CHAPTER 5 Analytic Trigonometry - Saddleback College

C H A P T E R 5Analytic TrigonometrySection 5.1Using Fundamental Identities. . . . . . . . . . . . . . . 438Section 5.2Verifying Trigonometric Identities . . . . . . . . . . . . . 450Section 5.3Solving Trigonometric EquationsSection 5.4Sum and Difference Formulas . . . . . . . . . . . . . . . 471Section 5.5Multiple-Angle and Product-to-Sum Formulas. . . . . . . . . . . . . 458. . . . . . 490Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524Practice Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

C H A P T E R 5Analytic TrigonometrySection 5.1 Using Fundamental IdentitiesYou should know the fundamental trigonometric identities.(a) Reciprocal Identitiessin u 1csc ucsc u 1sin 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 Identitiessin 2 u cos utan 2 u cot usec 2 u csc ucoscot 2 u sin u 2 u tan ucsc 2 u sec u(d) Even Odd 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 fundamental identities.Vocabulary Check1. tan u2. cos u3. cot u4. csc u5. cot2 u6. sec2 u7. cos u8. csc u9. cos u10. tan u438

Section 5.11. sin x tan x 32, cos x 1 x is in Quadrant II.2 3 2sin x 3cos x 1 2cot x 311 tan x3 3sec x 11 2cos x 1 2csc x 122 31 sin x 3 2 333. sec 2, sin cos 22 is in Quadrant IV. 21 1tan 1 2csc sin 5. tan x 513, sec x x is in1212121 sec x13sin x 1 cos2 x 3, cos x 32x is in Quadrant III. 1 23 sin x 2 csc x 1 2sin xsec x 212 3 3cos x3cot x 31 3 tan x3 14 21534. csc , tan 34sin 13 csc 5cos sin 3 tan 5sec 15 cos 4cot 14 tan 344 3 5 106. cot 3, sin 10 is in Quadrant II.Quadrant III.cos x 3 is in Quadrant I.11 sec 22sin 2 2 1tan 2 2cos cot 2. tan x Using Fundamental Identitiescos cot sin 1445 1 16913tan 11 cot 33 1010cot x 121 tan x5csc 1 10sin csc x 131 sin x5sec 10110 cos 3 10333 57. sec , csc is in Quadrant IV.258. cos 2 x 5, cos x 5, x is in Quadrant I.34sin 511 csc 3 5 53sin x 1 45 cos 211 sec 3 2 3tan x sin x3 cos x 5tan 5sin 5 3 cos 2 32csc x 15 sin x 3cot 112 52 tan 5 255sec x 15 cos x 4cot x 14 tan x 325 353 4 4439

440Chapter 59. sin x Analytic Trigonometry 211 sin x , tan x x is33410. sec x 4, sin x 0x is in Quadrant I.in Quadrant II.cos x 1 sin2 x 1 12 2 93cos x 11 sec x 4cot x 11 2 2tan x 2 4sin x 1 14 sec x 113 2 cos x 2 2 34tan x 15sin x cos x4csc x 11 3sin x 1 3csc x 144 15 sin x 1515cot x 1511 tan x 151511. tan 2, sin 0 is in Quadrant III.sec tan2 1 4 1 5cos 511 5sec 5 51 sin 211cot tan 2csc 13. sin 1, cot 0 3 2cos 1 sin2 0sec is undefined.4 1 1511 csc 5 1 51 2 2 65sin 1 cos 5sec 155 6 cos 122 6cot 112 2 6tan 6 65 122 614. tan is undefined, sin 0. 2sin is undefined cos 0cos sin 1 02 1csc 115. sec x cos x sec x 4tan tan tan is undefined. 15 is in Quadrant III.cos 1 51 25 2 5 5 12. csc 5, cos 0sin sin 1 cos2 21 1sec xThe expression is matched with (d).17. cot2 x csc2 x cot2 x 1 cot2 x 1The expression is matched with (b).csc 1 1sin sec 1is undefined.cos cot cos 0 0sin 116. tan x csc x sin xcos x11 sin x cos x sec xMatches (a).18. 1 cos2 x csc x sin2 x Matches f .1 sin xsin x

Section 5.119. sin xsin x tan xcos x cos x20.sin 2 x cos x cot xcos 2 x sin x21. sin x sec x sin x 22. cos2 x sec2 x 1 cos2 x tan2 x 1 tan xcos x23. sec4 x tan4 x sec2 x tan2 x sec2 x tan2 x sec2 x tan2 x 1 sec2 x tan2 x cos x sin2 x2The expression is matched with f . sin2 xMatches (c).24. cot x sec x cos xsin x11 cos x sin x csc x25.Matches (a).26.sin2 xsec2 x 1 tan2 x 22sin xsin xcos2 x1 sin2 x sec2 xThe expression is matched with (e).sin2 xsin xcos2 2 x sin x tan x sin xcos xcos xcos x27. cot sec cos sin 11 cos sin csc Matches (d).28. cos tan cos sin sin cos 29. sin csc sin sin 1 sin2 sin 1 sin2 cos2 30. sec2 x 1 sin2 x sec2 x sec2 x sin2 x sec2 x 1cos2 x sec2 x sin2 xcos2 x31.cos x sin xcot x csc x1 sin x sin2 x cos xsin x sin x1 cos x sec2 x tan2 x 132.csc 1 sin cos cot sec 1 cos sin 33.1 sin2 xcos2 xsin2 x cos2 x tan2 x cos2 x 222csc x 1cot xcos x sin2 x34.36.111 cos2 xtan2 x 1 sec2 x 1 cos2 x tan2 sin2 2sec cos2 sin2 cos2 1 sec2 1 1 cos2 441The expression is matched with (b).Matches (c).The expression is matched with (e). cos 2 xUsing Fundamental Identities35. sec37. cossin2 cos2 sin2 cos2 sintan 1 sin cotcos 1cos sin cossin 1 2 x sec x sin x sec x sin x cos x cos x tan x1sin x

442Chapter 5Analytic Trigonometry38. cot 2 x cos x tan x cos x cos x cos x sin x sin x39.cos2 y1 sin2 y 1 sin y1 sin y 40. cos t 1 tan2 t cos t sec2 t 41. sin tan cos sin 1 sin y 1 sin y 1 sin y1 sin y1cos t sec tcos2 t cos tsin cos cos 42. csc tan sec 1sin sin cos sec sin2 cos2 cos cos sin2 cos2 cos 2 sec 1cos 1 sec cos sec 43. cot u sin u tan u cos u cos usin u sin u cos u sin ucos u44. sin sec cos csc cos u sin usin cos cos sin sin2 cos2 cos sin 1cos sin sec csc 45. tan2 x tan2 x sin2 x tan2 x 1 sin2 x 46. sin2 x csc2 x sin2 x sin2 x csc2 x 1 sin2 x cot2 x tan2 x cos2 x sin2 xcos2 x sin2 x cos2 x cos2 x sin2 x47. sin2 x sec2 x sin2 x sin2 x sec2 x 1 cos2 xsin2 x48. cos2 x cos2 x tan2 x cos2 x 1 tan2 x cos2 x sec2 x sin2 x tan2 x cos2 x cos1 x 2 149.sec2 x 1 sec x 1 sec x 1 sec x 1sec x 150.cos2 x 4 cos x 2 cos x 2 cos x 2cos x 2 sec x 1 cos x 251. tan4 x 2 tan2 x 1 tan2 x 1 252. 1 2 cos2 x cos4 x 1 cos2 x 2 sec2 x 2 sec4 x sin2 x 2 sin4 x

Section 5.153. sin4 x cos4 x sin2 x cos2 x sin2 x cos2 x Using Fundamental Identities54. sec4 x tan4 x sec2 x tan2 x sec2 x tan2 x 1 sin2 x cos2 x sec2 x tan2 x 1 sin2 x cos2 x sec2 x tan2 x55. csc3 x csc2 x csc x 1 csc2 x csc x 1 1 csc x 1 csc2 x 1 csc x 1 cot2 x csc x 1 56. 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 57. sin x cos x 2 sin2 x 2 sin x cos x cos2 x58. cot x csc x cot x csc x cot2 x csc2 x 1 sin2 x cos2 x 2 sin x cos x 1 2 sin x cos x59. 2 csc x 2 2 csc x 2 4 csc2 x 461.60. 3 3 sin x 3 3 sin x 9 9 sin2 x 4 csc2 x 1 9 1 sin2 x 4 cot2 x 9 cos2 x111 cos x 1 cos x 1 cos x 1 cos x 1 cos x 1 cos x 62.11sec 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 x63.1 sin x cos2 x 1 sin x 2 cos2 x 1 2 sin x sin2 xcos x 1 sin xcos xcos x 1 sin x cos x 1 sin x 2 2 sin xcos x 1 sin x 2 1 sin x cos x 1 sin x 2cos x 2 sec x64. tan x sec2 x tan2 x sec2 x tan xtan x 1 cot xtan x65.1 cos2 ysin2 y 1 cos y1 cos y 1 cos y 1 cos y 1 cos y1 cos y443

44466.Chapter 55tan x sec xAnalytic Trigonometry tan x sec x 5 tan x sec x tan2 x sec2 x 5 tan x sec x 1tan x sec x67.3sec x tan x sec x tan x 3 sec x tan x sec2 x tan2 x 3 sec x tan x 1sec x tan x 3 sec x tan x 5 sec x tan x 68.csc x 1tan2 xcsc x 169. y1 cos csc x 1 2 x , y2tan2 x csc x 1 tan2 x csc x 1 tan2 x csc x 1 tan2 x tan4 x csc x 1 2csc x 1cot2 x sin 50.93200.9854Conclusion: y1 y2 2070. y1 sec x cos x, y2 sin x tan 1.31052.39735.7135 200y20.040371. y1 0.16460.38630.73861.31052.39735.7135It appears that y1 y2.cos x1 sin x, y 1 sin x 2cos 825.331911.6814 2072. y1 sec4 x sec2 x, y2 tan2 x tan4 xConclusion: y1 18418.308750.38691163.6143y21 t appears that y1 y2.73. y1 cos x cot x sin x csc xcos x cot x sin x cos xx sin x cossin x cos2 x sin2 x sin xsin x cos2 x sin2 x1 csc xsin xsin x4 2 2 4

Section 5.1Using Fundamental Identities74. y1 sec x csc x tan x cot xsec x csc x tan x 75. y1 1cos x16sin x sin x cos x 1sin2 x cos x sin x cos x sin x 1 sin2 xcos x sin x cos2 xcos x cot xcos x sin xsin x 2 6 11 cos x tan xsin x cos x 5 111cos x cos x sin x cos xsin x cos xsin x 76. y1 2 sin2 xsin x1 cos2 x tan xsin x cos xsin x cos x cos x1 1 sin cos 2 cos 1 sin 4 1 1 2 sin sin2 cos2 2 cos 1 sin 1 1 2 sin 12 cos 1 sin 12 2 sin 2 cos 1 sin 11 sin sec cos 1 sin cos 2 5cos 1 1 sin 1 sin 1 1 sin cos cos 2 cos 1 sin 2 cos 1 sin cos 1 sin 2 2 2 477. Let x 3 cos , then 9 x2 9 3 cos 2 9 9 cos2 9 1 cos2 9 sin2 3 sin .78. Let x 2 cos . 64 16x2 64 16 2 cos 279. Let x 3 sec , then x2 9 3 sec 2 9 64 1 cos2 9 sec2 9 64 sin2 9 sec2 1 8 sin 9 tan2 3 tan .445

446Chapter 5Analytic Trigonometry80. Let x 2 sec .81. Let x 5 tan , then x 4 2 sec 4 x2 25 5 tan 2 25 4 sec2 1 25 tan2 25 4 tan2 25 tan2 1 2 tan 25 sec2 22 5 sec .82. Let x 10 tan .83. Let x 3 sin , then 9 x2 3 becomes x2 100 10 tan 2 100 9 3 sin 2 3 100 tan2 1 9 9 sin2 3 100 sec2 9 1 sin2 3 10 sec 9 cos2 33 cos 3cos 1sin 1 cos2 1 1 2 0.84. x 6 sin 85. Let x 2 cos , then 16 4x2 2 2 becomes 16 4 2 cos 2 2 23 36 x2 36 6 sin 2 16 16 cos2 2 2 36 1 sin2 16 1 cos2 2 2 36 cos2 16 sin2 2 2 6 cos cos 4 sin 2 23 1 6 2sin 1 sin cos2 1 12 3 4 86. 22cos 1 sin2 1 211 22 3 2x 10 cos 22.87. sin 1 cos2 5 3 100 x2Let y1 sin x and y2 1 cos2 x, 0 x 2 . 100 10 cos 2y1 y2 for 0 x , so we have 100 1 cos2 sin 1 cos2 for 0 . 100 sin2 2 10 sin y2 35 3sin 102cos 1 sin 22 0y1 1 2 32 21 2

Section 5.1Using Fundamental Identities89. sec 1 tan2 88. cos 1 sin2 2Let y1 1and y2 1 tan2 x, 0 x 2 .cos x2 0447y1 y2 for 0 x 3 and x 2 , so we have22 2sec 1 tan2 for 0 3 22 3 and 2 .224y22 0y1 490. csc 1 cot2 91. ln cos x ln sin x ln20 cos x ln cot xsin x2 0 293. ln cot t ln 1 tan2 t ln cot t 1 tan2 t 92. ln sec x ln sin x ln sec x sin x ln1cos x ln cot t sec2 t sin x ln tan x94. ln cos2 t ln 1 tan2 t ln cos2 t 1 tan2 t ln cos2tsec2 ln cos2 t t 1cos2 t lncos tsin t ln1 ln csc t sec tsin t cos t95. (a) csc2 132 cot2 132(b) csc22 2 cot2771.8107 0.8107 11.6360 0.6360 1 ln 1 096. tan2 1 sec2 97. cos 346(a) tan 346 12 1 sec 346 cos 3462(b) sec 3.1 2 80 cos 90 80 sin 8020.9848 0.98481.0622 0.8(b) 3.1 tan 3.1 2 1 2 sin (a)1.06221.00173 cos 3.1 1cos21.001731 cos2 t 2 0.8 sin 0.80.7174 0.7174

448Chapter 5Analytic Trigonometry98. sin sin 99.W cos W sin 250(a) sin 250 0.9397 sin 250 0.9397 (b) 12 2 0.4794 0.4794sin sin112W sin tan W cos 100. csc x cot x cos x 101. True. For example, sin x sin x means that thegraph of sin x is symmetric about the origin. 1 cos x cos xsin x sin x cos x cos xsin2 x cos x sin2 x cos xsin2 x cos x 1 sin2 x sin2 x cos x cos2 x cos x cot2 xsin2 x102. False. A cofunction identity can be used to transforma tangent function so that it can be represented by acotangent function.104. As x 0 , cos x 1 and sec x 106. As x , sin x 0 and csc x 1 1.cos x1 sin x.103. As x , sin x 1 and csc x 1.2105. As x , tan x 2and cot x 0.107. cos 1 sin2 is not an identity.cos2 sin2 1 cos 1 sin2 108. The equation is not an identity.109.cot csc2 1sin k tan k cos k 111. sin csc 1 is an identity.110. The equation is not an identity. 1 111 sec 5 cos 5 cos 55 sec 112. The equation is not an identity. The angles are not the same.sin csc sin sin k tan is not an identity.cos k 1sin sin sin 1, in generalsin 1 sin 1, provided sin 0.

Section 5.1113. Let x, y be any point on the terminal side of . yr xr 2449114. Divide both sides of sin2 cos2 1 by cos2 :Then, r x2 y2 andsin2 cos2 Using Fundamental Identitiessin 2 cos 2 1 2cos cos 2 cos 2 2tan2 1 sec2 Divide both sides of sin2 cos2 1 by sin2 :y2 x2r2sin 2 cos 2 1 sin 2 sin 2 sin 2 r2r21 cot2 csc2 1.Discussion for remembering identities will vary, but onekey is first to learn the identities that concern the sineand cosine functions thoroughly, and then to use these asa basis to establish the other identities when necessary.115. x 5 x 5 x 5 2 x 25116. 2 z 3 2 2 z 2 2 z 3 3 222 4z 12 z 9117.1x x 8 x x 5 x 5 x 8 x 5 x 8 119.118.x2 6x 8 x 5 x 8 2x72x x 4 7 x2 4 x2 4 x 4 x2 4 x 4 121. f x 120.x2 6x 3x 4 3 2x 1 x 4xx2 x 5 x2x 25 x 5 x 5 x 5 x 5 x 5 2x2 8x 7x2 28 x2 4 x 4 x x3 5x2 x 5 x 5 5x2 8x 28 x2 4 x 4 x 1 x2 5x x 5 x 5 x x2 5x 1 x2 251sin x 2122. f x 2 tany 2x y32Amplitude: 21Amplitude:212 Period: 2 x1 13Period: 2 2 1 13 1 0, 0 , , , 1, 0 , , , 2, 0 2 22 2 3Two consecutive verticalasymptotes: x 1, x 1 2Key points: 6x36x3 x 4 4 x x 4 x 4Key points: 21, 2 , 0, 0 , 12, 2 1x1 1 2 33