CHAPTER 5 Analytic Trigonometry - Crunchy Math

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C H A P T E R 5Analytic TrigonometrySection 5.1Using Fundamental IdentitiesSection 5.2Verifying Trigonometric Identities . . . . . . . . . . . . . 391Section 5.3Solving Trigonometric EquationsSection 5.4Sum and Difference Formulas . . . . . . . . . . . . . . . 413Section 5.5Multiple-Angle and Product-to-Sum FormulasReview Exercises. . . . . . . . . . . . . 401. . . . . . 428. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 Houghton Mifflin Company. All rights reserved.Practice Test. . . . . . . . . . . . . . . 379

C H A P T E R 5Analytic TrigonometrySection 5.1 Using Fundamental IdentitiesYou should know the fundamental trigonometric identities.(a) Reciprocal Identities11sin u csc u csc usin ucos u 1sec usec u 1cos utan u 1sin u cot u cos ucot u 1cos u tan usin u(b) Pythagorean Identitiessin2 u cos2 u 11 tan2 u sec2 u1 cot2 u csc2 u(c) Cofunction Identities 2 u cos u tan u cot u2 sec u csc u2 Houghton Mifflin Company. All rights reserved.sin 2 u sin u cot u tan u2 csc u sec u2cos(d) Negative Angle Identitiessin x sin xcsc x csc xcos x cos xsec x sec xtan x tan xcot x cot x You should be able to use these fundamental identities to find function values.You should be able to convert trigonometric expressions to equivalent forms by using the fundamentalidentities.You should be able to check your answers with a graphing utility.Vocabulary Check1. sec u2. tan u3. cot u4. csc u5. tan 2 u6. csc 2 u7. sin u8. sec u9. tan u10. cos u379

Chapter 5Analytic Trigonometry 31, x is in Quadrant I.1. sin x , cos x 22tan x 31 21 3 3 2 3sin cot x 32 32 3 33 33 22 is in Quadrant IV., is in Quadrant I. 3cos cot sin 2sec 3. sec 2, sin 312cot csc x 2sec x 2. csc 2, tan 4. tan x 3 33 322 33, cos x 32, x is in Quadrant III. 1 23 cos 211 sec 22sin x tan sin 2 2 1cos 2 2csc x 1 2sin xcot 1 1tan sec x 122 3 cos x3 3cot x 13 3tan x 3csc 25. tan x cot x 7 25, sec x x is in Quadrant III.2424247cos x sin x 1 cos2 x csc x 6. cot 5, sin cos cot 2425725125 sin x77. sec 178, sin , is in Quadrant II.1517cos 1517csc 178tan 8 178 15 1715cot 158 2 2626 sin 14 12, is in Quadrant II 5 2626tan 11 cot 5csc 126 26sin 26sec 1 26 26 cos 5 265 Houghton Mifflin Company. All rights reserved.380

Section 5.18. cos 2 x 5, cos x 5, x is in Quadrant I.349. sin x sin x 1 45 tan x sin x3 cos x 5csc x 15 sin x 3cot x 51 tan x2sec x 15 cos x 4sec x 13 5 cos x5cot x 41 tan x 3csc x 31 sin x 22 5sin x 22 5sin x , tan x x is in Quadrant II.35353 4 4cos x 1 sin2 x cos 52 6 cot 6155 6 cos x 2 612cot x 1 2 6tan x1 2 4 2 2 13 2 3 2sin tan cos 2 2 2 2 3 2413. csc is undefined and cos 0 .sin 0cos 1tan 0cot is undefined.sec 1 5512csc tan2 sec 2 1 9 1 8 tan 8 2 2 1 51 45 25 25133 5 5 12. sec 3, tan 0, is in Quadrant II.csc 1sin 1 cos2 12sec x cot sec tan2 1 52 6cos x 5cos 1 94 3511. tan 2, sin 0 is in Quadrant III.11 csc x 5sin x1tan x cos x 538122 sin x 33sin x 10. csc x 5, cos x 0, x is in Quadrant I. Houghton Mifflin Company. All rights reserved.Using Fundamental Identities 52

382Chapter 5Analytic Trigonometry14. tan is undefined, sin 0. 2tan sin is undefined cos 0.cos sin 1 02 115. sec x cos x 1cos xcsc 1 1sin sec 1is undefined.cos cot cos 0 0sin 1 cos x 116. tan x csc x Matches (a).Matches (d).17. cot2 x csc2 x cot2 x 1 cot2 x 118. 1 cos2 x csc x sin2 xMatches (b).19.sin x 11 sec xcos x sin x cos x sin1 x sin xMatches (f). sin xsin x tan xcos x cos x20.cos xsin 2 x cot xcos 2 x sin xMatches (c).Matches (e).21. sin x sec x sin x cos1 x tan x22. cos2 x sec2 x 1 cos2 x tan2 x sin2 xMatches (c).Matches (b).23. sec4 x tan4 x sec2 x tan2 x sec2 x tan2 x 24. cot x sec x sec2 x tan2 x 1 cos xsin x11 cos x sin x csc x sec2 x tan2 xMatches (a).25.sec2 x 1 tan2 xsin2 x 22sin xsin xcos2 x1 sin2 x sec2 x26.cos2 2 x sin2 xsin x sin xcos xcos xcos x tan x sin xMatches (e).Matches (d).27. cot x sin x cos xsin x cos xsin x28. cos tan cos 29. sin csc sin sin csc sin2 sin 1 sin sin2 1 sin2 cos2 sin cos sin 30. sec2 x 1 sin2 x sec2 x sec2 x sin2 x sec2 x 1cos2 x sec2 x sin2 xcos2 x sin2 x sec2 x tan2 x 1 Houghton Mifflin Company. All rights reserved.Matches (f).

Section 5.131.csc x1 cot xsin x33. sec35. sinsintansin x1 cos x cos x sec x 1 sin cotcos 1cos sin cossin 32.1sec sin tan csc cos 34.tan2 sin2 sec2 cos2 1 2 x csc x cos x sin1 x cot xUsing Fundamental Identities36. cotsin2 cos2 1 sec2 1sin2 cos2 sin2 1cos2 cos2 2 x cos x tan x cos x 37.1 sin2 ycos2 y 1 sin y1 sin y 38.sin xcos x cos x sin x11 sin2 xcot2 x 1 csc2 x 1 sin y 1 sin y 1 sin y 1 sin y39. sin cos cot sin cos sin2 cos2 sin 1 csc sin Houghton Mifflin Company. All rights reserved.40. sec tan csc 1 41.cos sin cos cos 1 sin 1 sin 11 1 sin 1cos sin 11 1 sin 1 sin cos sin 1cos sin 1 sin2 1cos2 cot cos sin 1 sin 1 sin cos 1 sin 1 sin2 cos 1 sin cos2 1 sin sec tan cos 42.3831 csc 1 csc cot cos cos csc 1 1 sec cos

38443.44.Chapter 5Analytic Trigonometry1 cos sin 1 2 cos cos2 sin2 sin 1 cos sin 1 cos 2 2 cos sin 1 cos 2 1 cos sin 1 cos 2 2 csc sin sin cos cos sin 1 cot 1 tan sin cos cot tan 45. csc tan 47. 1 1sin sin cos sin sin cos cos2 sin2 sin cos 1 sec csc sin cos 1 cos cos sec 46. sin csc sin2 1 sin2 cos2 sin2 1 cos sin2 1 cos 1 cos cos2 cos 1 cos cos cos 1 1 cos 48.tan 1 sec tan2 1 2 sec sec2 1 sec tan 1 sec tan 2 sec2 2sec 1 sec tan 2 sec sec 1 1 sec tan 2 sec 2 csc tan cot cos sin 49.csc sin cos sin cos sin csc50. 2 tan sec csc tan Houghton Mifflin Company. All rights reserved. cos

Section 5.151. cot2 x cot2 x cos2 x cot2 x 1 cos2 x 53.55. tan4 x 2 tan2 x 1 tan2 x 1 238552. sec 2 x tan 2 x sec 2 x sec 2 x tan 2 x 1 sec 2 x sec 2 x sec 4 xcos2 x 2sin x cos2 xsin2 xcos2 x 4 cos x 2 cos x 2 cos x 2cos x 2cos x 2Using Fundamental Identities54.csc2 x 1 csc x 1 csc x 1 csc x 1csc x 1csc x 156. 1 2 sin2 x sin4 x 1 sin2 x 2 sec2 x 2 sec4 x cos2 x)2 cos4 x57. sin4 x cos4 x sin2 x cos2 x sin2 x cos2 x 1 sin2 x cos2 x sin2 x cos2 x58. sec4 x tan4 x sec2 x tan2 x sec2 x tan2 x sec2 x tan2 x59. csc3 x csc2 x csc x 1 csc2 x csc x 1 csc x 1 csc2 x 1 csc x 1 cot2 x csc x 1 60. sec3 x sec2 x sec x 1 sec2 x sec x 1 sec x 1 sec2 x 1 sec x 1 tan2 x sec x 1 61. sin x cos x 2 sin2 x 2 sin x cos x cos2 x62. tan x sec x tan x sec x tan2 x sec2 x sin2 x cos2 x 2 sin x cos x 1 1 2 sin x cos x Houghton Mifflin Company. All rights reserved.63. csc x 1 csc x 1 csc2 x 1 cot2 x64. 5 5 sin x 5 5 sin x 25 25 sin2 x 25 1 sin2 x 25 cos2 x65.111 cos x 1 cos x 1 cos x 1 cos x 1 cos x 1 cos x 66.11sec x 1 sec x 1 sec x 1 sec x 1 sec x 1 sec x 1 21 cos2 x sec x 1 sec x 1sec2 x 1 2sin2 x 2tan2 x 2 csc2 x 2 tan x 12 2 cot2 x

386Chapter 567. tan x 68.69.Analytic Trigonometrysec2 x tan2 x sec2 x 1 cot x tan xtan xtan xcos x1 sin x cos2 x 1 2 sin x sin2 x 1 sin xcos xcos x 1 sin x 2 2 sin xcos x 1 sin x 2 2 sec xcos x1 cos2 ysin2 y 1 cos y1 cos y 70.5tan x sec x tan x sec x 5 tan x sec x tan2 x sec2 x 5 tan x sec x 1tan x sec x 1 cos y 1 cos y 1 cos y 5 sec x tan x 1 cos y71.3sec x tan x sec x tan x 3 sec x tan x sec2 x tan2 x 3 sec x tan x 1sec x tan x72.tan2 xcsc x 1 csc x 1 tan2 x csc x 1 csc2 x 1 tan2 x csc x 1 cot2 xcsc x 1 tan2 x csc x 1 tan2 x 3 sec x tan x tan4 x csc x 1 2 x , y2 sin x1.0y1 50.93200.9854Conjecture: y1 y21.5074. y1 cos x sin x tan x, y2 sec 1.85082.75975.8835y1 y201.50y275. y1 re: y1 y2cos x1 sin x, y2 1 sin xcos x12y1 25.331911.681401.50Conjecture: y1 y2 Houghton Mifflin Company. All rights reserved.73. y1 cos

Section 5.1Using Fundamental Identities76. y1 sec4 x sec2 x, y2 tan2 x tan4 x1200y1 308750.38691163.614377. y1 cos x cot x sin x csc x01.50Conjecture: y1 y278. sin x cot x tan x sec x44 2 2 2p 42p 479. y1 sec x cos x tan x1 sin x2 2 2 280. y1 1 1 sin cos 2 cos 1 sin y1 and y2 sin y1 and y2 cos 4y1 and y2 tan 44y1y1y2 2 y2y2 2 2 2 2 2 y1 Houghton Mifflin Company. All rights reserved. 4y1 and y2 41 csc sin y1 and y2 4y11 sec cos y1 and y2 4y2 2 41 cot tan 4y1y1 y2 2 2 It appears that 4y2 2 2 4 41 1 sin cos sec .2 cos 1 sin 81. 25 x2 25 5 sin 2, x 5 sin 25 25 sin2 82. Let x 2 cos . 64 16x2 64 16 4 cos2 25 1 sin2 8 1 cos2 25 cos2 8 sin 5 cos 2 387

Analytic Trigonometry83. x2 9 3 sec 2 9, x 3 sec 9sec2 84. Let x 10 tan . x2 100 10 tan 2 100 9 9 sec2 1 100 tan2 1 9 tan2 100 sec2 3 tan 10 sec 85. x 3 sin , 0 286. x 2 cos , 0 9 x2 9 9 sin2 4 x2 4 4 cos2 9 cos2 3 cos 87. 2x 3 tan , 0 4 sin2 2 sin 288. 3x 2 tan , 0 4 sec2 2 sec 9 sec2 3 sec 290. 3x 5 sec , 0 25 tan2 5 tan 9 tan2 3 tan 292. x 5 cos , 0 5 sin2 5 sin 2 cos2 2 cos 94. cos 1 sin2 93. sin 1 cos2 Let y1 sin x and y2 1 cos2 x, 0 x 2 .y1 y2 for 0 x , so we haveLet y1 cos and y2 1 sin2 .y1 y2 forsin 1 cos2 for 0 . 3 .2222y1y22 02 y2y2y1 2 295. sec 1 tan2 Let y1 41and y2 1 tan2 x, 0 x 2 .cos x 3 y1 y2 for 0 x and x 2 , so we22have sec 1 tan2 for 0 2 5 x2 5 5 cos2 2 x2 2 2 sin2 0 2 9x2 25 25 sec2 25 16x2 9 9 sec2 991. x 2 sin , 0 2 9x2 4 4 tan2 4 4x2 9 9 tan2 989. 4x 3 sec , 0 23 and 2 .22y22 0y1 4 Houghton Mifflin Company. All rights reserved.Chapter 5388

Section 5.196. tan sec2 10 Using Fundamental Identities97. ln cos ln sin ln 3 , 2298. ln csc ln tan ln csc tan ln1 sin xsec x ln cos x 1 sin x 100. ln cot t ln 1 tan2 t ln cot t 1 tan2 t lncos ln cot sin 99. ln 1 sin x ln sec x ln ln sec 389 1 tan2 t tan t101. Let 7 . Then6cos cos 37 62 1 sin2 32.1tan2 t tan ttan t ln cot t tan t102. Let 2 . Then3tan tan103. Let 2 33 sec2 1 3.104. Let . Thensin sin105. Let sec sec 15 . Then3 1 tan2 1. 35 323 . Then4 Houghton Mifflin Company. All rights reserved.cot cot csc2 1 1.108. tan2 1 sec2 1.0622 1 sec 346 cos 3461.0622 sec 3.1 2 2 2 cot2771.6360 0.6360 1 2 sin 1.00173 cos 3.1 10.9848 0.9848(b) 0.8(b) 3.1cos21.00173.1.8107 0.8107 1cos 90 80 sin 8022 1 cot2 2.(a) 80 tan 346 2 1 tan 3.1 2 1(b) csc2109. cos(a) 34627 24107. (a) csc2 132 cot2 1323 14 37 . Then4csc csc106. Let 1 cos2 2 0.8 sin 0.80.7174 0.7174

390Chapter 5Analytic Trigonometry110. sin sin 111. csc x cot x cos x (a) 2501sin x cos x cos xsin x cos x csc2 x 1 sin 250 0.9397 sin 250 0.9397 cos x cot2 x12(b) sin 2 1 0.4794 12 0.4794 sin 112. sec x tan x sin x 1 sin x sin xcos x cos x sin x sec2 x 1 113. True for all sin sin x tan2 x114. Falsecos 0 sec 4n csc sin sin 11115. As x , sin x 1 and csc x 1.2117. As x , tan x 21116. As x 0 ,cos x 1 and sec x and cot x 0.1 1.cos x118. As x ,1 sin x.119. sin 120. cos cos 1 sin2 sin 1 cos2 tan sin sin cos 1 sin2 tan 1 cos2 sin cos cos csc 1sin csc 11 1 cos2 sin sec 1cos cot 1cos 1 cos2 tan sec 1 1 sin2 cot 1 sin2 sin The sign or depends on the choice of .The sign or depends on the choice of . Houghton Mifflin Company. All rights reserved.sin x 0 and csc x

Section 5.2121. sin oppadj, cos hyphyp391122. sin2 cos2 1sin2 cos2 1 2sin2 sin2 sin From the Pythagorean Theorem, opp 2 adj 2 hyp 21 cot2 csc2 sin2 cos2 1.hVerifying Trigonometric Identitiessin2 cos2 1sin2 cos2 1 2cos cos2 cos2 otan2 1 sec2 θa123. f x 1sin x2124. f x 2 tany x212 Period: 2 12 2 2Period:x 31Amplitude:2212 1232y4 1x 3 1 113 2 3 4125. f x 1 cot x 24 126. f x y4Period: 3cos x 32Amplitude: π 1π32y6x5 Houghton Mifflin Company. All rights reserved. 2431 2 πSection 5.2 πVerifying Trigonometric IdentitiesYou should know the difference between an expression, a conditional equation, and an identity.You should be able to solve trigonometric identities, using the following techniques.(a) Work with one side at a time. Do not “cross” the equal sign.(b) Use algebraic techniques such as combining fractions, factoring expressions, rationalizingdenominators, and squaring binomials.(c) Use the fundamental identities.(d) Convert all the terms into sines and cosines.π2πx

392Chapter 5Analytic TrigonometryVocabulary Check1. conditional2. identity3. cot u4. sin u5. tan u6. cos u7. cos2 u8. cot u9. sin u10. sec u1. sin t csc t sin t3.csc2 x1 2cot xsin x sin t 11sin x cos x sin x2. sec y cos y 14. cos x csc x sec x5. cos2 sin2 1 sin2 sin2 1cos y 1cos ysin2 tsin2 t cos2 t2 tan tsin2 tcos2 t6. cos2 sin2 cos2 1 cos2 2 cos2 1 1 2 sin2 7. tan2 6 tan2 1 58. 2 csc2 z 2 cot2 z 1 1 cot2 z sec2 59. 1 sin x 1 sin x 1 sin2 x cos2 xcos x1 cos x sec x tan xsin xcos2 x sin x 1 sin2 xsin .14090.02935y1 y21 sin xsin x0 csc x sin x1 1csc x 1sin x 12. y1 1 sin x1 sin x 1.5011 sin xsin x 1 sin .07291.0148101 sin x csc x y2y1 y201.50 Houghton Mifflin Company. All rights reserved.11.10. tan2 y csc2 y 1 tan2 y cot2 y 1

Section 5.213. csc x sin x 1 sin xsin x 1 sin2 xsin x cos2 xsin x cos x 90.14090.02935y1 y2cos xsin x0 cos x cot x1.501 cos xcos x1 cos2 x cos x14. y1 sec x cos x Verifying Trigonometric 861.31052.39735.71352sin xcos x sin x6 cos x sin xy1 y2 sin x tan x01.50 y215. sin x cos x cot x sin x cos x cos xsin xsin2 x cos2 xsin x1 sin x csc .07291.01485 Houghton Mifflin Company. All rights reserved.y1 y201.5016. y1 cos x sin x tan xsin2 x cos x cos xcos2 x sin2 x cos .75975.8835101 cos x sec x y2y1 y201.50

Chapter 539417.Analytic Trigonometry11cot x tan x tan x cot xtan x cot x cot x tan 67670.34690.14090.0293y1 y201.5018. y1 11 sin x csc x csc x sin x y210y1 y201.5019. The error is in line 1: cot x cot x.20. There are two errors in line 1:sec sec and sin sin .21. sin1 2 x cos x sin5 2 x cos x sin1 2 x cos x 1 sin2 x sin1 2 x cos x cos2 x cos3 x sin x22. sec6 x sec x tan x sec4 x sec x tan x sec4 x sec x tan x sec2 x 1 sec4 x sec x tan x tan2 x sec5 x tan3 x25. 2 x csc x tan x csc xsec 2 x csc x tan 2 x cot x sin xcos x sin x 1sin x 1 sec xcos x 1 sec xcos xcsc x 1 sin x sec x 1 cos x cos x sin x 24.cos x sin x cot x1sin x cos x26. 1 sin y 1 sin y 1 sin y 1 sin y 1 sin2 y cos2 y Houghton Mifflin Company. All rights reserved.23. cot

Section 5.227.cos cos 1 sin 1 sin 1 sin 1 sin 28.Verifying Trigonometric Identities1 csc 1 csc cos cot cos cot cos 1 sin 1 sin2 cos 1 sin cos2 1 sin cos 1sin cos cos 1 csc 1cos 1 sin 1 csc cos 1 csc 1cos sec sec tan sin x cos ycos x sin y sin x cos y cos x sin ytan x tan ycos x cos y cos x cos y29. cos x cos y sin x sin ycos x cos ysin x sin y1 tan x tan y cos x cos y cos x cos y11 tan x tan ycot x cot y30. 1 tan x tan y111 cot x cot y31.cot x cot y cot x cot y cot y cot xcot x cot y 1cos x cos ysin x sin y cos x cos y cos x cos y sin x sin y sin x sin y sin x sin ycos x cos y sin x sin y cos x cos y cos2 x cos2 y sin2 x sin2 y sin x sin y cos x cos y 1 1 sin x sin y cos x cos y Houghton Mifflin Company. All rights reserved. 032.33.11tan x cot y tan y cot xtan x cot ycot y tan x 1 sin 1 sin 11 sinsin 1 sin 1 sin 1 sin 1 sin 1 sin cos 2222 1 sin cos Note: Check your answer with a graphing utility.What happens if you leave off the absolute value?34. 1 cos 1 cos 11 coscos 1 cos 1 cos 1 cos 1 cos 1 cos sin 2222 1 cos sin 395

396Chapter 535. sin2Analytic Trigonometry 2 x sin37. sin x csc2x cos2 x sin2 x 136. sec2 y cot2 2 x sin x sec x sin x38. sec2 2 y sec2 2 x 1 csc2y tan2 y 1x 1 cot2 x cos1 x tan x39. 2 sec2 x 2 sec2 x sin2 x sin2 x cos2 x 2 sec2 x 1 sin2 x sin2 x cos2 x 2 sec2 x cos2 x 1 21 cos2 x cos2 x 1 2 1 140. csc x csc x sin x sin x cos xcos x cot x csc2 x csc x sin x 1 cot xsin xsin x csc2 x 1 1 cot x cot x csc2 x41.cot x tan x1 csc xsin xsin x42.1 csc 1cos cot cos 1 sec sin sin cos sin 1 1 sin cos 44. sin x 1 2 cos2 x cos4 x sin x 1 cos2 x 2 sin x sin2 x 2 cot2 x 2 cot4 x sin5 x45. sec4 tan4 sec2 tan2 sec2 tan2 46. csc4 cot4 csc2

Analytic Trigonometry Section 5.1 Using Fundamental Identities 379 You should know the fundamental trigonometric identities. (a) Reciprocal Identities (b) Pythagorean Identities (c) Cofunction Identities (d) Negative Angle Identities You should be able to

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