20 −2 −20 Figure 5.18 x +3 2 2 General solution where n is an integer. This solution is confirmed graphically in Figure 5.18. Now try Exercise 71. y = 3 tan 333353_0503.qxp 1/15/07 374 Chapter 5 2:46 PM Page 374 Analytic Trigonometry Using Inverse Functions Example 10 Using Inverse Functions Find all solutions of sec2 x 2 tan x 4. Solution sec2 x 2 tan x 4 Write original equation. 1 tan2 x 2 tan x 4 0 Pythagorean identity tan2 x 2 tan x 3 0 Combine like terms. 共tan x 3兲共tan x 1兲 0 Factor. Setting each factor equal to zero, you obtain two solutions in the interval 共 兾2, 兾2兲. [Recall that the range of the inverse tangent function is 共 兾2, 兾2兲.] tan x 3 tan x 1 and x arctan共1兲 x arctan 3 y = sec2 x − 2 tan x − 4 4 6 Finally, because tan x has a period of , add multiples of to obtain x arctan 3 n x n General solution and 4 − 2 2 −4 where n is an integer. This solution is confirmed graphically in Figure 5.19. Figure 5.19 Now try Exercise 47. With some trigonometric equations, there is no reasonable way to find the solutions algebraically. In such cases, you can still use a graphing utility to approximate the solutions. Example 11 Approximating Solutions Approximate the solutions of x 2 sin x in the interval 关 , 兴. Solution Use a graphing utility to graph y x 2 sin x in the interval 关 , 兴. Using the zero or root feature or the zoom and trace features, you can see that the solutions are x ⬇ 1.8955, x 0, and x ⬇ 1.8955. (See Figure 5.20.) 3 − − −3 Figure 5.20 3 3 −3 y ⴝ x ⴚ 2 sin x Now try Exercise 85. − −3 333353_0503.qxp 1/15/07 2:46 PM Page 375 Section 5.3 375 Solving Trigonometric Equations Example 12 Surface Area of a Honeycomb θ The surface area of a honeycomb is given by the equation 3 冪3 cos S 6hs s 2 , 2 sin 冢 冣 0 < ≤ 90 where h 2.4 inches, s 0.75 inch, and is the angle indicated in Figure 5.21. h = 2.4 in. a. What value of gives a surface area of 12 square inches? b. What value of gives the minimum surface area? Solution s = 0.75 in. a. Let h 2.4, s 0.75, and S 12. Figure 5.21 3 冪3 cos S 6hs s2 2 sin 冢 冣 冪3 cos 3 12 6共2.4兲共0.75兲 共0.75兲2 2 sin 冢 12 10.8 0.84375 0 0.84375 冢 冢 冪3 cos sin 冪3 cos sin 冣 冣 TECHNOLOGY SUPPORT For instructions on how to use the degree mode and the minimum feature, see Appendix A; for specific keystrokes, go to this textbook’s Online Study Center. 冣 1.2 Using a graphing utility set to degree mode, you can graph the function y 0.84375 冢 冪3 cos x sin x 冣 1.2. Using the zero or root feature or the zoom and trace features, you can determine that ⬇ 49.9 and ⬇ 59.9. (See Figure 5.22.) 0.05 y = 10.8 + 0.84375 90 0 y ⴝ 0.84375 冪3 ⴚ cos x 冸 sin x 冹 ⴚ 1.2 b. From part (a), let h 2.4 and s 0.75 to obtain S 10.8 0.84375 冢 冪3 cos sin 冣. Graph this function using a graphing utility set to degree mode. Use the minimum feature or the zoom and trace features to approximate the minimum point on the graph, which occurs at ⬇ 54.7, as shown in Figure 5.23. Now try Exercise 93. 150 11 −0.02 Figure 5.22 ( 90 0 −0.02 3 − cos x sin x 14 0.05 0 ( Figure 5.23 STUDY TIP By using calculus, it can be shown that the exact minimum value is arccos 冢冪3冣 ⬇ 54.7356. 1 333353_0503.qxp 376 1/15/07 Chapter 5 2:47 PM Page 376 Analytic Trigonometry 5.3 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises. Vocabulary Check Fill in the blanks. 1. The equation 2 cos x 1 0 has the solutions x 5 2n and x 2n, 3 3 which are called _______ solutions. 2. The equation tan2 x 5 tan x 6 0 is an equation of _______ type. 3. A solution of an equation that does not satisfy the original equation is called an _______ solution. In Exercises 1– 6, verify that each x-value is a solution of the equation. 1. 2 cos x 1 0 (a) x 3 2. sec x 2 0 (a) x 3 5 (b) x 3 (b) x 5 3 5 (b) x 12 (b) x 2 3 (b) x 1 2 7 6 5 3 8. cos x 1 9. cos x 2 冪3 冪3 15. cot x 1 2 24. csc x 冪2 25. 2 cos x 1 0 26. 冪2 sin x 1 0 27. 冪3 sec x 2 0 28. cot x 1 0 10. sin x 14. sin x 16. sin x csc2 x40 sin2 x 3 30. 3 cot2 x 1 0 32. cos x共cos x 1兲 0 cos2 x 35. tan x 冪3 0 36. 2 sin x 1 0 37. csc2 x 2 0 38. tan2 x 1 0 tan3 x tan x 40. 2 sin2 x 2 cos x 41. sec2 x sec x 2 42. sec x csc x 2 csc x 43. 2 sin x csc x 0 44. sec x tan x 1 45. cos x sin x tan x 2 46. sin2 x cos x 1 0 47. sec2 x tan x 3 48. 2 cos2 x cos x 1 0 2 冪2 2 In Exercises 13–24, find all solutions of the equation in the interval [0, 2冈. 13. cos x 23. tan x 1 39. 3 12. tan x 冪3 11. tan x 1 22. sec x 2 In Exercises 35–48, find all solutions of the equation in the interval [0, 2冈 algebraically. Use the table feature of a graphing utility to check your answers numerically. In Exercises 7–12, find all solutions of the equation in the interval [0ⴗ, 360ⴗ冈. 7. sin x 2 34. 共3 tan2 x 1兲共tan2 x 3兲 0 6. sec4 x 3 sec2 x 4 0 (a) x 冪2 20. sec x 冪2 33. 7 (b) x 8 2 18. cos x 21. cot x 冪3 31. 4 cos2 x 1 0 5. 2 sin2 x sin x 1 0 (a) x 3 19. csc x 2 29. 3 4. 4 cos2 2x 2 0 (a) x 8 冪3 In Exercises 25–34, solve the equation. 3. 3 tan2 2x 1 0 (a) x 12 17. tan x 1 2 冪3 2 In Exercises 49–56, use a graphing utility to approximate the solutions of the equation in the interval [0, 2冈 by setting the equation equal to 0, graphing the new equation, and using the zero or root feature to approximate the x-intercepts of the graph. 49. 2 sin2 x 3 sin x 1 0 50. 2 sec2 x tan2 x 3 0 51. 4 sin2 x 2 cos x 1 52. csc2 x 3 csc x 4 333353_0503.qxp 1/15/07 2:47 PM Page 377 Section 5.3 53. csc x cot x 1 Solving Trigonometric Equations In Exercises 77–84, use a graphing utility to approximate the solutions of the equation in the interval [0, 2冈. 54. 4 sin x cos x 2 55. cos x cot x 3 1 sin x 77. 2 cos x sin x 0 56. cos x 1 sin x 4 cos x 1 sin x 79. x tan x 1 0 78. 2 sin x cos x 0 80. 2x sin x 2 0 In Exercises 57–60, (a) use a graphing utility to graph each function in the interval [0, 2冈, (b) write an equation whose solutions are the points of intersection of the graphs, and (c) use the intersect feature of the graphing utility to find the points of intersection (to four decimal places). 58. y cos x, y x x2 59. y sin2 x, y ex 4x 60. y cos2 x, y ex x 1 62. sin 63. sin 4x 1 冪3 88. 2 sec2 x tan x 6 0, 69. tan 3x共tan x 1兲 0 70. cos 2x共2 cos x 1兲 0 冪2 x 2 2 72. tan x 1 3 In Exercises 73–76, approximate the x-intercepts of the graph. Use a graphing utility to check your solutions. x 1 2 74. y sin x cos x 3 2 −1 −2 3 4 −1 75. y tan2 冤 2 , 2 冥 冤 2 , 2 冥 冤 2 , 2 冥 66. sec 4x 2 2 68. tan2 3x 3 73. y sin 84. 3 tan2 x 4 tan x 4 0 87. 4 cos2 x 2 sin x 1 0, x 0 2 67. 2 sin2 2x 1 71. cos 83. 12 sin2 x 13 sin x 3 0 86. cos2 x 2 cos x 1 0, 关0, 兴 64. cos 2x 1 65. sin 2x 82. csc2 x 0.5 cot x 5 85. 3 tan2 x 5 tan x 4 0, In Exercises 61–72, solve the multiple-angle equation. x 0 4 81. sec2 x 0.5 tan x 1 In Exercises 85–88, use a graphing utility to approximate the solutions (to three decimal places) of the equation in the given interval. 57. y sin 2x, y x2 2x 61. cos 377 In Exercises 89–94, (a) use a graphing utility to graph the function and approximate the maximum and minimum points (to four decimal places) of the graph in the interval [0, 2 兴, and (b) solve the trigonometric equation and verify that the x-coordinates of the maximum and minimum points of f are among its solutions (the trigonometric equation is found using calculus). Function Trigonometric Equation 89. f 共x兲 sin 2x 2 cos 2x 0 90. f 共x兲 cos 2x 2 sin 2x 0 91. f 共x兲 sin2 x cos x 2 sin x cos x sin x 0 92. f 共x兲 cos2 x sin x 2 sin x cos x cos x 0 93. f 共x兲 sin x cos x cos x sin x 0 94. f 共x兲 2 sin x cos 2x 2 cos x 4 sin x cos x 0 −2 x 3 6 冢 冣 76. y sec4 2 −3 冢 冣 4 3 −4 Fixed Point In Exercises 95 and 96, find the smallest positive fixed point of the function f. [A fixed point of a function f is a real number c such that f 冇c冈 ⴝ c.] x 4 8 95. f 共x兲 tan −3 3 −4 x 4 96. f 共x兲 cos x 333353_0503.qxp 378 1/15/07 2:47 PM Chapter 5 Page 378 Analytic Trigonometry 97. Graphical Reasoning 100. Position of the Sun Cheyenne, Wyoming has a latitude of 41 N. At this latitude, the position of the sun at sunrise can be modeled by Consider the function 1 f 共x兲 cos x D 31 sin and its graph shown in the figure. y 2 1 −π π x 2 t 1.4冣 冢365 where t is the time (in days) and t 1 represents January 1. In this model, D represents the number of degrees north or south of due east at which the sun rises. Use a graphing utility to determine the days on which the sun is more than 20 north of due east at sunrise. 101. Harmonic Motion A weight is oscillating on the end of a spring (see figure). The position of the weight relative to the point of equilibrium is given by −2 (a) What is the domain of the function? (b) Identify any symmetry or asymptotes of the graph. (c) Describe the behavior of the function as x → 0. (d) How many solutions does the equation cos共1兾x兲 0 have in the interval 关1, 1兴? Find the solutions. 1 y 12 共cos 8t 3 sin 8t兲 where y is the displacement (in meters) and t is the time (in seconds). Find the times at which the weight is at the point of equilibrium 共y 0兲 for 0 ≤ t ≤ 1. (e) Does the equation cos共1兾x兲 0 have a greatest solution? If so, approximate the solution. If not, explain. 98. Graphical Reasoning f 共x兲 Consider the function Equilibrium sin x x y and its graph shown in the figure. y 3 2 −π −1 −2 −3 π x (a) What is the domain of the function? (b) Identify any symmetry or asymptotes of the graph. (c) Describe the behavior of the function as x → 0. (d) How many solutions does the equation 共sin x)/x 0 have in the interval 关8, 8兴? Find the solutions. 99. Sales The monthly sales S (in thousands of units) of lawn mowers are approximated by S 74.50 43.75 cos 102. Damped Harmonic Motion The displacement from equilibrium of a weight oscillating on the end of a spring is given by y 1.56e0.22t 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 ≤ 10. Find the time beyond which the displacement does not exceed 1 foot from equilibrium. 103. Projectile Motion A batted baseball leaves the bat at an angle of with the horizontal and an initial velocity of v0 100 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 1 2 v sin 2. 32 0 t 6 where t is the time (in months), with t 1 corresponding to January. Determine the months during which sales exceed 100,000 units. θ r = 300 ft Not drawn to scale 333353_0503.qxp 1/15/07 2:47 PM Page 379 Section 5.3 g共x兲 0.45x 2 5.52x 13.70. A 2x cos x, 0 ≤ x ≤ . 2 Use a graphing utility to graph f and g in the same viewing window. Describe the result. y (c) Use the Quadratic Formula to find the zeros of g. Compare the zero in the interval 关0, 6兴 with the result of part (a). y = cos x π 2 x π 2 x Synthesis True or False? In Exercises 107–109, determine whether the statement is true or false. Justify your answer. −1 (a) Use a graphing utility to graph the area function, and approximate the area of the largest inscribed rectangle. (b) Determine the values of x for which A ≥ 1. 105. Data Analysis: Unemployment Rate The table shows the unemployment rates r in the United States for selected years from 1990 through 2004. The time t is measured in years, with t 0 corresponding to 1990. (Source: U.S. Bureau of Labor Statistics) Time, t Rate, r Time, t Rate, r 0 2 4 6 5.6 7.5 6.1 5.4 8 10 12 14 4.5 4.0 5.8 5.5 (a) Use a graphing utility to create a scatter plot of the data. (b) Which of the following models best represents the data? Explain your reasoning. 107. All trigonometric equations have either an infinite number of solutions or no solution. 108. The solutions of any trigonometric equation can always be found from its solutions in the interval 关0, 2兲. 109. If you correctly solve a trigonometric equation down to the statement sin x 3.4, then you can finish solving the equation by using an inverse trigonometric function. 110. Writing Describe the difference between verifying an identity and solving an equation. Skills Review In Exercises 111–114, convert the angle measure from degrees to radians. Round your answer to three decimal places. 111. 124 112. 486 113. 0.41 114. 210.55 In Exercises 115 and 116, solve for x. 115. 116. (1) r 1.24 sin共0.47t 0.40兲 5.45 14 (2) r 1.24 sin共0.47t 0.01兲 5.45 x (4) r 896 sin共0.57t 2.05兲 6.48 (c) What term in the model gives the average unemployment rate? What is the rate? (d) Economists study the lengths of business cycles, such as unemployment rates. Based on this short span of time, use the model to determine the length of this cycle. (e) Use the model to estimate the next time the unemployment rate will be 5% or less. Consider the function f 共x兲 3 sin共0.6x 2兲. (a) Approximate the zero of the function in the interval 关0, 6兴. 10 30° (3) r sin共0.10t 5.61兲 4.80 106. Quadratic Approximation 379 (b) A quadratic approximation agreeing with f at x 5 is 104. Area The area of a rectangle inscribed in one arc of the graph of y cos x (see figure) is given by − Solving Trigonometric Equations x 70° 117. Distance From the 100-foot roof of a condominium on the coast, a tourist sights a cruise ship. The angle of depression is 2.5. How far is the ship from the shoreline? 118. Make a Decision To work an extended application analyzing the normal daily high temperatures in Phoenix and in Seattle, visit this textbook’s Online Study Center. (Data Source: NOAA) 333353_0504.qxp 380 1/15/07 2:48 PM Chapter 5 Page 380 Analytic Trigonometry 5.4 Sum and Difference Formulas What you should learn Using Sum and Difference Formulas In this section and the following section, you will study the uses of several trigonometric identities and formulas. Sum and Difference Formulas (See the proofs on page 404.) sin共u v兲 sin u cos v cos u sin v tan共u v兲 tan u tan v 1 tan u tan v tan共u v兲 tan u tan v 1 tan u tan v sin共u v兲 sin u cos v cos u sin v cos共u v兲 cos u cos v sin u sin v cos共u v兲 cos u cos v sin u sin v 䊏 Use sum and difference formulas to evaluate trigonometric functions, verify trigonometric identities, and solve trigonometric equations. Why you should learn it You can use sum and difference formulas to rewrite trigonometric expressions. For instance, Exercise 79 on page 385 shows how to use sum and difference formulas to rewrite a trigonometric expression in a form that helps you find the equation of a standing wave. Exploration Use a graphing utility to graph y1 cos共x 2兲 and y2 cos x cos 2 in the same viewing window. What can you conclude about the graphs? Is it true that cos共x 2兲 cos x cos 2? Use a graphing utility to graph y1 sin共x 4兲 and y2 sin x sin 4 in the same viewing window. What can you conclude about the graphs? Is it true that sin共x 4兲 sin x sin 4? Examples 1 and 2 show how sum and difference formulas can be used to find exact values of trigonometric functions involving sums or differences of special angles. Richard Megna/Fundamental Photographs Example 1 Evaluating a Trigonometric Function Find the exact value of cos 75. Solution To find the exact value of cos 75, use the fact that 75 30 45. Consequently, the formula for cos共u v兲 yields cos 75 cos共30 45兲 cos 30 cos 45 sin 30 sin 45 冪3 冪2 1 冪2 2 冢 2 冣 2冢 2 冣 冪6 冪2 4 . Try checking this result on your calculator. You will find that cos 75 ⬇ 0.259. Now try Exercise 1. Prerequisite Skills To review sines, cosines, and tangents of special angles, see Section 4.3. 333353_0504.qxp 1/15/07 2:48 PM Page 381 Section 5.4 381 Sum and Difference Formulas Example 2 Evaluating a Trigonometric Function Find the exact value of sin . 12 Solution Using the fact that 兾12 兾3 兾4 together with the formula for sin共u v兲, you obtain sin sin sin cos cos sin 12 3 4 3 4 3 4 冢 y 冣 冪3 冪2 1 冪2 2 冢 2 冣 2冢 2 冣 冪6 冪2 4 . 5 u 4 x Now try Exercise 3. 52 − 42 = 3 Example 3 Evaluating a Trigonometric Expression Find the exact value of sin共u v兲 given 4 sin u , where 0 < u < 5 2 12 and cos v , where < v < . 13 2 Figure 5.24 y Solution Because sin u 4兾5 and u is in Quadrant I, cos u 3兾5, as shown in Figure 5.24. Because cos v 12兾13 and v is in Quadrant II, sin v 5兾13, as shown in Figure 5.25. You can find sin共u v兲 as follows. 13 2 − 12 2 = 5 13 v 12 sin共u v兲 sin u cos v cos u sin v x 12 15 33 冢45冣冢 13 冣 冢35冣冢135 冣 48 65 65 65 Now try Exercise 35. Figure 5.25 Example 4 An Application of a Sum Formula Write cos共arctan 1 arccos x兲 as an algebraic expression. Solution This expression fits the formula for cos共u v兲. Angles u arctan 1 and v arccos x are shown in Figure 5.26. 2 1 u 1 cos共u v兲 cos共arctan 1兲 cos共arccos x兲 sin共arctan 1兲 sin共arccos x兲 1 冪2 x 1 冪2 冪1 x 2 Now try Exercise 43. x 冪1 x2 . 冪2 1 v x Figure 5.26 1 − x2 333353_0504.qxp 1/15/07 382 2:48 PM Chapter 5 Page 382 Analytic Trigonometry Example 5 Proving a Cofunction Identity Prove the cofunction identity cos 冢 2 x冣 sin x. Solution Using the formula for cos共u v兲, you have cos 冢 2 x冣 cos 2 cos x sin 2 sin x 共0兲共cos x兲 共1兲共sin x兲 sin x. Now try Exercise 63. Sum and difference formulas can be used to derive reduction formulas involving expressions such as 冢 sin n 2 冣 and 冢 cos n , where n is an integer. 2 冣 Example 6 Deriving Reduction Formulas Simplify each expression. 冢 a. cos 3 2 冣 b. tan共 3兲 Solution a. Using the formula for cos共u v兲, you have 冢 cos 3 3 3 cos cos sin sin 2 2 2 冣 共cos 兲共0兲 共sin 兲共1兲 sin . b. Using the formula for tan共u v兲, you have tan共 3兲 tan tan 3 1 tan tan 3 tan 0 1 共tan 兲共0兲 tan . Note that the period of tan is , so the period of tan共 3兲 is the same as the period of tan . Now try Exercise 67. 333353_0504.qxp 1/15/07 2:48 PM Page 383 Section 5.4 383 Sum and Difference Formulas Example 7 Solving a Trigonometric Equation 冢 Find all solutions of sin x sin x 1 in the interval 关0, 2兲. 4 4 冣 冢 冣 Algebraic Solution Graphical Solution Using sum and difference formulas, rewrite the equation as Use a graphing utility set in radian mode to sin x 1, graph y sin x 4 4 as shown in Figure 5.27. Use the zero or root feature or the zoom and trace features to approximate the x-intercepts in the interval 关0, 2兲 to be sin x cos 冢 cos x sin sin x cos cos x sin 1 4 4 4 4 2 sin x cos 2共sin x兲 1 4 冪2 冢 2 冣 1 sin x sin x x ⬇ 3.927 ⬇ 冣 冢 冣 5 7 . and x ⬇ 5.498 ⬇ 4 4 1 冪2 冪2 2 ( π4 ( + sin (x − π4 ( + 1 y = sin x + . 3 So, the only solutions in the interval 关0, 2兲 are x 5 4 and x 7 . 4 2 0 −1 Now try Exercise 71. Figure 5.27 The next example was taken from calculus. It is used to derive the formula for the derivative of the cosine function. Example 8 An Application from Calculus cos共x h兲 cos x cos h 1 sin h 共cos x兲 共sin x兲 , h 0. Verify h h h 冢 冣 冢 冣 Solution Using the formula for cos共u v兲, you have cos共x h兲 cos x cos x cos h sin x sin h cos x h h cos x共cos h 1兲 sin x sin h h 共cos x兲 冢 cos h 1 sin h 共sin x兲 . h h Now try Exercise 93. 冣 冢 冣 TECHNOLOGY SUPPORT For instructions on how to use the zero or root feature and the zoom and trace features, see Appendix A; for specific keystrokes, go to this textbook’s Online Study Center. 333353_0504.qxp 1/15/07 384 2:49 PM Chapter 5 Page 384 Analytic Trigonometry 5.4 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises. Vocabulary Check Fill in the blank to complete the trigonometric formula. 1. sin共u v兲 _______ 2. cos共u v兲 _______ 3. tan共u v兲 _______ 4. sin共u v兲 _______ 5. cos共u v兲 _______ 6. tan共u v兲 _______ In Exercises 1–6, find the exact value of each expression. 1. (a) cos共240 0兲 (b) cos 240 cos 0 2. (a) sin共405 120兲 (b) sin 405 sin 120 3. (a) cos 4 3 (b) cos cos 4 3 (b) sin 5 2 sin 3 6 冢 4. (a) sin 2 冢3 冣 5 6 冣 5. (a) sin共315 60兲 (b) sin 315 sin 60 7 6. (a) sin 6 3 (b) sin 冢 冣 7 sin 6 3 In Exercises 7– 22, find the exact values of the sine, cosine, and tangent of the angle. 7. 105 60 45 8. 165 135 30 9. 195 225 30 10. 255 300 45 11. 11 3 12 4 6 13. 12 6 4 12. 17 7 12 6 4 14. 19 2 9 12 3 4 15. 75 16. 15 17. 225 18. 165 13 19. 12 5 20. 12 7 21. 12 13 22. 12 28. cos 0.96 cos 0.42 sin 0.96 sin 0.42 29. cos cos sin sin 9 7 9 7 30. sin 4 4 cos cos sin 9 8 9 8 Numerical, Graphical, and Algebraic Analysis In Exercises 31–34, use a graphing utility to complete the table and graph the two functions in the same viewing window. Use both the table and the graph as evidence that y1 y2 . Then verify the identity algebraically. 0.2 x 0.4 0.6 0.8 1.0 1.2 1.4 y1 y2 31. y1 sin 冢 6 x冣, 32. y1 cos 5 冢4 y2 1 共cos x 冪3 sin x兲 2 冣 x , y2 33. y1 cos共x 兲 cos共x 兲, 34. y1 sin共x 兲 sin共x 兲, 冪2 2 共cos x sin x兲 y2 cos 2 x y2 sin2 x In Exercises 35–38, find the exact value of the trigonometric 5 function given that sin u 13 and cos v 35. (Both u and v are in Quadrant II.) 35. sin共u v兲 36. cos共v u兲 In Exercises 23 – 30, write the expression as the sine, cosine, or tangent of an angle. 37. tan共u v兲 38. sin共u v兲 23. cos 60 cos 20 sin 60 sin 20 In Exercises 39–42, find the exact value of the trigonometric 8 function given that sin u 17 and cos v 45. (Both u and v are in Quadrant III.) 24. sin 110 cos 80 cos 110 sin 80 25. tan 325 tan 86 1 tan 325 tan 86 26. tan 154 tan 49 1 tan 154 tan 49 27. sin 3.5 cos 1.2 cos 3.5 sin 1.2 39. cos共u v兲 40. tan共u v兲 41. sin共v u兲 42. cos共u v兲 333353_0504.qxp 1/15/07 2:49 PM Page 385 Section 5.4 In Exercises 43–46, write the trigonometric expression as an algebraic expression. 43. sin共arcsin x arccos x兲 44. cos共arccos x arcsin x兲 45. sin共arctan 2x arccos x兲 46. cos共arcsin x arctan 2x兲 In Exercises 47– 54, find the value of the expression without using a calculator. 47. sin共sin1 1 cos1 1兲 In Exercises 71–74, find the solution(s) of the equation in the interval [0, 2冈. Use a graphing utility to verify your results. 冢 sin x 1 3 3 冢 cos x 1 6 6 71. sin x 72. cos x 冢 冣 1 52. cos冤 cos 冢 冣 sin 1冥 2 1 53. tan冢sin 0 sin 2冣 冪2 54. tan冢cos sin 0冣 2 冢 冣 3 tan共 x兲 0 2 冣 冢 cos x 1 4 4 冢 3 cos x 0 2 2 75. cos x 1 76. sin x 1 1 冣 In Exercises 75–78, use a graphing utility to approximate the solutions of the equation in the interval [0, 2冈. 1 1 51. sin sin1 cos1 2 2 1 冣 74. 2 sin x 50. cos关cos1共1兲 cos1 1兴 冢 73. tan共x 兲 2 sin共x 兲 0 49. sin关sin1 1 cos1共1兲兴 1 冣 冢 48. cos关sin1共1兲 cos1 0兴 冣 冢 冣 冣 冢 冣 冢 77. tan共x 兲 cos x 1 冢 0 2 冣 78. tan共 x兲 2 cos x In Exercises 55– 58, evaluate the trigonometric function without using a calculator. 55. sin sin1共1兲 2 56. sin关cos 共1兲 兴 57. cos共 sin1 1兲 58. cos关 cos1共1兲兴 冤 冥 1 In Exercises 59– 62, use right triangles to evaluate the expression. 冢 60. cos冢sin 61. sin冢tan 62. tan冢sin 59. sin cos1 3 5 sin1 5 13 冣 1 12 8 cos1 13 17 1 3 3 sin1 4 5 1 y1 A cos 2 冣 冢 t x 2 A cos 2 冢T 冣 t 2 x 2 t cos . T y1 + y2 y2 t=0 冣 y1 64. sin共3 x兲 sin x 1 tan 4 1 tan 冣 67. sin共x y兲 sin共x y兲 2 sin x cos y 68. cos共x y兲 cos共x y兲 2 cos x cos y 69. cos共x y兲 cos共x y兲 cos2 x sin2 y 70. sin共x y兲 sin共x y兲 sin2 x sin2 y y1 + y2 y2 t = 18 T 65. tan共x 兲 tan共 x兲 2 tan x 66. tan 冢T 冣 and y y1 y2 2A cos y1 冢2 x冣 cos x 冣 show that In Exercises 63– 70, verify the identity. 63. sin 3 0 2 79. Standing Waves The equation of a standing wave is obtained by adding the displacements of two waves traveling in opposite directions (see figure). Assume that each of the waves has amplitude A, period T, and wavelength . If the models for these waves are 冣 4 5 cos1 5 13 385 Sum and Difference Formulas y1 t = 28 T y1 + y2 y2 x 333353_0504.qxp 386 1/15/07 2:49 PM Chapter 5 Page 386 Analytic Trigonometry 80. Harmonic Motion A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by y 93. Verify the following identity used in calculus. sin共x h兲 sin x cos x sin h sin x共1 cos h兲 h h h 94. Exploration Let x 兾3 in the identity in Exercise 93 and define the functions f and g as follows. 1 1 sin 2t cos 2t 3 4 sin共兾3 h兲 sin共兾3兲 h where y is the distance from equilibrium (in feet) and t is the time (in seconds). f 共h兲 (a) Use a graphing utility to graph the model. g共h兲 cos (b) Use the identity a sin B b cos 冪a2 b2 sin共B C兲 where C arctan共b兾a兲, a > 0, to write the model in the form y 冪a2 b2 sin共Bt C兲. Use a graphing utility to verify your result. (c) Find the amplitude of the oscillations of the weight. sin h 1 cos h sin 3 h 3 h 冢 冣 冢 冣 (a) What are the domains of the functions f and g? (b) Use a graphing utility to complete the table. 0.01 h 0.02 0.05 0.1 0.2 0.5 f 共h兲 (d) Find the frequency of the oscillations of the weight. g共h兲 Synthesis (c) Use a graphing utility to graph the functions f and g. True or False? In Exercises 81 and 82, determine whether the statement is true or false. Justify your answer. 81. cos共u ± v兲 cos u ± cos v 95. Conjecture Three squares of side s are placed side by side (see figure). Make a conjecture about the relationship between the sum u v and w. Prove your conjecture by using the identity for the tangent of the sum of two angles. 11 82. sin x cos x 2 冢 (d) Use the table and graph to make a conjecture about the values of the functions f and g as h → 0. 冣 In Exercises 83– 86, verify the identity. 83. cos共n 兲 共1兲n cos , n is an integer. 84. sin共n 兲 共1兲n sin , n is an integer. s 85. a sin B b cos B 冪a 2 b2 sin共B C兲, where C arctan共b兾a兲 and a > 0. 86. a sin B b cos B C arctan共a兾b兲 and b > 0. 冪a 2 b2 cos共B C兲, where (a) 冪a2 1 b2 sin冇B 1 C 冈 (b) 冪a2 1 b2 cos冇B C 冈 87. sin cos 88. 3 sin 2 4 cos 2 89. 12 sin 3 5 cos 3 90. sin 2 cos 2 In Exercises 91 and 92, use the formulas given in Exercises 85 and 86 to write the trigonometric expression in the form a sin B 1 b cos B. 冢 2 冣 v s w s s 96. (a) Write a sum formula for sin共u v w兲. In Exercises 87– 90, use the formulas given in Exercises 85 and 86 to write the expression in the following forms. Use a graphing utility to verify your results. 91. 2 sin u 冢 92. 5 cos 4 冣 (b) Write a sum formula for tan共u v w兲. Skills Review In Exercises 97–100, find the x- and y-intercepts of the graph of the equation. Use a graphing utility to verify your results. 97. y 12共x 10兲 14 ⱍ ⱍ 99. y 2x 9 5 98. y x2 3x 40 100. y 2x冪x 7 In Exercises 101–104, evaluate the expression without using a calculator. 冢 23冣 101. arccos 103. sin1 1 冪 102. arctan共 冪3兲 104. tan1 0 333353_0505.qxp 1/15/07 2:50 PM Page 387 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 387 5.5 Multiple-Angle and Product-to-Sum Formulas What you should learn Multiple-Angle Formulas In this section, you will study four additional categories of trigonometric identities. 1. The first category involves functions of multiple angles such as sin ku and cos ku. 2. The second category involves squares of trigonometric functions such as sin2 u. 3. The third category involves functions of half-angles such as sin共u兾2兲. 4. The fourth category involves products of trigonometric functions such as sin u cos v. You should learn the double-angle formulas below because they are used most often. Double-Angle Formulas (See the proofs on page 405.) 䊏 䊏 䊏 䊏 Use multiple-angle formulas to rewrite and evaluate trigonometric functions. Use power-reducing formulas to rewrite and evaluate trigonometric functions. Use half-angle formulas to rewrite and evaluate trigonometric functions. Use product-to-sum and sum-to-product formulas to rewrite and evaluate trigonometric functions. Why you should learn it You can use a variety of trigonometric formulas to rewrite trigonometric functions in more convenient forms.For instance, Exercise 130 on page 398 shows you how to use a halfangle formula to determine the apex angle of a sound wave cone caused by the speed of an airplane. cos 2u cos2 u sin2 u sin 2u 2 sin u cos u 2 cos2 u 1 2 tan u tan 2u 1 tan2 u 1 2 sin2 u Example 1 Solving a Multiple-Angle Equation Solve 2 cos x sin 2x 0. NASA-Liaison/Getty Images Solution Begin by rewriting the equation so that it involves functions of x (rather than 2x). Then factor and solve as usual. 2 cos x sin 2x 0 Write original equation. 2 cos x 2 sin x cos x 0 Double-angle formula 2 cos x共1 sin x兲 0 cos x 0 x Factor. 1 sin x 0 3 , 2 2 x 3 2 Set factors equal to zero. Solutions in 关0, 2兲 So, the general solution is x 2n 2 and x 3 2n 2 General solution where n is an integer. Try verifying this solution graphically. Now try Exercise 3. 333353_0505.qxp 388 1/15/07 2:50 PM Chapter 5 Page 388 Analytic Trigonometry Example 2 Using Double-Angle Formulas to Analyze Graphs Analyze the graph of y 4 cos2 x 2 in the interval 关0, 2兴. Solution Using a double-angle formula, you can rewrite the original function as y 4 cos2 x 2 2共2 cos2 x 1兲 2 cos 2x. Using the techniques discussed in Section 4.5, you can recognize that the graph of this function has an amplitude of 2 and a period of . The key points in the interval 关0, 兴 are as follows. Maximum 共0, 2兲 Intercept ,0 4 冢 冣 Minimum , 2 2 冢 Intercept 3 ,0 4 冣 冢 冣 Maximum 共, 2兲 Two cycles of the graph are shown in Figure 5.28. y = 4 cos2 x − 2 3 2 0 −3 Figure 5.28 Now try Exercise 7. Example 3 Evaluating Functions Involving Double Angles Use the following to find sin 2, cos 2, and tan 2. cos 5 , 13 3 < < 2 2 Solution In Figure 5.29, you can see that sin y兾r 12兾13. Consequently, using each of the double-angle formulas, you can write the double angles as follows. 冢 冣冢 冣 sin 2 2 sin cos 2 cos 2 2 cos2 1 2 12 13 5 120 13 169 冢169冣 1 169 25 119 2 tan 2共12兾5兲 120 tan 2 1 tan2 1 共12兾5兲2 119 Now try Exercise 13. and θ −4 x −2 2 4 −2 −4 −6 13 −8 −10 The double-angle formulas are not restricted to the angles 2 and . Other double combinations, such as 4 and 2 or 6 and 3, are also valid. Here are two examples. sin 4 2 sin 2 cos 2 y cos 6 cos2 3 sin2 3 By using double-angle formulas together with the sum formulas derived in the preceding section, you can form other multiple-angle formulas. −12 Figure 5.29 (5, −12) 6 333353_0505.qxp 1/15/07 2:50 PM Page 389 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 389 Example 4 Deriving a Triple-Angle Formula sin 3x sin共2x x兲 Rewrite as a sum. sin 2x cos x cos 2x sin x Sum formula 2 sin x cos x cos x 共1 2 sin2 x兲 sin x Double-angle formula 2 sin x Multiply. cos2 x sin x 2 sin3 x 2 sin x共1 sin2 x兲 sin x 2 sin3 x Pythagorean identity 2 sin x 2 Multiply. sin3 x sin x 2 sin3 x 3 sin x 4 sin3 x Simplify. Now try Exercise 19. Power-Reducing Formulas The double-angle formulas can be used to obtain the following power-reducing formulas. Power-Reducing Formulas sin2 u 1 cos 2u 2 (See the proofs on page 405.) cos2 u 1 cos 2u 2 tan2 u 1 cos 2u 1 cos 2u Example 5 Reducing a Power Rewrite sin4 x as a sum of first powers of the cosines of multiple angles. Solution sin4 x 共sin2 x兲2 冢 1 cos 2x 2 Property of exponents 冣 2 Power-reducing formula 1 共1 2 cos 2x cos2 2x兲 4 Expand binomial. 1 1 cos 4x 1 2 cos 2x 4 2 1 1 1 1 cos 2x cos 4x 4 2 8 8 Distributive Property 3 1 1 cos 2x cos 4x 8 2 8 Simplify. 冢 1 共3 4 cos 2x cos 4x兲 8 Now try Exercise 23. 冣 Power-reducing formula Factor. STUDY TIP Power-reducing formulas are often used in calculus. Example 5 shows a typical power reduction that is used in calculus. Note the repeated use of power-reducing formulas. 333353_0505.qxp 390 1/15/07 2:50 PM Chapter 5 Page 390 Analytic Trigonometry Half-Angle Formulas You can derive some useful alternative forms of the power-reducing formulas by replacing u with u兾2. The results are called half-angle formulas. Half-Angle Formulas 冪1 2cos u sin u ± 2 tan u 1 cos u sin u 2 sin u 1 cos u The signs of sin cos u ± 2 冪1 2cos u u u u and cos depend on the quadrant in which lies. 2 2 2 Example 6 Using a Half-Angle Formula Find the exact value of sin 105. STUDY TIP y 105° 210° x To find the exact value of a trigonometric function with an angle in DM S form using a half-angle formula, first convert the angle measure to decimal degree form. Then multiply the angle measure by 2. Figure 5.30 Solution Begin by noting that 105 is half of 210. Then, using the half-angle formula for sin共u兾2兲 and the fact that 105 lies in Quadrant II (see Figure 5.30), you have 冪 1 共cos 30兲 冪 2 1 共 3兾2兲 冪2 冪 2 2 sin 105 1 cos 210 2 冪 冪3 . The positive square root is chosen because sin is positive in Quadrant II. Now try Exercise 39. TECHNOLOGY TIP Use your calculator to verify the result obtained in Example 6. That is, evaluate sin 105 and 共冪2 冪3 兲兾2. You will notice that both expressions yield the same result. 333353_0505.qxp 1/15/07 2:50 PM Page 391 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 391 Example 7 Solving a Trigonometric Equation Find all solutions of 2 sin2 x 2 cos2 x in the interval 关0, 2兲. 2 Algebraic Solution Graphical Solution 2 sin2 x 2 cos 2 冢冪 2 sin2 x 2 ± 2 sin2 x 2 x 2 1 cos x 2 冢 Write original equation. 2 冣 1 cos x 2 冣 2 sin2 x 1 cos x 2 共1 cos 2 cos 2 x兲 1 cos x x cos x 0 cos x共cos x 1兲 0 Half-angle formula Simplify. Simplify. Pythagorean identity Use a graphing utility set in radian mode to graph y 2 sin2 x 2 cos2 共x兾2兲, as shown in Figure 5.31. Use the zero or root feature or the zoom and trace features to approximate the x-intercepts in the interval 关0, 2兲 to be x 0, x ⬇ 1.5708 ⬇ 3 , and x ⬇ 4.7124 ⬇ . 2 2 These values are the approximate solutions of x 2 sin2 x 2 cos2 in the interval 关0, 2兲. 2 Simplify. y = 2 − sin2 x − 2 cos2 Factor. 3 By setting the factors cos x and 共cos x 1兲 equal to zero, you find that the solutions in the interval 关0, 2兲 are x 3 , x , and x 0. 2 2 − 2 2 −1 Now try Exercise 57. Figure 5.31 Product-to-Sum Formulas Each of the following product-to-sum formulas is easily verified using the sum and difference formulas discussed in the preceding section. Product-to-Sum Formulas 1 sin u sin v 关cos共u v兲 cos共u v兲兴 2 1 cos u cos v 关cos共u v兲 cos共u v兲兴 2 1 sin u cos v 关sin共u v兲 sin共u v兲兴 2 1 cos u sin v 关sin共u v兲 sin共u v兲兴 2 Product-to-sum formulas are used in calculus to evaluate integrals involving the products of sines and cosines of two different angles. x 2 333353_0505.qxp 392 1/15/07 2:50 PM Chapter 5 Page 392 Analytic Trigonometry Example 8 Writing Products as Sums TECHNOLOGY TIP You can use a graphing utility to verify the solution in Example 8. Graph y1 cos 5x sin 4x and y2 12 sin 9x 12 sin x in the same viewing window. Notice that the graphs coincide. So, you can conclude that the two expressions are equivalent. Rewrite the product as a sum or difference. cos 5x sin 4x Solution 1 cos 5x sin 4x 关sin共5x 4x兲 sin共5x 4x兲兴 2 1 1 sin 9x sin x 2 2 Now try Exercise 63. Occasionally, it is useful to reverse the procedure and write a sum of trigonometric functions as a product. This can be accomplished with the following sum-to-product formulas. Sum-to-Product Formulas sin u sin v 2 sin 冢 sin u sin v 2 cos uv uv cos 2 2 冣 冢 冣 uv uv sin 2 2 冣 冢 cos u cos v 2 cos (See the proof on page 406.) 冣 冢 冢 uv uv cos 2 2 cos u cos v 2 sin 冣 冢 冢 冣 uv uv sin 2 2 冣 冢 冣 Example 9 Using a Sum-to-Product Formula Find the exact value of cos 195 cos 105. Solution Using the appropriate sum-to-product formula, you obtain cos 195 cos 105 2 cos 冢 195 105 195 105 cos 2 2 冣 冢 2 cos 150 cos 45 冢 2 冪3 冪6 2 冪2 2 冣冢 2 冣 . Now try Exercise 81. 冣 333353_0505.qxp 1/15/07 2:51 PM Page 393 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 393 Example 10 Solving a Trigonometric Equation Find all solutions of sin 5x sin 3x 0 in the interval 关0, 2兲. Algebraic Solution Graphical Solution sin 5x sin 3x 0 2 sin 冢 5x 3x 5x 3x cos 0 2 2 冣 冢 冣 2 sin 4x cos x 0 Write original equation. Sum-to-product formula Simplify. By setting the factor sin 4x equal to zero, you can find that the solutions in the interval 关0, 2兲 are 3 5 3 7 x 0, , , , , , , . 4 2 4 4 2 4 Moreover, the equation cos x 0 yields no additional solutions. Note that the general solution is x Use a graphing utility set in radian mode to graph y sin 5x sin 3x, as shown in Figure 5.32. Use the zero or root feature or the zoom and trace features to approximate the x-intercepts in the interval 关0, 2兲 to be x ⬇ 0, x ⬇ 0.7854 ⬇ , x ⬇ 1.5708 ⬇ , 4 2 3 5 , x ⬇ 3.1416 ⬇ , x ⬇ 3.9270 ⬇ , 4 4 3 7 x ⬇ 4.7124 ⬇ , x ⬇ 5.4978 ⬇ . 2 4 x ⬇ 2.3562 ⬇ These values are the approximate solutions of sin 5x sin 3x 0 in the interval 关0, 2兲. n 4 3 y = sin 5x + sin 3x where n is an integer. − 4 2 −3 Now try Exercise 85. Figure 5.32 Example 11 Verifying a Trigonometric Identity Verify the identity sin t sin 3t tan 2t. cos t cos 3t Algebraic Solution Numerical Solution Using appropriate sum-to-product formulas, you have Use the table feature of a graphing utility set in radian mode to create a table that shows the values of y1 共sin x sin 3x兲兾共cos x cos 3x兲 and y2 tan 2x for different values of x, as shown in Figure 5.33. In the table, you can see that the values appear to be identical, so 共sin x sin 3x兲兾共cos x cos 3x兲 tan 2x appears to be an identity. sin t sin 3t 2 sin 2t cos共t兲 cos t cos 3t 2 cos 2t cos共t兲 sin 2t cos 2t tan 2t. Now try Exercise 105. Figure 5.33 333353_0505.qxp 394 1/15/07 2:51 PM Chapter 5 Page 394 Analytic Trigonometry 5.5 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises. Vocabulary Check Fill in the blank to complete the trigonometric formula. 1. sin 2u _______ 2. cos2 u _______ 3. _______ 1 2 sin2 u 4. _______ 5. tan 2u _______ 6. cos u cos v _______ 7. _______ 1 cos 2u 2 sin u 1 cos u 8. _______ ± 9. sin u cos v _______ 冪1 2cos u 10. sin u sin v _______ In Exercises 1 and 2, use the figure to find the exact value of each trigonometric function. In Exercises 13–18, find the exact values of sin 2u, cos 2u, and tan 2u using the double-angle formulas. 3 13. sin u 5, 0 < u < 兾2 2 14. cos u 7, 兾2 < u < 1. 1 15. tan u 2, < u < 3兾2 3 16. cot u 6, 3兾2 < u < 2 θ 5 17. sec u 2, 兾2 < u < 4 (a) sin (b) cos (c) cos 2 (d) sin 2 (e) tan 2 (f) sec 2 (g) csc 2 (h) cot 2 18. csc u 3, 兾2 < u < In Exercises 19–22, use a double-angle formula to rewrite the expression. Use a graphing utility to graph both expressions to verify that both forms are the same. 19. 8 sin x cos x 2. 20. 4 sin x cos x 1 θ 5 21. 6 12 sin2 x 22. 共cos x sin x兲共cos x sin x兲 12 (a) sin (b) cos (c) sin 2 (d) cos 2 (e) tan 2 (f) cot 2 (g) sec 2 (h) csc 2 In Exercises 3–12, use a graphing utility to approximate the solutions of the equation in the interval [0, 2冈. If possible, find the exact solutions algebraically. 3. sin 2x sin x 0 4. sin 2x cos x 0 5. 4 sin x cos x 1 6. sin 2x sin x cos x 7. cos 2x cos x 0 8. tan 2x cot x 0 9. sin 4x 2 sin 2x 11. cos 2x sin x 0 10. 共sin 2x cos 2x兲2 1 12. tan 2x 2 cos x 0 In Exercises 23–36, rewrite the expression in terms of the first power of the cosine. Use a graphing utility to graph both expressions to verify that both forms are the same. 23. cos4 x 24. sin4 x 25. sin2 x cos2 x 27. sin2 x cos4 x 29. sin2 2x 31. cos2 x 2 33. sin2 2x cos2 2x 35. sin4 x 2 26. cos 6 x 28. sin4 x cos2 x 30. cos2 2x x 2 x x 34. sin2 cos2 2 2 32. sin2 36. cos4 x 2 333353_0505.qxp 1/15/07 2:51 PM Page 395 Section 5.5 Multiple-Angle and Product-to-Sum Formulas In Exercises 37 and 3 8, use the figure to find the exact value of each trigonometric function. 8 49. tan u 5, 37. 51. csc u 53, < u < 3兾2 3兾2 < u < 2 50. cot u 7, < u < 3兾2 52. sec u 72, 兾2 < u < 8 θ In Exercises 53–56, use the half-angle formulas to simplify the expression. 15 (a) cos 2 (b) sin 2 53. (c) tan 2 6x 冪1 cos 2 (d) sec 2 54. (e) csc 2 (f) cot 2 4x 冪1 cos 2 55. 8x 冪11 cos cos 8x 56. 冪1 cos2共x 1兲 (g) 2 sin cos 2 2 (h) 2 cos tan 2 2 38. θ 7 24 (a) sin (c) tan (e) sec 2 (b) cos 2 (d) cot 2 (f) csc (g) 2 sin cos 2 2 2 2 2 In Exercises 57–60, find the solutions of the equation in the interval [0, 2冈. Use a graphing utility to verify your answers. 57. sin x cos x 0 2 58. sin x cos x 1 0 2 59. cos x sin x 0 2 60. tan x sin x 0 2 (h) cos 2 In Exercises 61–72, use the product-to-sum formulas to write the product as a sum or difference. In Exercises 39–46, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. 61. 6 sin cos 3 3 39. 15 40. 165 62. 4 sin 41. 112 30 42. 157 30 5 cos 3 6 43. 8 44. 12 45. 3 8 46. 7 12 In Exercises 47–52, find the exact values of sin 冇u/2冈, cos 冇u/ 2冈, and tan 冇u/ 2冈 using the half-angle formulas. 5 47. sin u 13 , 395 兾2 < u < 7 48. cos u 25 , 0 < u < 兾2 63. sin 5 cos 3 64. 5 sin 3 sin 4 65. 10 cos 75 cos 15 66. 6 sin 45 cos 15 67. 5 cos共5兲 cos 3 68. cos 2 cos 4 69. sin共x y兲 sin共x y兲 70. sin共x y兲 cos共x y兲 71. cos共 兲 sin共 兲 72. sin共 兲 sin共 兲 333353_0505.qxp 396 1/15/07 2:52 PM Chapter 5 Page 396 Analytic Trigonometry sec2 2 sec2 In Exercises 73–80, use the sum-to-product formulas to write the sum or difference as a product. 94. sec 2 73. sin 5 sin 95. cos2 2 sin2 2 cos 4 74. sin 3 sin 96. cos4 x sin4 x cos 2x 75. cos 6x cos 2x 97. 共sin x cos x兲2 1 sin 2x 76. sin x sin 7x 98. sin 77. sin共 兲 sin共 兲 99. 1 cos 10y 2 cos2 5y 78. cos共 2兲 cos 冢 cos 2 2 冢 sin x 2 2 79. cos 80. sin x 冣 冢 冣 冢 2 1 cos sin 3 3 2 3 冣 100. 冣 In Exercises 81–84, use the sum-to-product formulas to find the exact value of the expression. 81. sin 195 sin 105 cos 3 1 4 sin2 cos 冪tan 2u tan sinu u 101. sec u ± 2 102. tan u csc u cot u 2 103. cos 3 cos3 3 sin2 cos 104. sin 4 4 sin cos 共1 2 sin2 兲 82. cos 165 cos 75 105. cos 4x cos 2x sin x 2 sin 3x 106. cos 3x cos x tan 2x sin 3x sin x 107. cos 4x cos 2x cot 3x sin 4x sin 2x In Exercises 85– 88, find the solutions of the equation in the interval [0, 2冈. Use a graphing utility to verify your answers. 108. cos t cos 3t cot t sin 3t sin t 85. sin 6x sin 2x 0 109. sin 5 83. cos cos 12 12 84. sin 11 7 sin 12 12 86. cos 2x cos 6x 0 冢6 x冣 sin冢6 x冣 cos x 冢3 x冣 cos冢3 x冣 cos x cos 2x 87. 10 sin 3x sin x 110. cos 88. sin2 3x sin2 x 0 In Exercises 111–114, rewrite the function using the powerreducing formulas. Then use a graphing utility to graph the function. In Exercises 89–92, use the figure and trigonometric identities to find the exact value of the trigonometric function in two ways. 111. f 共x兲 sin2 x 112. f 共x兲 cos2 x 113. f 共x兲 cos 4 x 3 5 β 4 α 114. f 共x兲 sin 3 x In Exercises 115–120, write the trigonometric expression as an algebraic expression. 12 89. sin2 90. cos2 91. sin cos 92. cos sin 115. sin共2 arcsin x兲 116. cos共2 arccos x兲 117. cos共2 arcsin x兲 In Exercises 93–110, verify the identity algebraically. Use a graphing utility to check your result graphically. 118. sin共2 arccos x兲 csc 93. csc 2 2 cos 120. sin共2 arctan x兲 119. cos共2 arctan x兲 333353_0505.qxp 1/15/07 2:52 PM Page 397 Section 5.5 In Exercises 121–124, (a) use a graphing utility to graph the function and approximate the maximum and minimum points of the graph in the interval [0, 2], and (b) solve the trigonometric equation and verify that the x-coordinates of the maximum and minimum points of f are among its solutions (calculus is required to find the trigonometric equation). Function x 121. f 共x兲 4 sin cos x 2 Trigonometric Equation x 2 cos sin x 0 2 122. f 共x兲 cos 2x 2 sin x 2 cos x 共2 sin x1兲 0 123. f 共x兲 2 cos x sin 2x 2 2 cos 2x sin 124. f 共x兲 2 sin x 2 10 sin 2x 冢 5 cos 2x 冢 4 x 0 2 x cos 0 4 2 冣 冣 In Exercises 125 and 126, the graph of a function f is shown over the interval [0, 2]. (a) Find the x-intercepts of the graph of f algebraically. Verify your solutions by using the zero or root feature of a graphing utility. (b) The x-coordinates of the extrema or turning points of the graph of f are solutions of the trigonometric equation (calculus is required to find the trigonometric equation). Find the solutions of the equation algebraically. Verify the solutions using the maximum and minimum features of a graphing utility. 125. Function: f 共x兲 sin 2x sin x Trigonometric equation: 2 cos 2x cos x 0 3 f 2 0 Multiple-Angle and Product-to-Sum Formulas 127. Projectile Motion The range of a projectile fired at an angle with the horizontal and with an initial velocity of v0 feet per second is given by r 1 2 v sin 2 32 0 where r is measured in feet. (a) Rewrite the expression for the range in terms of . (b) Find the range r if the initial velocity of a projectile is 80 feet per second at an angle of 42. (c) Find the initial velocity required to fire a projectile 300 feet at an angle of 40. (d) For a given initial velocity, what angle of elevation yields a maximum range? Explain. 128. Geometry The length of each of the two equal sides of an isosceles triangle is 10 meters (see figure). The angle between the two sides is . 10 m θ 10 m (a) Write the area of the triangle as a function of 兾2. (b) Write the area of the triangle as a function of and determine the value of such that the area is a maximum. 129. Railroad Track When two railroad tracks merge, the overlapping portions of the tracks are in the shape of a circular arc (see figure). The radius of each arc r (in feet) and the angle are related by x 2r sin2 . 2 2 Write a formula for x in terms of cos . −3 126. Function: f 共x兲 cos 2x sin x Trigonometric equation: 2 sin 2x cos x 0 r 3 r θ 2 0 f −3 397 θ x 333353_0505.qxp 398 1/15/07 2:52 PM Chapter 5 Page 398 Analytic Trigonometry 130. Mach Number The mach number M of an airplane is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane (see figure). The mach number is related to the apex angle of the cone by sin (d) Rewrite the result of part (c) in terms of the sine of a double angle. Use a graphing utility to rule out incorrectly rewritten functions. 1 . 2 M (e) When you rewrite a trigonometric expression, the result may not be the same as a friend’s. Does this mean that one of you is wrong? Explain. θ (a) Find the angle that corresponds to a mach number of 1. (b) Find the angle that corresponds to a mach number of 4.5. (c) The speed of sound is about 760 miles per hour. Determine the speed of an object having the mach numbers in parts (a) and (b). (d) Rewrite the equation as a trigonometric function of . Synthesis 135. Writing Describe how you can use a double-angle formula or a half-angle formula to derive a formula for the area of an isosceles triangle. Use a labeled sketch to illustrate your derivation. Then write two examples that show how your formula can be used. 136. (a) Write a formula for cos 3. (b) Write a formula for cos 4. Skills Review In Exercises 137–140, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment connecting the points. 137. 共5, 2兲, 共1, 4兲 True or False? In Exercises 131 and 132, determine whether the statement is true or false. Justify your answer. 131. sin (c) Add a trigonometric term to the function so that it becomes a perfect square trinomial. Rewrite the function as a perfect square trinomial minus the term that you added. Use a graphing utility to rule out incorrectly rewritten functions. 冪 x 2 1 cos x , 2 ≤ x ≤ 2 138. 共4, 3兲, 共6, 10兲 1 4 5 139. 共0, 2 兲, 共3, 2 兲 140. 共13, 23 兲, 共1, 32 兲 132. The graph of y 4 8 sin2 x has a maximum at 共, 4兲. In Exercises 141–144, find (if possible) the complement and supplement of each angle. 133. Conjecture Consider the function 141. (a) 55 (b) 162 冢 142. (a) 109 (b) 78 143. (a) 18 (b) 144. (a) 0.95 (b) 2.76 f 共x兲 2 sin x 2 cos 2 冣 x 1 . 2 (a) Use a graphing utility to graph the function. (b) Make a conjecture about the function that is an identity with f. (c) Verify your conjecture algebraically. 134. Exploration Consider the function f 共x兲 sin x cos4 x. 4 (a) Use the power-reducing formulas to write the function in terms of cosine to the first power. (b) Determine another way of rewriting the function. Use a graphing utility to rule out incorrectly rewritten functions. 9 20 145. Find the radian measure of the central angle of a circle with a radius of 15 inches that intercepts an arc of length 7 inches. 146. Find the length of the arc on a circle of radius 21 centimeters intercepted by a central angle of 35. In Exercises 147–150, sketch a graph of the function. (Include two full periods.) Use a graphing utility to verify your graph. 147. f 共x兲 3 cos 2x 2 148. f 共x兲 1 5 sin x 2 2 149. f 共x兲 1 tan 2x 2 150. f 共x兲 1 x sec 4 2 333353_050R.qxp 1/15/07 2:53 PM Page 399 Chapter Summary 399 What Did You Learn? Key Terms sum and difference formulas, p. 380 reduction formulas, p. 382 double-angle formulas, p. 387 power-reducing formulas, p. 389 half-angle formulas, p. 390 Key Concepts 5.1 䊏 Use the fundamental trigonometric identities The fundamental trigonometric identities can be used to evaluate trigonometric functions, simplify trigonometric expressions, develop additional trigonometric identities, and solve trigonometric equations. 5.2 䊏 Verify trigonometric identities 1. Work with one side of the equation at a time. It is often better to work with the more complicated side first. 5.5 product-to-sum formulas, p. 391 sum-to-product formulas, p. 392 䊏 Use multiple-angle formulas, power-reducing formulas, half-angle formulas, product-tosum formulas, and sum-to-product formulas Double-Angle Formulas: sin 2u 2 sin u cos u tan 2u 2 tan u 1 tan2 u fractions, square a binomial, or create a monomial denominator. 3. Look for opportunities to use the fundamental identi- Half-Angle Formulas: ties. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. 4. Try converting all terms to sines and cosines. 5. Always try something. 5.3 䊏 Solve trigonometric equations 1. Use algebraic techniques, such as collecting like terms, extracting square roots, factoring, and the Quadratic Formula to isolate the trigonometric function involved in the equation. 2. If there is no reasonable way to find the solution(s) of a trigonometric equation algebraically, use a graphing utility to approximate the solution(s). 5.4 䊏 Use sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations sin共u ± v兲 sin u cos v ± cos u sin v cos共u ± v兲 cos u cos v sin u sin v tan共u ± v兲 tan u ± tan v 1 tan u tan v 2 cos2 u 1 1 2 sin2 u Power-Reducing Formulas: 1 cos 2u 2 1 cos 2u cos2 u 2 1 cos 2u tan2 u 1 cos 2u 2. Look for opportunities to factor an expression, add cos 2u cos2 u sin2 u sin2 u 冪 u 1 cos u u ± cos ± 2 2 2 u 1 cos u sin u tan 2 sin u 1 cos u sin 冪1 2cos u Product-to-Sum Formulas: sin u sin v 12关cos共u v兲 cos共u v兲兴 cos u cos v 12关cos共u v兲 cos共u v兲兴 sin u cos v 12关sin共u v兲 sin共u v兲兴 cos u sin v 12关sin共u v兲 sin共u v兲兴 Sum-to-Product Formulas: 冢u 2 v冣 cos冢u 2 v冣 uv uv sin u sin v 2 cos冢 sin 2 冣 冢 2 冣 uv uv cos u cos v 2 cos冢 cos冢 2 冣 2 冣 uv uv cos u cos v 2 sin冢 sin 2 冣 冢 2 冣 sin u sin v 2 sin 333353_050R.qxp 400 1/15/07 2:53 PM Chapter 5 Page 400 Analytic Trigonometry Review Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises. 5.1 In Exercises 1–10, name the trigonometric function that is equivalent to the expression. 5.2 In Exercises 25–36, verify the identity. 25. cos x共tan2 x 1兲 sec x 1 1. cos x 1 2. sin x 26. sec2 x cot x cot x tan x 1 3. sec x 1 4. tan x 28. cot2 x cos2 x cot2 x cos2 x 5. 冪1 cos2 x 6. 冪1 tan2 x 7. csc 冢2 x冣 8. cot 9. sec共x兲 27. sin3 sin cos 2 sin 冢2 x冣 29. sin5 x cos2 x 共cos2 x 2 cos 4 x cos6 x兲 sin x 30. cos3 x sin2 x 共sin2 x sin4 x兲 cos x 31. 10. tan共x兲 1 sin 冪11 sin sin ⱍcos ⱍ 32. 冪1 cos x In Exercises 11–14, use the given values to evaluate (if possible) the remaining trigonometric functions of the angle. 4 11. sin x , 5 cos x 2 12. tan , 3 sec 3 5 冪13 3 冢 14. csc 冣 冢2 冣 3, sin 17. 1 tan2 x 1 sin2 cos 2 sin2 sin cos 19. tan2 共csc2 1兲 21. tan 冢2 x冣 sec x 34. 1 sec共x兲 csc x sin共x兲 tan共x兲 16. 18. 2 x 冢2 x冣 sec x csc x 37. 2 sin x 1 0 38. tan x 1 0 39. sin x 冪3 sin x 40. 4 cos x 1 2 cos x 41. 3冪3 tan x 3 42. x1 sec x 1 43. 3 csc x 4 44. 4 tan2 x 1 tan2 x 45. 4 cos2 x 3 0 46. sin x共sin x 1兲 0 sin3 cos3 sin cos 47. sin x tan x 0 48. csc x 2 cot x 0 2 sec2 20. csc 2 x共1 cos2 x兲 22. 冢2 x冣 1 tan 5.3 In Exercises 37–48, solve the equation. In Exercises 15 – 22, use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. 15. csc共x兲 cot x sec共x兲 36. tan 2冪2 3 冪1 cos x 33. 35. csc2 1 1 13. sin x , sin x 2 冪2 冪2 ⱍsin xⱍ sin共x兲 cot x sin x 2 冢 冣 23. Rate of Change The rate of change of the function f 共x兲 2冪sin x is given by the expression sin1兾2 x cos x. Show that this expression can also be written as cot x冪sin x. 24. Rate of Change The rate of change of the function f 共x兲 csc x cot x is given by the expression csc2 x csc x cot x. Show that this expression can also be written as 共1 cos x兲兾sin2 x. 1 2 sec x 1 0 In Exercises 49–52, find all solutions of the equation in the interval [0, 2冈. Use a graphing utility to check your answers. 49. 2 cos2 x cos x 1 50. 2 sin2 x 3 sin x 1 51. cos2 x sin x 1 52. sin2 x 2 cos x 2 In Exercises 53–58, find all solutions of the multiple-angle equation in the interval [0, 2冈. 53. 2 sin 2x 冪2 0 54. 冪3 tan 3x 0 55. cos 4x共cos x 1兲 0 56. 3 csc2 5x 4 57. cos 4x 7 cos 2x 8 58. sin 4x sin 2x 0 333353_050R.qxp 1/15/07 2:54 PM Page 401 Review Exercises In Exercises 59–62, solve the equation. 59. 2 sin 2x 1 0 60. 2 cos 4x 冪3 0 61. 2 sin2 3x 1 0 62. 4 cos2 2x 3 0 In Exercises 63 – 66, use the inverse functions where necessary to find all solutions of the equation in the interval [0, 2冈. 63. sin2 x 2 sin x 0 65. tan2 3 tan 10 0 66. 冢 sin x 冪2 2 2 冢 cos x 1 4 4 91. sin x 92. cos x 冣 冢 冣 冣 冢 5 93. sin u , 7 x 6 tan x 4 0 5.4 In Exercises 67– 70, find the exact values of the sine, cosine, and tangent of the angle. 67. 285 315 30 68. 345 300 45 31 11 3 69. 12 6 4 70. 13 11 3 12 6 4 In Exercises 71–74, write the expression as the sine, cosine, or tangent of an angle. 71. sin 130 cos 50 cos 130 sin 50 冣 < u < 3 < u < 2 2 4 94. cos u , 5 < u < 2 2 95. tan u , 9 96. cos u 3 2 2 冪5 , < u < 2 In Exercises 97–100, use double-angle formulas to verify the identity algebraically. Use a graphing utility to check your result graphically. 97. 6 sin x cos x 3 sin 2x 72. cos 45 cos 120 sin 45 sin 120 tan 25 tan 10 73. 1 tan 25 tan 10 In Exercises 91 and 92, find the solutions of the equation in the interval [0, 2冈. 5.5 In Exercises 93–96, find the exact values of sin 2u, cos 2u, and tan 2u using the double-angle formulas. 64. 3 cos2 x 5 cos x 0 sec2 401 tan 63 tan 118 74. 1 tan 63 tan 118 In Exercises 75– 80, find the exact value of the trigonometric 7 function given that sin u ⴝ 35 and cos v ⴝ ⴚ 25 . (Both u and v are in Quadrant II.) 75. sin共u v兲 76. tan共u v兲 77. tan共u v兲 78. sin共u v兲 79. cos共u v兲 80. cos共u v兲 In Exercises 81– 84, find the value of the expression without using a calculator. 98. 4 sin x cos x 2 2 sin 2x 2 99. 1 4 sin2 x cos2 x cos2 2x 100. sin 4x 8 cos3 x sin x 4 cos x sin x 101. Projectile Motion A baseball leaves the hand of the first baseman at an angle of with the horizontal and with an initial velocity of v0 80 feet per second. The ball is caught by the second baseman 100 feet away. Find if the 1 2 range r of a projectile is given by r 32 v0 sin 2. 102. Projectile Motion Use the equation in Exercise 101 to find when a golf ball is hit with an initial velocity of v0 50 feet per second and lands 77 feet away. 83. cos关cos1 1 sin1共1兲兴 In Exercises 103–106, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. 84. tan关cos1共1兲 cos1 1兴 103. sin6 x 81. sin关sin1 0 cos1共1兲兴 82. cos共cos1 1 sin1 0兲 105. cos4 2x 104. cos4 x sin4 x 106. sin4 2x In Exercises 85– 90, verify the identity. 冢 85. cos x 87. cot sin x 2 冣 冢 2 x冣 tan x 89. cos 3x 4 cos3 x 3 cos x 90. sin共 兲 tan tan cos cos 冢 86. sin x 3 cos x 2 冣 88. sin共 x兲 sin x In Exercises 107–110, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. 107. 105 109. 7 8 108. 112 30 110. 11 12 333353_050R.qxp 1/15/07 402 2:54 PM Chapter 5 Page 402 Analytic Trigonometry 冢 sin x 4 4 冢 cos x 6 6 In Exercises 111–114, find the exact values of sin冇u/ 2冈, cos冇u/ 2冈, and tan 冇u/ 2冈 using the half-angle formulas. 125. sin x 2 126. cos x 3 111. sin u , 5 112. tan u 0 < u < 21 , 20 < u < 3 2 2 113. cos u , 7 < u < 2 114. sec u 6, < u < 2 In Exercises 115 and 116, use the half-angle formulas to simplify the expression. 115. 116. 冪 1 cos 8x 2 冣 冢 冣 冣 冢 冣 Harmonic Motion In Exercises 127–130, a weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position. This motion is described by the model y ⴝ 1.5 sin 8t ⴚ 0.5 cos 8t where y is the distance from equilibrium in feet and t is the time in seconds. 127. Write the model in the form y 冪a 2 b2 sin共Bt C兲. 128. Use a graphing utility to graph the model. sin 10x 1 cos 10x 129. Find the amplitude of the oscillations of the weight. 130. Find the frequency of the oscillations of the weight. Geometry In Exercises 117 and 118, a trough for feeding cattle is 4 meters long and its cross sections are isosceles triangles with two equal sides of 12 meter (see figure). The angle between the equal sides is . Synthesis True or False? In Exercises 131–134, determine whether the statement is true or false. Justify your answer. 131. If < < , then cos < 0. 2 2 132. sin共x y兲 sin x sin y 4m 133. 4 sin共x兲 cos共x兲 2 sin 2x 1 2m 134. 4 sin 45 cos 15 1 冪3 θ 1 2 135. List the reciprocal identities, quotient identities, and Pythagorean identities from memory. m 117. Write the trough’s volume as a function of 兾2. 136. Is cos 冪1 sin2 an identity? Explain. 118. Write the volume of the trough as a function of and determine the value of such that the volume is maximum. In Exercises 137 and 138, use the graphs of y1 and y2 to determine how to change y2 to a new function y3 such that y1 ⴝ y3 . In Exercises 119–122, use the product-to-sum formulas to write the product as a sum or difference. 137. y1 sec2 119. 6 sin cos 4 4 121. sin 5 sin 4 冢 2 x冣 cos 3x cos x y2 共2 sin x兲2 y2 cot2 x 120. 4 sin 15 sin 45 5 5 y2 122. cos 6 sin 8 y1 In Exercises 123–126, use the sum-to-product formulas to write the sum or difference as a product. 123. cos 5 cos 4 138. y1 124. sin 3 sin 2 − −1 y2 − y1 −4 333353_050R.qxp 1/15/07 2:54 PM Page 403 403 Chapter Test 5 Chapter Test See www.CalcChat.com for worked-out solutions to odd-numbered exercises. Take this test as you would take a test in class. After you are finished, check your work against the answers given in the back of the book. 1. If tan 32 and cos < 0, use the fundamental identities to evaluate the other five trigonometric functions of . 2. Use the fundamental identities to simplify csc2 共1 cos2 兲. 3. Factor and simplify 4. Add and simplify sec4 x tan4 x . sec2 x tan2 x sin cos . sin cos 5. Determine the values of , 0 ≤ < 2, for which tan 冪sec2 1 is true. 6. Use a graphing utility to graph the functions y1 sin x cos x cot x and y2 csc x. Make a conjecture about y1 and y2. Verify your result algebraically. In Exercises 7–12, verify the identity. 7. sin sec tan 8. sec2 x tan2 x sec2 x sec4 x 9. csc sec cot tan sin cos 冢 10. cos x sin x 2 冣 11. sin共n 兲 共1兲n sin , n is an integer. 12. 共sin x cos x兲2 1 sin 2x y 13. Find the exact value of tan 105 . 14. Rewrite sin4 x tan2 x in terms of the first power of the cosine. (1, 2) sin 4 15. Use a half-angle formula to simplify the expression . 1 cos 4 16. Write 4 cos 2 sin 4 as a sum or difference. u x 17. Write sin 3 sin 4 as a product. In Exercises 18 – 21, find all solutions of the equation in the interval [0, 2冈. 18. tan2 x tan x 0 19. sin 2 cos 0 20. 4 cos x 3 0 21. csc2 x csc x 2 0 2 22. Use a graphing utility to approximate the solutions of the equation 3 cos x x 0 accurate to three decimal places. Figure for 23 Air θ 23. Use the figure to find the exact values of sin 2u, cos 2u, and tan 2u. 24. The index of refraction n of a transparent material is the ratio of the speed of light in a vacuum to the speed of light in the material. For the triangular glass prism in the figure, n 1.5 and 60 . Find the angle for the glass prism if n sin共兾2 兾2兲 . sin共兾2兲 α ht Lig Prism Figure for 24 333353_050R.qxp 404 1/15/07 Chapter 5 2:54 PM Page 404 Analytic Trigonometry Proofs in Mathematics Sum and Difference Formulas (p. 380) sin共u v兲 sin u cos v cos u sin v sin共u v兲 sin u cos v cos u sin v cos共u v兲 cos u cos v sin u sin v cos共u v兲 cos u cos v sin u sin v tan共u v兲 tan u tan v 1 tan u tan v tan共u v兲 tan u tan v 1 tan u tan v Proof You can use the figures at the right for the proofs of the formulas for cos共u ± v兲. In the top figure, let A be the point 共1, 0兲 and then use u and v to locate the points B 共x1, y1兲, C 共x2, y2兲, and D 共x3, y3兲 on the unit circle. So, x i2 y i2 1 for i 1, 2, and 3. For convenience, assume that 0 < v < u < 2. In the bottom figure, note that arcs AC and BD have the same length. So, line segments AC and BD are also equal in length, which implies that y B = (x1, y1 ) C = (x2, y2 ) u−v v A = (1, 0) x u 冪共x2 1兲 共 y2 0兲 冪共x3 x1兲 共 y3 y1兲 2 2 2 2 x22 2x2 1 y22 x32 2x1x3 x12 y32 2y1 y3 y12 D = (x3, y3 ) 共x22 y22兲 1 2x2 共x32 y32兲 共x12 y12兲 2x1x3 2y1y3 1 1 2x2 1 1 2x1 x3 2y1 y3 x2 x3 x1 y3 y1. Finally, by substituting the values x2 cos共u v兲, x3 cos u, x1 cos v, y3 sin u, and y1 sin v, you obtain cos共u v兲 cos u cos v sin u sin v. The formula for cos共u v兲 can be established by considering u v u 共v兲 and using the formula just derived to obtain y C = (x2, y2 ) cos共u v兲 cos关u 共v兲兴 cos u cos共v兲 sin u sin 共v兲 A = (1, 0) x cos u cos v sin u sin v. You can use the sum and difference formulas for sine and cosine to prove the formulas for tan共u ± v兲. tan共u ± v兲 sin共u ± v兲 cos共u ± v兲 Quotient identity sin u cos v ± cos u sin v cos u cos v sin u sin v Sum and difference formulas sin u cos v ± cos u sin v cos u cos v cos u cos v sin u sin v cos u cos v Divide numerator and denominator by cos u cos v. B = (x1, y1 ) D = (x3, y3 ) 333353_050R.qxp 1/15/07 2:55 PM Page 405 Proofs in Mathematics sin u cos v cos u sin v ± cos u cos v cos u cos v cos u cos v sin u sin v cos u cos v cos u cos v Write as separate fractions. sin v sin u ± cos u cos v sin u sin v 1 cos u cos v Product of fractions tan u ± tan v 1 tan u tan v Double-Angle Formulas tan 2u Quotient identity (p. 387) sin 2u 2 sin u cos u Trigonometry and Astronomy cos 2u cos2 u sin2 u 2 cos2 u 1 2 tan u 1 tan2 u 1 2 sin2 u Proof To prove all three formulas, let v u in the corresponding sum formulas. sin 2u sin共u u兲 sin u cos u cos u sin u 2 sin u cos u cos 2u cos共u u兲 cos u cos u sin u sin u cos2 u sin2 u tan 2u tan共u u兲 tan u tan u 2 tan u 1 tan u tan u 1 tan2 u Power-Reducing Formulas sin2 u 1 cos 2u 2 (p. 389) cos2 u 1 cos 2u 2 tan2 u 1 cos 2u 1 cos 2u Proof To prove the first formula, solve for sin2 u in the double-angle formula cos 2u 1 2 sin2 u, as follows. cos 2u 1 2 sin2 u 2 sin2 u 1 cos 2u sin2 u 1 cos 2u 2 405 Write double-angle formula. Subtract cos 2u from and add 2 sin2 u to each side. Divide each side by 2. Trigonometry was used by early astronomers to calculate measurements in the universe. Trigonometry was used to calculate the circumference of Earth and the distance from Earth to the moon. Another major accomplishment in astronomy using trigonometry was computing distances to stars. 333353_050R.qxp 406 1/15/07 Chapter 5 2:55 PM Page 406 Analytic Trigonometry In a similar way you can prove the second formula, by solving for cos2 u in the double-angle formula cos 2u 2 cos2 u 1. To prove the third formula, use a quotient identity, as follows. tan2 u sin2 u cos2 u 1 cos 2u 2 1 cos 2u 2 1 cos 2u 1 cos 2u Sum-to-Product Formulas sin u sin v 2 sin (p. 392) 冢u 2 v冣 cos冢u 2 v冣 sin u sin v 2 cos 冢u 2 v冣 sin冢u 2 v冣 cos u cos v 2 cos 冢u 2 v冣 cos冢u 2 v冣 cos u cos v 2 sin 冢u 2 v冣 sin冢u 2 v冣 Proof To prove the first formula, let x u v and y u v. Then substitute u 共x y兲兾2 and v 共x y兲兾2 in the product-to-sum formula. 1 sin u cos v 关sin共u v兲 sin共u v兲兴 2 sin 2 sin 冢x 2 y冣 cos冢x 2 y冣 12 共sin x sin y兲 冢x 2 y冣 cos冢x 2 y冣 sin x sin y The other sum-to-product formulas can be proved in a similar manner.