688 CHAPTER 7 Trigonometric Identities and Equations Work each problem. 79. Let cos x = 1 5. Find all possible values of sec x - tan x sin x . 80. Let csc x = -3. Find all possible values of sin x + cos x sec x . Use a graphing calculator to make a conjecture about whether each equation is an identity. 81. cos 2x = 1 - 2 sin2 x 82. 2 sinx = sin2x 83. sinx = 21 - cos2 x 84. cos 2x = cos2 x - sin2 x Relating Concepts For individual or collaborative investigation (Exercises 85 –90) Previously we graphed functions of the form y = c + a # ƒ3b1x - d24 with the assumption that b 70. To see what happens when b 60, work Exercises 85 –90 in order. 85. Use an even-odd identity to write y = sin1-2x2 as a function of 2x. 86. How is the answer to Exercise 85 related to y = sin2x? 87. Use an even-odd identity to write y = cos1-4x2 as a function of 4x. 88. How is the answer to Exercise 87 related to y = cos 4x? 89. Use the results from Exercises 85–88 to rewrite the following with a positive value of b. (a) y = sin1-4x2 (b) y = cos1-2x2 (c) y = -5 sin1-3x2 90. Write a short response to this statement, which is often used by one of the authors of this text in trigonometry classes: Students who tend to ignore negative signs should enjoy graphing functions involving the cosine and the secant. 7.2 Verifying Trigonometric Identities ■ Strategies ■ Verifying Identities by Working with One Side ■ Verifying Identities by Working with Both Sides Strategies One of the skills required for more advanced work in mathematics, especially in calculus, is the ability to use identities to write expressions in alternative forms. We develop this skill by using the fundamental identities to verify that a trigonometric equation is an identity (for those values of the variable for which it is defined). CAUTION The procedure for verifying identities is not the same as that for solving equations. Techniques used in solving equations, such as adding the same term to each side, and multiplying each side by the same term, should not be used when working with identities.
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