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Trigonometric Identities a n d Conditional Equations Outline Application 6-1 Basic Identities and Their Use The roof for a 24-foot-wide industrial building is formed by bending a 30-foot-wide panel of corrugated steel into a circular arc (see the figure). How high (to the nearest tenth of a foot) is the building? 6-2 Sum, Difference, and Cofunction Identities 6-3 Double-Angle and Half-Angle Identities –Sum and 6-4 Product– –Product Identities Sum– 30 ft 6-5 Trigonometric Equations Chapter 6 Group Activity: From M sin Bt ⴙ N cos Bt to —A Harmonic A sin (Bt ⴙ C)— Analysis Tool Chapter 6 Review 10 ft 24 ft rigonometric functions are widely used in solving real-world problems as well as in the development of mathematics. Whatever their use, it is often of value to be able to change a trigonometric expression from one form to an equivalent more useful form. This involves the use of identities. Recall that an equation in one or more variables is said to be an identity if the left side is equal to the right side for all replacements of the variables for which both sides are defined. For example, the equation T x2 ⫺ 2x ⫺ 8 ⫽ (x ⫺ 4)(x ⫹ 2) is an identity, but x2 ⫺ 2x ⫺ 8 ⫽ 0 is not. The latter is called a conditional equation, since it holds only for certain values of x and not for all values for which both sides are defined. The first four sections of the chapter deal with trigonometric identities and the last section with conditional trigonometric equations. Preparing for This Chapter Before getting started on this chapter, review the following concepts: Operations on Polynomials (Appendix A, Sections 2 and 3) Rational Expressions (Appendix A, Section 4) Cartesian Coordinate System (Chapter 1, Section 1) Quadratic Equations (Chapter 2, Section 5) Basic Identities (Chapter 5, Section 2) Section 6-1 Basic Identities and Their Use Basic Identities Establishing Other Identities In this section we review the basic identities introduced in Section 5-2 and show how they are used to verify other identities. Basic Identities In the following box we list for convenient reference the basic identities introduced in Section 5-2. These identities will be used very frequently in the work that follows and should be memorized. 452 6-1 Basic Identities and Their Use 453 BASIC TRIGONOMETRIC IDENTITIES Reciprocal Identities csc x ⫽ 1 sin x sec x ⫽ 1 cos x cot x ⫽ cos x sin x cot x ⫽ 1 tan x Quotient Identities tan x ⫽ sin x cos x Identities for Negatives sin (⫺x) ⫽ ⫺sin x cos (⫺x) ⫽ cos x tan (⫺x) ⫽ ⫺tan x Pythagorean Identities sin2 x ⫹ cos2 x ⫽ 1 tan2 x ⫹ 1 ⫽ sec2 x 1 ⫹ cot2 x ⫽ csc2 x All these identities were established in Section 5-2 (the second and third Pythagorean identities were established in Problems 87 and 88 in Exercise 5-2). Explore/Discuss 1 Discuss an easy way to recall the second and third Pythagorean identities from the first. [Hint: Divide through the first Pythagorean identity by appropriate expressions.] Establishing Other Identities As indicated above, when working with trigonometric expressions it is often desirable to convert one form to an equivalent form that may be more useful. This section is designed to give you experience in this process. In addition to using the basic identities and other verified identities, we will often use basic algebraic operations such as multiplication, factoring, combining and reducing fractions, and so on. The following examples illustrate some of the techniques used to verify certain identities. The steps illustrated are not necessarily unique—often, there is more than one path to a desired goal. To become proficient in the use of identities, it is important that you work out many problems on your own. EXAMPLE 1 Identity Verification Verify the identity cos x tan x ⫽ sin x. 454 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Verification Generally, we proceed by starting with the more complicated of the two sides, and transform that side into the other side in one or more steps using basic identities, algebra, or other established identities. Thus, cos x tan x ⫽ cos x sin x cos x ⫽ sin x MATCHED PROBLEM Quotient identity Algebra Verify the identity sin x cot x ⫽ cos x. 1 Explore/Discuss 2 EXAMPLE 2 Verification Graph the left and right sides of the identity in Example 1 in a graphing utility by letting y1 ⫽ cos x tan x and y2 ⫽ sin x. Use TRACE, moving back and forth between the graphs of y1 and y2, to compare values of y for given values of x. What does this investigation illustrate? Identity Verification Verify the identity sec (⫺x) ⫽ sec x. sec (⫺x) ⫽ ⫽ 1 cos (⫺x) Reciprocal identity 1 cos x Identity for negatives ⫽ sec x MATCHED PROBLEM Reciprocal identity Verify the identity csc (⫺x) ⫽ ⫺csc x. 2 EXAMPLE 3 Verification Identity Verification Verify the identity cot x cos x ⫹ sin x ⫽ csc x. cos x cos x ⫹ sin x sin x Quotient identity ⫽ cos2 x ⫹ sin x sin x Algebra ⫽ cos2 x ⫹ sin2 x sin x Algebra ⫽ 1 sin x Pythagorean identity cot x cos x ⫹ sin x ⫽ ⫽ csc x Reciprocal identity 6-1 Basic Identities and Their Use 455 Key Algebraic Steps in Example 3 a a2 a2 ⫹ b2 a⫹b⫽ ⫹b⫽ b b b MATCHED PROBLEM Verify the identity tan x sin x ⫹ cos x ⫽ sec x. 3 To verify an identity, proceed from one side to the other, or from both sides to the middle, making sure all steps are reversible. Do not use properties of equality to perform the same operation on both sides of the equation. Even though there is no fixed method of verification that works for all identities, there are certain steps that help in many cases. SUGGESTED STEPS IN VERIFYING IDENTITIES 1. Start with the more complicated side of the identity, and transform it into the simpler side. 2. Try algebraic operations such as multiplying, factoring, combining fractions, and splitting fractions. 3. If other steps fail, express each function in terms of sine and cosine functions, and then perform appropriate algebraic operations. 4. At each step, keep the other side of the identity in mind. This often reveals what you should do in order to get there. EXAMPLE 4 Verification Identity Verification Verify the identity cos x 1 ⫹ sin x ⫹ ⫽ 2 sec x. cos x 1 ⫹ sin x 1 ⫹ sin x cos x (1 ⫹ sin x)2 ⫹ cos2 x ⫹ ⫽ cos x 1 ⫹ sin x cos x (1 ⫹ sin x) Algebra ⫽ 1 ⫹ 2 sin x ⫹ sin2 x ⫹ cos2 x cos x (1 ⫹ sin x) ⫽ 1 ⫹ 2 sin x ⫹ 1 cos x (1 ⫹ sin x) ⫽ 2 ⫹ 2 sin x cos x (1 ⫹ sin x) Algebra ⫽ 2(1 ⫹ sin x) cos x (1 ⫹ sin x) Algebra ⫽ 2 cos x Algebra ⫽ 2 sec x Algebra Pythagorean identity Reciprocal identity 456 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Key Algebraic Steps in Example 4 a b a2 ⫹ b2 ⫹ ⫽ (1 ⫹ c)2 ⫽ 1 ⫹ 2c ⫹ c2 b a ba MATCHED PROBLEM 4 EXAMPLE 5 Verification Verify the identity m(a ⫹ b) m ⫽ n(a ⫹ b) n sin x 1 ⫹ cos x ⫹ ⫽ 2 csc x. sin x 1 ⫹ cos x Identity Verification sin2 x ⫹ 2 sin x ⫹ 1 1 ⫹ sin x ⫽ . cos2 x 1 ⫺ sin x sin2 x ⫹ 2 sin x ⫹ 1 (sin x ⫹ 1)2 ⫽ Algebra cos2 x cos2 x Verify the identity ⫽ (sin x ⫹ 1)2 1 ⫺ sin2 x Pythagorean identity ⫽ (1 ⫹ sin x)2 (1 ⫺ sin x)(1 ⫹ sin x) Algebra ⫽ 1 ⫹ sin x 1 ⫺ sin x Algebra Key Algebraic Steps in Example 5 a2 ⫹ 2a ⫹ 1 ⫽ (a ⫹ 1)2 1 ⫺ b2 ⫽ (1 ⫺ b)(1 ⫹ b) MATCHED PROBLEM Verify the identity sec4 x ⫺ 2 sec2 x tan2 x ⫹ tan4 x ⫽ 1. 5 EXAMPLE 6 Verification Identity Verification tan x ⫺ cot x ⫽ 1 ⫺ 2 cos2 x. Verify the identity tan x ⫹ cot x sin x cos x ⫺ tan x ⫺ cot x cos x sin x ⫽ tan x ⫹ cot x sin x cos x ⫹ cos x sin x Change to sines and cosines (quotient identities). ⫺ 冢 cos x sin x 冣 ⫽ sin x cos x (sin x)(cos x)冢 ⫹ cos x sin x 冣 (sin x)(cos x) sin x ⫽ sin2 x ⫺ cos2 x sin2 x ⫹ cos2 x ⫽ 1 ⫺ cos2 x ⫺ cos2 x 1 ⫽ 1 ⫺ 2 cos2 x cos x Multiply numerator and denominator by (sin x)(cos x), and use algebra to transform the compound fraction into a simple fraction. Pythagorean identity Algebra 6-1 Basic Identities and Their Use 457 Key Algebraic Steps in Example 6 a b a b ⫺ ab ⫺ b a b a a2 ⫺ b2 ⫽ ⫽ 2 a b a b a ⫹ b2 ⫹ ab ⫹ b a b a 冢 冢 冣 冣 Verify the identity cot x ⫺ tan x ⫽ MATCHED PROBLEM 6 2 cos2 x ⫺ 1 . sin x cos x Just observing how others verify identities won’t make you good at it. You must verify a large number on your own. With practice the process will seem less complicated. EXAMPLE Testing Identities Using a Graphing Utility 7 Use a graphing utility to test whether each of the following is an identity. If an equation appears to be an identity, verify it. If the equation does not appear to be an identity, find a value of x for which both sides are defined but are not equal. (A) tan x ⫹ 1 ⫽ (sec x)(sin x ⫺ cos x) (B) tan x ⫺ 1 ⫽ (sec x)(sin x ⫺ cos x) (A) Graph each side of the equation in the same viewing window (Fig. 1). The equation is not an identity, since the graphs do not match. Solutions FIGURE 1 4 ⫺2 2 ⫺4 Try x ⫽ 0. Left side: tan 0 ⫹ 1 ⫽ 1 Right side: (sec 0)(sin 0 ⫺ cos 0) ⫽ ⫺1 Finding one value of x for which both sides are defined, but are not equal, is enough to verify that the equation is not an identity. (B) Graph each side of the equation in the same viewing window (Fig. 2). The equation appears to be an identity, which we now verify: FIGURE 2 4 (sec x) (sin x ⫺ cos x) ⫺2 2 ⫺4 冢cos1 x 冣(sin x ⫺ cos x) sin x cos x ⫺ ⫽冢 cos x cos x 冣 ⫽ ⫽ tan x ⫺ 1 458 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS MATCHED PROBLEM 7 Use a graphing utility to test whether each of the following is an identity. If an equation appears to be an identity, verify it. If the equation does not appear to be an identity, find a value of x for which both sides are defined but are not equal. sin x sin x (A) ⫽ csc x (B) ⫽ sec x 2 1 ⫺ cos x 1 ⫺ cos2 x Answers to Matched Problems In the following identity verifications, other correct sequences of steps are possible—the process is not unique. cos x 1 1 1. sin x cot x ⫽ sin x 2. csc (⫺x) ⫽ ⫽ cos x ⫽ ⫽ ⫺csc x sin x sin (⫺x) ⫺sin x sin2 x ⫹ cos2 x 1 sin2 x 3. tan x sin x ⫹ cos x ⫽ ⫹ cos x ⫽ ⫽ ⫽ sec x cos x cos x cos x 1 ⫹ cos x sin x (1 ⫹ cos x)2 ⫹ sin2 x 1 ⫹ 2 cos x ⫹ cos2 x ⫹ sin2 x 2(1 ⫹ cos x) 4. ⫹ ⫽ ⫽ ⫽ ⫽ 2 csc x sin x 1 ⫹ cos x sin x (1 ⫹ cos x) sin x (1 ⫹ cos x) sin x (1 ⫹ cos x) 5. sec4 x ⫺ 2 sec2 x tan2 x ⫹ tan4 x ⫽ (sec2 x ⫺ tan2 x)2 ⫽ 12 ⫽ 1 cos x sin x cos2 x ⫺ sin2 x cos2 x ⫺ (1 ⫺ cos2 x) 2 cos2 x ⫺ 1 6. cot x ⫺ tan x ⫽ ⫺ ⫽ ⫽ ⫽ sin x cos x sin x cos x sin x cos x sin x cos x 7. (A) An identity: 4 ⫺2 2 ⫺4 sin x 1 sin x ⫽ ⫽ ⫽ csc x 1 ⫺ cos2 x sin2 x sin x (B) Not an identity: the left side is not equal to the right side for x ⫽ 1, for example. 4 ⫺2 2 ⫺4 6-1 Basic Identities and Their Use 459 29. ⫺ ⱕ x ⱕ EXERCISE 6-1 (A) y ⫽ A cos x cot x sin x (B) y ⫽ 1 30. ⫺ ⱕ x ⱕ Verify that Problems 1–26 are identities. 1. sin sec ⫽ tan 2. cos csc ⫽ cot 3. cot u sec u sin u ⫽ 1 4. tan csc cos ⫽ 1 5. sin (⫺x) ⫽ ⫺tan x cos (⫺x) 7. sin ␣ ⫽ 6. cot (⫺x) tan x ⫽ ⫺1 tan ␣ cot ␣ csc ␣ 8. tan ␣ ⫽ cos ␣ sec ␣ cot ␣ 9. cot u ⫹ 1 ⫽ (csc u)(cos u ⫹ sin u) 10. tan u ⫹ 1 ⫽ (sec u)(sin u ⫹ cos u) 11. cos x ⫺ sin x ⫽ csc x ⫺ sec x sin x cos x 12. cos x ⫺ sin x ⫽ cot x ⫺ tan x sin x cos x 2 13. 15. (A) y ⫽ sin x cos x tan x B Verify that Problems 31–60 are identities. 31. 1 ⫺ (sin x ⫺ cos x)2 ⫽ 2 cos x sin x 32. 1 ⫺ cos2 y ⫽ tan2 y (1 ⫺ sin y)(1 ⫹ sin y) 33. cos ⫹ sin ⫽ cot ⫹ 1 csc 34. sin ⫹ cos ⫽ tan ⫹ 1 sec 2 sin2 t ⫹ cos t ⫽ sec t cos t 14. cos x ⫽ sec x 1 ⫺ sin2 x 16. cos2 t ⫹ sin t ⫽ csc t sin t sin u ⫽ csc u 1 ⫺ cos2 u 17. (1 ⫺ cos u)(1 ⫹ cos u) ⫽ sin2 u 20. (sin x ⫹ cos x)2 ⫽ 1 ⫹ 2 sin x cos x 21. (sec t ⫹ 1)(sec t ⫺ 1) ⫽ tan2 t cos2 y 1 ⫹ sin y 39. csc ⫽ cos cot ⫹ tan 43. ln (cot x) ⫽ ⫺ln (tan x) 24. sec2 u ⫺ tan2 u ⫽ 1 45. cos x ⫹ tan x 25. cot x ⫹ sec x ⫽ sin x 26. sin m (csc m ⫺ sin m) ⫽ cos2 m In Problems 27–30, graph all parts of each problem in the same viewing window in a graphing utility. 27. ⫺ ⱕ x ⱕ (B) y ⫽ cos x 2 (C) y ⫽ sin2 x ⫹ cos2 x 28. ⫺ ⱕ x ⱕ (B) y ⫽ tan2 x (C) y ⫽ sec x ⫺ tan x 2 40. 1 ⫹ sec ⫽ csc sin ⫹ tan 42. ln (cot x) ⫽ ln (cos x) ⫺ ln (sin x) 23. csc2 x ⫺ cot2 x ⫽ 1 2 36. 1 ⫺ sin y ⫽ 41. ln (tan x) ⫽ ln (sin x) ⫺ ln (cos x) 22. (csc t ⫺ 1)(csc t ⫹ 1) ⫽ cot2 t (A) y ⫽ sec2 x 1 ⫹ cos y sin2 y ⫽ 1 ⫺ cos y (1 ⫺ cos y)2 38. sec2 x ⫹ csc2 x ⫽ sec2 x csc2 x 19. cos2 x ⫺ sin2 x ⫽ 1 ⫺ 2 sin2 x (A) y ⫽ sin x 35. 37. tan2 x ⫺ sin2 x ⫽ tan2 x sin2 x 18. (1 ⫺ sin t)(1 ⫹ sin t) ⫽ cos2 t 2 (B) y ⫽ 1 1 ⫺ cos A sec A ⫺ 1 ⫽ 1 ⫹ cos A sec A ⫹ 1 44. ln (csc x) ⫽ ⫺ln (sin x) 46. 1 ⫺ csc y sin y ⫺ 1 ⫽ 1 ⫹ csc y sin y ⫹ 1 47. sin4 w ⫺ cos4 w ⫽ 1 ⫺ 2 cos2 w 48. sin4 x ⫹ 2 sin2 x cos2 x ⫹ cos4 x ⫽ 1 49. sec x ⫺ cos x ⫽ tan x 1 ⫹ sin x 50. csc n ⫺ sin n ⫽ cot n 1 ⫹ cos n 51. cos2 z ⫺ 3 cos z ⫹ 2 2 ⫺ cos z ⫽ sin2 z 1 ⫹ cos z 52. sin2 t ⫹ 4 sin t ⫹ 3 3 ⫹ sin t ⫽ cos2 t 1 ⫺ sin t 460 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS 53. cos3 ⫺ sin3 ⫽ 1 ⫹ sin cos cos ⫺ sin 77. tan ␣ ⫹ cot  ⫽ 54. cos3 u ⫹ sin3 u ⫽ 1 ⫺ sin u cos u cos u ⫹ sin u 78. 55. (sec x ⫺ tan x)2 ⫽ 1 ⫺ sin x 1 ⫹ sin x 56. (cot u ⫺ csc u)2 ⫽ 1 ⫺ cos u 1 ⫹ cos u tan  ⫹ cot ␣ tan  cot ␣ tan ␣ ⫹ tan  cot ␣ ⫹ cot  ⫽ cot ␣ cot  ⫺ 1 1 ⫺ tan ␣ tan  From the graph of y1 ⫽ f(x) in a graphing utility, find a simpler function of the form g(x) ⫽ k ⫹ AT(x), where T(x) is one of the six trigonometric functions that has the same graph as y1 ⫽ f(x). Verify the identity f(x) ⫽ g(x). 57. csc4 x ⫺ 1 ⫽ 2 ⫹ cot2 x cot2 x 58. sec4 x ⫺ 1 ⫽ 2 ⫹ tan2 x tan2 x 79. f(x) ⫽ 1 ⫺ sin2 x ⫹ sin x cos x tan x 59. 1 ⫹ sin v cos v ⫽ cos v 1 ⫺ sin v 60. 1 ⫹ cos x sin x ⫽ 1 ⫺ cos x sin x 80. f(x) ⫽ 1 ⫹ sin x cos x ⫺ 2 cos x 2 ⫹ 2 sin x 81. f(x) ⫽ cos2 x 1 ⫹ sin x ⫺ cos2 x 82. f(x) ⫽ tan x sin x 1 ⫺ cos x 83. f(x) ⫽ 1 ⫹ cos x ⫺ 2 cos2 x sin2 x ⫺ 1 ⫺ cos x 1 ⫹ cos x 84. f(x) ⫽ 3 sin x ⫺ 2 sin x cos x 1 ⫹ cos x ⫺ 1 ⫺ cos x sin x Use a graphing utility to test whether each of the following is an identity. If an equation appears to be an identity, verify it. If the equation does not appear to be an identity, find a value of x for which both sides are defined but are not equal. 61. cos (⫺x) sin (⫺x) ⫽ ⫺1 62. ⫽1 cos (⫺x) tan (⫺x) sin x cot (⫺x) 63. sin x ⫽ ⫺1 cos x tan (⫺x) 64. cos x ⫽1 sin (⫺x) cot (⫺x) 65. sin x ⫹ cos2 x ⫽ sec x sin x 66. 1 ⫺ tan2 x ⫽ tan2 x 1 ⫺ cot2 x 67. sin x ⫹ cos2 x ⫽ csc x sin x 68. tan2 x ⫺ 1 ⫽ tan2 x 1 ⫺ cot2 x tan x 1 ⫽ 69. sin x ⫺ 2 tan x cos x ⫺ 2 70. cos x cos x ⫹ ⫽ 2 sec x 1 ⫺ sin x 1 ⫹ sin x 71. tan x 1 ⫽ sin x ⫹ 2 tan x cos x ⫺ 2 72. cos x cos x ⫺ ⫽ 2 csc x sin x ⫹ 1 sin x ⫺ 1 C Verify that Problems 73–78 are identities. 73. 2 sin2 x ⫹ 3 cos x ⫺ 3 2 cos x ⫺ 1 ⫽ sin2 x 1 ⫹ cos x 74. 3 cos2 z ⫹ 5 sin z ⫺ 5 3 sin z ⫺ 2 ⫽ cos2 z 1 ⫹ sin z 75. tan u ⫹ sin u sec u ⫹ 1 ⫺ ⫽0 tan u ⫺ sin u sec u ⫺ 1 tan x ⫹ tan y sin x cos y ⫹ cos x sin y ⫽ 76. cos x cos y ⫺ sin x sin y 1 ⫺ tan x tan y Each of the equations in Problems 85–92 is an identity in certain quadrants associated with x. Indicate which quadrants. 85. 兹1 ⫺ cos2 x ⫽ ⫺sin x 86. 兹1 ⫺ sin2 x ⫽ cos x 87. 兹1 ⫺ cos2 x ⫽ sin x 88. 兹1 ⫺ sin2 x ⫽ ⫺cos x 89. 兹1 ⫺ sin2 x ⫽ 兩cos x兩 90. 兹1 ⫺ cos2 x ⫽ 兩sin x兩 91. sin x ⫽ tan x 兹1 ⫺ sin2 x 92. sin x ⫽ ⫺tan x 兹1 ⫺ sin2 x In calculus, trigonometric substitutions provide an effective way to rationalize the radical forms 兹a2 ⫺ u2 and 兹a2 ⫹ u2, which in turn leads to the solution to an important class of problems. Problems 93–96 involve such transformations. [Recall: 兹x2 ⫽ 兩x兩 for all real numbers x.] 93. In the radical form 兹a2 ⫺ u2, a ⬎ 0, let u ⫽ a sin x, ⫺/2 ⬍ x ⬍ /2. Simplify, using a basic identity, and write the final form free of radicals. 94. In the radical form 兹a2 ⫺ u2, a ⬎ 0, let u ⫽ a cos x, 0 ⬍ x ⬍ . Simplify, using a basic identity, and write the final form free of radicals. 95. In the radical form 兹a2 ⫹ u2, a ⬎ 0, let u ⫽ a tan x, 0 ⬍ x ⬍ /2. Simplify, using a basic identity, and write the final form free of radicals. 96. In the radical form 兹a2 ⫹ u2, a ⬎ 0, let u ⫽ a cot x, 0 ⬍ x ⬍ /2. Simplify, using a basic identity, and write the final form free of radicals.