5.4 Sum and Difference Formulas

Transcription

1 380 Capter 5 Analtic Trigonometr 5. Sum and Difference Formulas Using Sum and Difference Formulas In tis section and te following section, ou will stud te uses of several trigonometric identities and formulas. Sum and Difference Formulas See te proofs on page 0.) sinu v sin u cos v cos u sin v sinu v sin u cos v cos u sin v cosu v cos u cos v sin u sin v cosu v cos u cos v sin u sin v tanu v tanu v tan u tan v 1 tan u tan v tan u tan v 1 tan u tan v Wat ou sould learn Use sum and difference formulas to evaluate trigonometric functions, verif trigonometric identities, and solve trigonometric equations. W ou sould learn it You can use sum and difference formulas to rewrite trigonometric expressions. For instance, Exercise 79 on page 385 sows ow to use sum and difference formulas to rewrite a trigonometric expression in a form tat elps ou find te equation of a standing wave. Exploration Use a graping utilit to grap 1 cosx and cos x cos in te same viewing window. Wat can ou conclude about te graps? Is it true tat cosx cos x cos? Use a graping utilit to grap 1 sinx and sin x sin in te same viewing window. Wat can ou conclude about te graps? Is it true tat sinx sin x sin? Examples 1 and sow ow sum and difference formulas can be used to find exact values of trigonometric functions involving sums or differences of special angles. Example 1 Evaluating a Trigonometric Function Find te exact value of cos 75. To find te exact value of cos 75, use te fact tat Consequentl, te formula for cosu v ields cos 75 cos30 5 cos 30 cos 5 sin 30 sin Tr cecking tis result on our calculator. You will find tat cos Now tr Exercise 1. Ricard Megna/Fundamental Potograps Prerequisite Skills To review sines, cosines, and tangents of special angles, see Section.3.

2 Section 5. Sum and Difference Formulas 381 Example Evaluating a Trigonometric Function Find te exact value of sin 1. Using te fact tat 1 3 togeter wit te formula for sinu v, ou obtain sin 1 sin sin 3 cos cos 3 sin Now tr Exercise u 5 = 3 x Example 3 Evaluating a Trigonometric Expression Find te exact value of sinu v given sin u were and cos v 1 were 5, 0 < u < 13, < v <. Figure 5. Because sin u 5 and u is in Quadrant I, cos u 35, as sown in Figure 5.. Because cos v 113 and v is in Quadrant II, sin v 513, as sown in Figure 5.5. You can find sinu v as follows. sinu v sin u cos v cos u sin v Now tr Exercise = 5 Figure v 1 x Example An Application of a Sum Formula Write cosarctan 1 arccos x as an algebraic expression. 1 Tis expression fits te formula for cosu v. Angles u arctan 1 and v arccos x are sown in Figure 5.6. u 1 cosu v cosarctan 1cosarccos x sinarctan 1sinarccos x 1 x 1 1 x Now tr Exercise 3. x 1 x. 1 1 x v x Figure 5.6

3 38 Capter 5 Analtic Trigonometr Example 5 Proving a Cofunction Identit Prove te cofunction identit Using te formula for cosu v, ou ave cos x cos 0cos x 1sin x sin x. Now tr Exercise 63. cos x sin x. cos x sin sin x Sum and difference formulas can be used to derive reduction formulas involving expressions suc as sin n and cos n were n is an integer., Example 6 Deriving Reduction Formulas Simplif eac expression. a. cos 3 b. tan 3 a. Using te formula for cosu v, ou ave cos 3 cos cos 3 sin sin 3 cos 0 sin 1 sin. b. Using te formula for tanu v, ou ave tan 3 tan tan 3 1 tan tan 3 tan 0 1 tan 0 tan. Note tat te period of tan is, so te period of is te same as te period of tan. Now tr Exercise 67. tan 3

4 Example 7 Solving a Trigonometric Equation Find all solutions of sin x in te interval 0,. sin x 1 Algebraic Using sum and difference formulas, rewrite te equation as sin x cos cos x sin sin x cos cos x sin 1 sin x cos 1 sin x 1 Section 5. Sum and Difference Formulas 383 Grapical Use a graping utilit set in radian mode to grap sin x as sown in Figure 5.7. Use te zero or root feature or te zoom and trace features to approximate te x-intercepts in te interval 0, to be 5 x 3.97 and sin x 1, x So, te onl solutions in te interval 0, are 5 x and x 7. Now tr Exercise 71. sin x 1 sin x. π π = sin x + + sin x Figure 5.7 Te next example was taken from calculus. It is used to derive te formula for te derivative of te cosine function. Example 8 Verif An Application from Calculus cosx cos x Using te formula for cosu v, ou ave cosx cos x cos 1 cos x sin x sin, 0. For instructions on ow to use te zero or root feature and te zoom and trace features, cos x cos sin x sin cos x cos xcos 1 sin x sin cos 1 cos x sin x sin. Now tr Exercise 93. TECHNOLOGY SUPPORT see Appendix A; for specific kestrokes, go to tis textbook s Online Stud Center.

5 38 Capter 5 Analtic Trigonometr 5. Exercises See for worked-out solutions to odd-numbered exercises. Vocabular Ceck Fill in te blank to complete te trigonometric formula. 1. sinu v. cosu v 3. tanu v. sinu v 5. cosu v 6. tanu v In Exercises 1 6, find te exact value of eac expression. 1. a) cos0 0 b) cos 0 cos 0. a) sin05 10 b) sin 05 sin a) cos b) cos cos a) b) sin 5 sin sin a) sin b) sin 315 sin a) sin b) sin 7 sin In Exercises 7, find te exact values of te sine, cosine, and tangent of te angle In Exercises 3 30, write te expression as te sine, cosine, or tangent of an angle. 3. cos 60 cos 0 sin 60 sin 0. sin 110 cos 80 cos 110 sin 80 tan 35 tan tan 35 tan 86 tan 15 tan tan 15 tan 9 7. sin 3.5 cos 1. cos 3.5 sin Numerical, Grapical, and Algebraic Analsis In Exercises 31 3, use a graping utilit to complete te table and grap te two functions in te same viewing window. Use bot te table and te grap as evidence tat 1. Ten verif te identit algebraicall cos 0.96 cos 0. sin 0.96 sin 0. cos 9 cos sin 7 9 sin 7 sin 9 cos 8 1 sin 6 x, 1 cos 5 x, cos 9 sin 1 cos x 3 sin x cosx cosx, cos x 3. 1 sinx sinx, sin x x In Exercises 35 38, find te exact value of te trigonometric 5 function given tat sin u and cos v Bot u and v are in Quadrant II.) 35. sinu v 36. cosv u 37. tanu v 38. sinu v cos x sin x In Exercises 39, find te exact value of te trigonometric function given tat 17 and cos v 5. Bot u and v are in Quadrant III.) cosu v tanu v sinv u. cosu v 8

6 Section 5. Sum and Difference Formulas 385 In Exercises 3 6, write te trigonometric expression as an algebraic expression. 3. sinarcsin x arccos x. cosarccos x arcsin x 5. sinarctan x arccos x 6. cosarcsin x arctan x In Exercises 7 5, find te value of te expression witout using a calculator. 7. sinsin 1 1 cos cossin 1 1 cos sinsin 1 1 cos In Exercises 55 58, evaluate te trigonometric function witout using a calculator. 55. sin sin sincos In Exercises 59 6, use rigt triangles to evaluate te expression. 59. sin cos sin coscos 1 1 cos 1 1 sin sin cos1 cos cos 1 1 sin1 1 tan sin 1 0 sin 1 1 tan 1 cos sin1 0 cos sin 1 1 cos sin cos1 sin tan 1 3 sin1 3 5 tan sin 1 5 cos cos cos 1 1 In Exercises 71 7, find te solutions) of te equation in te interval [0,. Use a graping utilit to verif our results In Exercises 75 78, use a graping utilit to approximate te solutions of te equation in te interval [0, sin x cos x sin x tanx cos x 0 tan x cos x Standing Waves Te equation of a standing wave is obtained b adding te displacements of two waves traveling in opposite directions see figure). Assume tat eac of te waves as amplitude A, period T, and wavelengt. If te models for tese waves are 1 A cos t T x sow tat 1 A cos t T t = 0 3 sin x cos x 6 cos x 6 1 tanx sinx 0 sin x 3 tan x 0 cos x cos x 0 1 and 0 x cos. A cos t T x + 1 In Exercises 63 70, verif te identit sin 6. x cos x 65. tanx tan x tan x 66. tan 1 tan 1 tan 67. sinx sinx sin x cos 68. cosx cosx cos x cos 69. cosx cosx cos x sin 70. sinx sinx sin x sin sin3 x sin x t t = 1 8 = 8 T T 1 + 1

7 386 Capter 5 Analtic Trigonometr 80. Harmonic Motion A weigt is attaced to a spring suspended verticall from a ceiling. Wen a driving force is applied to te sstem, te weigt moves verticall from its equilibrium position, and tis motion is modeled b were is te distance from equilibrium in feet) and t is te time in seconds). a) Use a graping utilit to grap te model. b) Use te identit were C arctanba, a > 0, to write te model in te form a b sinbt C. Use a graping utilit to verif our result. c) Find te amplitude of te oscillations of te weigt. d) Find te frequenc of te oscillations of te weigt. Sntesis True or False? In Exercises 81 and 8, determine weter te statement is true or false. Justif our answer sin t 1 cos t a sin B b cos a b sinb C cosu ± v cos u ± cos v sin x 11 cos x In Exercises 83 86, verif te identit n cos, n is an integer n sin, n is an integer. 85. a sin B b cos B a b sinb C, were C arctanba and a > a sin B b cos B a b cosb C, were C arctanab and b > 0. cosn sinn In Exercises 87 90, use te formulas given in Exercises 85 and 86 to write te expression in te following forms. Use a graping utilit to verif our results. a) a 1 b sinb 1 C b) a 1 b cosb C 87. sin cos sin cos sin 3 5 cos sin cos In Exercises 91 and 9, use te formulas given in Exercises 85 and 86 to write te trigonometric expression in te form a sin B 1 b cos B. 91. sin 9. 5 cos 93. Verif te following identit used in calculus. sinx sin x 9. Exploration Let x 3 in te identit in Exercise 93 and define te functions f and g as follows. f sin3 sin3 g cos 3 sin a) Wat are te domains of te functions f and g? b) Use a graping utilit to complete te table. c) Use a graping utilit to grap te functions f and g. d) Use te table and grap to make a conjecture about te values of te functions f and g as Conjecture Tree squares of side s are placed side b side see figure). Make a conjecture about te relationsip between te sum u v and w. Prove our conjecture b using te identit for te tangent of te sum of two angles. s 96. a) Write a sum formula for sinu v w. b) Write a sum formula for tanu v w. Skills Review cos x sin sin x1 cos sin u v w s s s In Exercises , find te x- and -intercepts of te grap of te equation. Use a graping utilit to verif our results x xx 7 x 9 5 x 3x 0 In Exercises , evaluate te expression witout using a calculator arccos arctan sin tan cos f g