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CHAPTER 2.4 CHAPTER 2 ANALYTICAL TRIGONOMETRY PART 4 –Sum and Difference Formulas TRIGONOMETRY MATHEMATICS CONTENT STANDARDS: 9.0 - Students compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points. 10.0 - Students demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use those formulas to prove and/or simplify other trigonometric identities. OBJECTIVE(S): Students will learn the sum and difference formulas. Students will learn how to evaluate trigonometric expressions using the sum and difference formulas. Students will learn how to solve trigonometric equations using the sum and difference formulas. Using Sum and Difference Formulas In this and the following section, you will study the uses of several trigonometric identities and formulas. Sum and Difference Formulas b g sin u v sin u cos v cos u sin v b g 1tan tanu utantanvv tan u v b g sin u v sin u cos v cos u sin v b g cos u v cos u cos v sin u sin v b g cos u v cos u cos v sin u sin v b g 1tan tanu utantanvv tan u v CHAPTER 2.4 EXAMPLE 1: Evaluating a Trigonometric Function Find the exact value of cos750 . To find the exact value of cos750 , use the fact that 750 = __________ + _________. Consequently, the formula for cos u v yields b g cos750 = = = = Try checking this result on your calculator: cos750 EXAMPLE 2: Evaluating a Trigonometric Function Find the exact value of sin . 12 = ___________ - _______________ together with the formula 12 for sin u v , you obtain Using the fact that b g sin = 12 = = = = CHAPTER 2.4 EXAMPLE 3: Evaluating a Trigonometric Expression Find the exact value of sin 420 cos 120 cos 420 sin 120 Recognizing that this expression fits the formula ___________________, you can write sin 420 cos 120 cos 420 sin 120 = = = EXAMPLE 4: An Application of a Sum Formula Evaluate cos arctan 1 arccos x as an algebraic expression. b g b g This expression fits the formula for cos u v . Angles u = _________________ and v = ______________. b g cos u v = = = CHAPTER 2.4 1.) Find the exact value: a. sin150 c. tan1950 DAY 1 b. cos 5 12 CHAPTER 2.4 EXAMPLE 5: Proving a Cofunction Identity Prove the cofunction identity cos x sin x . 2 Using the formula for cosu v , you have cos x 2 = = = Sum and difference formulas can be used to rewrite expressions such as n sin and 2 n cos , where n is an integer 2 as expressions involving only sin or cos . The resulting formulas are called reduction formulas. EXAMPLE 6: Deriving Reduction Formulas Simplify each expression. 3 a.) cos 2 Using the formula for cosu v , you have 3 cos = 2 = = CHAPTER 2.4 b.) tan 3 Using the formula for tan u v , you have tan 3 = = = EXAMPLE 7: Solving a Trigonometric Equation Find all solutions of sin x sin x 1 in the interval 0,2 . 4 4 Using the sum and difference formulas, rewrite the equation as sin x sin x = -1 4 4 Therefore, the only solutions in the interval 0,2 are: x = ____________and x = _____________ The next example was taken from calculus. It is used to derive the formula for the derivative of the sine function. CHAPTER 2.4 EXAMPLE 8: An Application from Calculus Verify that sin x h sin x sinh 1 cosh cos x sin x h h h where h 0 . Using the formula for sin u v , you have sin x h sin x h = = = CHAPTER 2.4 2.) Rewrite in a simpler form: a. cos 200 cos 300 sin 200 sin 300 DAY 2 b. sin 2 2 cos cos sin 9 10 9 10 CHAPTER 2.4 3.) Verify that the two functions are equivalent: 5 y1 cos x 4 y2 F IJ G H K 2 bcos x sin xg 2 4.) If sin u 7 4 3 v 2 , find cos u v . , where u , and cosv , where 25 2 5 2 b g CHAPTER 2.4 b g b g 5.) Verify sin x y sin x y sin 2 x sin 2 y . 6.) Write as an algebraic expression: b cos arcsin x arctan 2 x g CHAPTER 2.4 g 7.) Find all solutions on 0,2 . I F G H 2J K 3 tanb xg 0 2 sin x DAY 3