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C H A P T E R Trigonometry 4 Section 4.1 Radian and Degree Measure . . . . . . . . . . . . . . . . 335 Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . . 344 Section 4.3 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 350 Section 4.4 Trigonometric Functions of Any Angle Section 4.5 Graphs of Sine and Cosine Functions . . . . . . . . . . . 373 Section 4.6 Graphs of Other Trigonometric Functions . . . . . . . . . 386 Section 4.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . 397 Section 4.8 Applications and Models . . . . . . . . . . . . . . . . . . 410 . . . . . . . . . . 360 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 C H A P T E R Trigonometry Section 4.1 4 Radian and Degree Measure You should know the following basic facts about angles, their measurement, and their applications. ■ Types of Angles: (a) Acute: Measure between 0 and 90. (b) Right: Measure 90. (c) Obtuse: Measure between 90 and 180. (d) Straight: Measure 180. ■ and are complementary if 90. They are supplementary if 180. ■ Two angles in standard position that have the same terminal side are called coterminal angles. ■ To convert degrees to radians, use 1 180 radians. ■ To convert radians to degrees, use 1 radian 180. ■ 1 one minute 160 of 1. ■ 1 one second 160 of 1 13600 of 1. ■ The length of a circular arc is s r where is measured in radians. ■ Linear speed ■ Angular speed t srt arc length s time t Vocabulary Check 1. Trigonometry 2. angle 3. coterminal 4. radian 5. acute; obtuse 6. complementary; supplementary 7. degree 8. linear 10. A 12r 2 9. angular 1. 2. 3. The angle shown is approximately 3 radians. The angle shown is approximately 2 radians. 4. The angle shown is approximately 5.5 radians. 6. 5. The angle shown is approximately 6.5 radians. The angle shown is approximately 4 radians. The angle shown is approximately 1 radian. 335 336 Chapter 4 7. (a) Since 0 < (b) Since < Trigonometry < ; lies in Quadrant I. 5 2 5 8. (a) Since < 7 3 7 < ; lies in Quadrant III. 5 2 5 (b) Since < lies in Quadrant IV. < < 0; 2 12 12 10. (a) Since (b) Since < 2 < ; 2 lies in Quadrant III. 2 (b) Since 9. (a) Since 11. (a) Since < 3.5 < (b) Since 13. (a) 3 ; 3.5 lies in Quadrant III. 2 12. (a) Since < 2.25 < ; 2.25 lies in Quadrant II. 2 5 4 14. (a) y 9 3 9 < ; lies in Quadrant III. 8 2 8 < 1 < 0; 1 lies in Quadrant IV. 2 11 3 11 lies in Quadrant II. < < , 2 9 9 3 < 6.02 < 2 ; 6.02 lies in Quadrant IV. 2 (b) Since 7 4 11 3 11 < ; lies in Quadrant III. 8 2 8 3 < 4.25 < ; 4.25 lies in Quadrant II. 2 15. (a) y 11 6 y 5π 4 11π 6 x x x 7π − 4 (b) 2 3 (b) 5 2 y (b) 3 y y 5π 2 x x x −3 2π − 3 16. (a) 4 (b) 7 y y 7π 4 x x Section 4.1 17. (a) Coterminal angles for 6 18. (a) 13 2 6 6 7 19 2 6 6 Radian and Degree Measure 19. (a) Coterminal angles for (b) (b) Coterminal angles for 5 6 2 4 2 3 3 11 2 6 6 11 23 2 6 6 (b) Coterminal angles for 5 17 2 6 6 25 2 12 12 5 7 2 6 6 23 2 12 12 20. (a) (b) 9 2 4 4 21. (a) Complement: 9 7 4 4 4 22. (a) Complement: Supplement: 5 2 12 12 Supplement: 2 3 3 (b) Complement: Not possible, 2 32 2 15 15 23. (a) Complement: 12 2 3 6 Supplement: 2 28 2 15 15 2 3 2 8 2 3 3 7 5 2 6 6 11 2 6 6 337 3 is greater than . 4 2 3 4 4 1 0.57 2 Supplement: 1 2.14 11 12 12 (b) Complement: Not possible, 2 is greater than . 2 11 (b) Complement: Not possible, is greater than . 12 2 Supplement: 2 1.14 11 Supplement: 12 12 24. (a) Complement: Not possible, 3 is greater than . 2 25. Supplement: 3 0.14 (b) Complement: 1.5 0.07 2 The angle shown is approximately 210º . Supplement: 1.5 1.64 27. 26. The angle shown is approximately 120. 28. The angle shown is approximately 60º. The angle shown is approximately 330. 338 Chapter 4 Trigonometry 29. 30. The angle shown is approximately 165. 31. (a) Since 90 < 130 < 180, 130 lies in Quadrant II. (b) Since 270 < 285 < 360, 285 lies in Quadrant IV. 33. (a) Since 180 < 132 50 < 90, 132 50 lies in Quadrant III. (b) Since 360 < 336 < 270, 336 lies in Quadrant I. The angle shown is approximately 10. 32. (a) Since 0 < 8.3 < 90, 8.3 lies in Quadrant I. (b) Since 180 < 257 30 < 270, 257 30 lies in Quadrant III. 34. (a) Since 270 < 260 < 180, 260 lies in Quadrant II. (b) Since 90 < 3.4 < 0, 3.4 lies in Quadrant IV. 36. (a) 270 35. (a) 30 y y 30° x x − 270° (b) 120 (b) 150 y y 150° x x − 120° 38. (a) 750 37. (a) 405 y y 405° x x − 750° (b) 600 (b) 480 y y 480° − 600° x x Section 4.1 39. (a) Coterminal angles for 45 Radian and Degree Measure 40. (a) 120 360 480 45 360 405 41. (a) Coterminal angles for 240 120 360 240 45 360 315 (b) Coterminal angles for 36 240 360 600 (b) 420 720 300 240 360 120 420 360 60 36 360 324 339 (b) Coterminal angles for 180 180 360 180 36 360 396 180 360 540 43. (a) Complement: 90 18 72 42. (a) 420 720 300 Supplement: 180 18 162 420 360 60 (b) Complement: Not possible, 115 is greater than 90 . (b) 230 360 590 Supplement: 1180 115 65 230 360 130 44. (a) Complement: 90 3 87 45. (a) Complement: 90 79 11 Supplement: 180 3 177 Supplement: 180 79 101 (b) Complement: 90 64 26 (b) Complement: Not possible, 150 is greater than 90. Supplement: 180 64 116 Supplement: 180 150 30 47. (a) 30 30 46. (a) Complement: Not possible, 130 is greater than 90. Supplement: 180 130 50 (b) 150 150 (b) Complement: Not possible, 170 is greater than 90. 180 6 180 5 6 Supplement: 180 170 10 48. (a) 315 315 7 4 2 3 180 (b) 120 120 180 49. (a) 20 20 (b) 240 240 51. (a) 3 3 180 270 2 2 52. (a) (b) 7 7 180 210 6 6 (b) 54. (a) 11 11 180 330 6 6 (b) 34 34 180 408 15 15 59. 532 532 180 3.776 radians 180 9.285 radians 50. (a) 270 270 4 180 3 (b) 144 144 7 7 180 105 12 12 53. (a) 180 20 9 9 55. 115 115 57. 216.35 216.35 180 9 4 5 11 11 180 66 30 30 180 1.525 radians 58. 48.27 48.27 60. 345 345 180 56. 87.4 87.4 2.007 radians 3 7 7 180 420 3 3 (b) 180 180 2 180 0.842 radians 180 6.021 radians 340 Chapter 4 61. 0.83 0.83 Trigonometry 180 0.014 radian 62. 0.54 0.54 5 5 180 81.818 11 11 63. 180 25.714 7 7 64. 66. 13 13 180 1170.000 2 2 67. 4.2 4.2 180 0.009 radians 65. 180 15 15 180 337.500 8 8 68. 4.8 4.8 864.000 180 756.000 69. 2 2 180 70. 0.57 0.57 114.592 71. (a) 54 45 54 45 60 54.75 32.659 180 72. (a) 24510 245 10 60 245 0.167 245.167 (b) 212 2 12 60 2 0.2 2.2 (b) 128 30 128 30 60 128.5 18 30 73. (a) 85 18 30 85 60 3600 85.308 36 74. (a) 135 36 135 3600 25 (b) 330 25 330 3600 330.007 135 0.01 135.01 16 20 (b) 408 16 20 408 60 3600 408 0.2667 0.0056 408.272 76. (a) 345.12 345 0.1260 75. (a) 240.6 240 0.660 240 36 345 7 0.260 (b) 145.8 145 0.860 145 48 345 7 12 (b) 0.45 0 0.4560 0 27 0 27 78. (a) 0.355 0 0.35560 77. (a) 2.5 2 30 0 21 0.360 (b) 3.58 3 0.5860 0 21 18 0 21 18 3 34 0.860 (b) 0.7865 0 0.786560 3 34 48 0 47 0.1960 0 47 11.4 0 47 11.4 79. s r 6 5 6 5 radians 82. s r 60 75 60 4 radians 75 5 Because the angle represented is clockwise, this angle is 45 radians. 80. s r 29 10 29 10 radians 83. s r 6 27 6 2 radians 27 9 81. s r 32 7 4 32 7 4 7 radians 84. r 14 feet, s 8 feet s 8 4 radians r 14 7 Section 4.1 85. 86. r 80 kilometers, s 160 kilometers s r 25 14.5 88. r 9 feet, 60 3 s 15180 90. r 20 centimeters, s 31 3 meters 180 15 inches 47.12 inches 89. s r, in radians 3 3 feet 341 87. s r, in radians s 160 2 radians r 80 25 50 radians 14.5 29 s r 9 Radian and Degree Measure s r 20 9.42 feet 4 4 5 centimeters 15.71 centimeters 1 91. A r 2 2 92. r 12 mm, 8 1 square inches A 42 2 3 3 4 1 93. A r 2 2 1 A 2.52225 2 180 1 1 A r 2 122 2 2 4 8.38 square inches 12.27 square feet 18 mm2 56.55 mm2 95. 41 15 50 32 47 39 94. r 1.4 miles, 330 21.56 1 330 A 1.42 5.64 square miles 2 180 12 96. r 4000 miles 47 37 18 37 47 36 9 49 42 8.46972 0.14782 radian s r 40000.14782 591.3 miles 97. 450 s 0.071 radian 4.04 r 6378 99. s 2.5 25 5 radian r 6 60 12 0.1715 radian s r 40000.1715 686.2 miles 98. r 3189 kilometers s r 400 6378 400 6378 0.062716 The difference in latitude is about 0.062716 radians 3.59. 100. s 24 180 4.8 radians 4.8 275 r 5 101. (a) 65 miles per hour 655280 5720 feet per minute 60 The circumference of the tire is C 2.5 feet. The number of revolutions per minute is 5720 728.3 revolutions per minute r 2.5 (b) The angular speed is . t 5720 2 4576 radians 2.5 Angular speed 4576 radians 4576 radians per minute 1 minute 342 Chapter 4 Trigonometry 102. Linear velocity for either pulley: 17002 3400 inches per minute (a) Angular speed of motor pulley: v 3400 3400 radians per minute r 1 v 3400 1700 radians per minute r 2 Angular speed of the saw arbor: (b) Revolutions per minute of the saw arbor: 103. (a) Angular speed 1700 850 revolutions per minute 2 52002 radians 1 minute 104. (a) 4 rpm 42 radians per minute 8 radians per minute 10,400 radians per minute 25.13 radians per minute 32,672.56 radians per minute (b) Linear speed (b) r 25 ft 1 ft in. 52002 7.25 2 12 in. r 200 feet per minute t 1 minute Linear speed 2525.13274 feet per minute 2 3141 feet per minute 3 628.32 feet per minute 9869.84 feet per minute 105. (a) 2002 ≤ Angular speed ≤ 5002 radians per minute Interval: 400, 1000 radians per minute (b) 62002 ≤ Linear speed ≤ 65002 centimeters per minute Interval: 2400, 6000 centimeters per minute 1 106. A R 2 r 2 2 R 25 25 r 25 14 11 A 1 107. A r 2 2 1 125 2 180 1 352140 2 180 125° r 14 252 112 175 549.8 square inches 476.39 square meters 1496.62 square meters 108. (a) Arc length of larger sprocket in feet: s r 2 1 feet s 2 3 3 Therefore, the chain moves 23 feet, as does the smaller rear sprocket. Thus, the angle of the smaller sprocket is r 2 inches 212 feet. s 23 feet 4 and the arc length of the tire in feet is: r 212 feet s r s 4 Speed 14 feet 14 12 3 s 143 14 feet per second t 1 second 3 14 feet 3600 seconds 1 mile 10 miles per hour 3 seconds 1 hour 5280 feet —CONTINUED— 140° 35 Section 4.1 Radian and Degree Measure 343 108. —CONTINUED— (b) Since the arc length of the tire is 143 feet and the cyclist is pedaling at a rate of one revolution per second, we have: Distance feet 1 mile 7 n revolutions n miles 143 revolutions 5280 feet 7920 (c) Distance Rate Time 7 1 mile t seconds t miles 143 feet per second5280 feet 7920 (d) The functions are both linear. 109. False. An angle measure of 4 radians corresponds to two complete revolutions from the initial to the terminal side of an angle. 110. True. If and are coterminal angles, then n360 where n is an integer. The difference between and is n360 2n. 111. False. The terminal side of 1260 lies on the negative x-axis. 112. (a) An angle is in standard position if its vertex is at the origin and its initial side is on the positive x-axis. (b) A negative angle is generated by a clockwise rotation of the terminal side. (c) Two angles in standard position with the same terminal sides are coterminal. (d) An obtuse angle measures between 90° and 180°. 113. Increases, since the linear speed is proportional to the radius. 114. 1 radian 180 57.3, so one radian is much larger than one degree. 116. The area of a circle is A r 2 ⇒ C2 C 115. The arc length is increasing. In order for the angle to remain constant as the radius r increases, the arc length s must increase in proportion to r, as can be seen from the formula s r . A . The circumference of a circle is C 2 r. r2 r r A 2 2A r Cr A 2 For a sector, C s r. Thus, A 117. 4 4 42 42 2 2 r r 1 2 r for a sector. 2 2 42 2 8 2 119. 22 62 4 36 40 4 118. 10 210 5 55 210 2 105 25 12 2 5 2 2 2 120. 172 92 289 81 208 16 13 413 52 4 344 Chapter 4 Trigonometry 121. f x x 25 y Graph of y x 5 shifted to the right by two units 122. f x x5 4 y = x5 y 4 Vertical shift four units downward 3 2 2 1 −3 x −2 2 3 4 x −2 1 y = x5 − 4 −2 −6 y = (x − 2)5 −3 123. f x 2 x 5 124. f x x 35 y Graph of y x 5 reflected in x-axis and shifted upward by two units 6 4 y = x5 3 x −1 1 −2 2 3 ■ ■ ■ y = x5 1 x 1 2 −1 −3 Trigonometric Functions: The Unit Circle You should know the definition of the trigonometric functions in terms of the unit circle. Let t be a real number and x, y the point on the unit circle corresponding to t. sin t y csc t 1 , y y0 cos t x sec t 1 , x x0 y tan t , x 0 x cot t x , y y0 The cosine and secant functions are even. cost cos t ■ 2 −2 y = 2 − x5 −3 Section 4.2 y 3 −5 −4 −3 1 −2 y = −(x + 3)5 Reflection in the x-axis and a horizontal shift three units to the left 5 −3 3 −2 y = x5 −1 2 sect sec t The other four trigonometric functions are odd. sint sin t csct csc t tan t tan t cott cot t Be able to evaluate the trigonometric functions with a calculator. Vocabulary Check 1. unit circle 2. periodic 3. period 4. odd; even Section 4.2 1. x 8 15 ,y 17 17 sin y 5 13 csc 1 13 y 5 8 17 sec 17 1 x 8 cos x 12 13 sec 1 13 x 12 y 15 x 8 cot x 8 y 15 tan y 5 x 12 cot x 12 y 5 4 3 4. x , y 5 5 12 5 ,y 13 13 sin y cos x tan 12 5 ,y 13 13 1 17 y 15 15 17 cos x 3. x 2. x csc sin y tan Trigonometric Functions: The Unit Circle 5 13 12 13 5 y x 12 3 5 csc 1 5 y 3 4 5 sec 1 5 x 4 cot x 4 y 3 csc 13 1 y 5 sin y sec 1 13 x 12 cos x cot 12 x y 5 tan 5. t 2 2 , . corresponds to 4 2 2 7. t 3 7 1 , . corresponds to 6 2 2 9. t y 3 x 4 1 3 , x y , 3 2 2 2 2 5 , x, y , 4 2 2 8. t 3 4 1 . corresponds to , 3 2 2 10. t 11. t 3 corresponds to 0, 1. 2 12. t , x, y 1, 0 13. t 2 2 , . corresponds to 4 2 2 sin t y cos t x tan t 15. t 2 3 1 , . corresponds to 6 2 2 tan t cos 1 3 2 tan 32 3 3 12 4 22 3 cos 2 3 1 y x 3 3 sin 4 2 2 22 1 4 22 tan 2 2 16. t , x, y , 4 2 2 1 2 sin t y cos t x 1 3 , x, y , 3 2 2 3 3 2 2 y 1 x 5 1 3 , x, y , 3 2 2 sin 2 14. t 2 6. t 345 346 Chapter 4 17. t 2 2 7 , . corresponds to 4 2 2 2 y 1 x tan t 21. t 3 3 1 y x 3 3 y is undefined. x 25. t 5 1 3 , x, y , 3 2 2 sin 3 5 3 2 cos 5 1 3 2 tan 5 32 3 3 12 tan2 2 2 cos t x cos2 1 2 2 3 , . corresponds to 4 2 2 tan t 32 4 3 3 12 sin2 0 cos t x 0 sin t y 22. t 2, x, y 1, 0 sin t y 1 23. t 4 1 3 2 2 3 corresponds to 0, 1. 2 tan t 20. t 1 2 3 4 3 2 tan 3 1 11 corresponds to , . 6 2 2 cos t x cos 2 sin t y 4 1 3 , x, y , 3 2 2 sin 2 cos t x tan t 18. t 2 sin t y 19. t Trigonometry 2 2 y 1 x 24. t 0 0 1 3 1 5 , x, y , 6 2 2 csc t 1 2 y sin 5 1 6 2 csc 5 1 2 6 sin t sec t 1 2 x cos 3 5 6 2 sec 5 1 23 6 cos t 3 cot t x 1 y tan 3 12 5 6 3 32 cot 5 1 3 6 tan t corresponds to 0, 1. 2 26. t 3 , x, y 0, 1 2 sin t y 1 csc t 1 1 y sin 3 1 2 csc 3 1 1 2 sin t cos t x 0 sec t 1 is undefined. x cos 3 0 2 sec 3 is undefined. 2 cot t x 0 y tan 3 is undefined. 2 cot 0 3 0 2 1 tan t y is undefined. x Section 4.2 27. t 3 4 1 . corresponds to , 3 2 2 sin 2 7 4 2 csc 7 1 2 4 sin t 1 2 sec t 1 2 x cos 7 2 4 2 sec 7 1 2 4 cos t y 3 x cot t x 3 y 3 tan 7 22 1 4 22 cot 1 7 1 4 tan t 2 30. cos 5 cos 1 29. sin 5 sin 0 9 2 sin 4 4 2 9 7 2 31. cos 2 1 8 cos 3 3 2 7 1 19 sin 6 6 2 15 cos 0 2 2 34. sin 8 4 1 cos 3 3 2 37. sin t 33. cos 4 sin 4 2 35. sin 2 2 7 , x, y , 4 2 2 1 23 y 3 3 cos t x 32. sin 28. t 347 csc t sin t y tan t Trigonometric Functions: The Unit Circle 36. cos 1 3 (a) sint sin t 1 3 (b) csc t csc t 3 38. sint 3 8 39. cos t (a) sin t sint (b) csc t 41. sin t 3 8 (b) sec t 4 5 42. cos t 4 5 (b) sint sin t 44. tan 1.7321 3 47. cos1.7 0.1288 50. sec 1.8 1 4.4014 0051.8 53. (a) sin 5 y 1 (b) cos 2 x 0.4 40. cos t (a) cos t cost 8 1 sin t 3 (a) sin t sin t 1 5 4 5 1 5 (a) cost cos t 1 5 cos t (b) sect sec t 4 5 43. sin (a) cos t cos t 4 5 (b) cost cos t 4 5 45. csc 1.3 1 1.0378 sin 1.3 1 1.4486 cos 22.8 46. cot 1 (b) cos 2.5 x 0.8 1 4 cos t 3 1 0.6421 tan 1 1 1.3940 sin 0.8 52. sin0.9 0.7833 54. (a) sin 0.75 y 0.7 3 4 0.7071 4 49. csc 0.8 48. cos2.5 0.8011 51. sec 22.8 3 4 348 Chapter 4 Trigonometry 56. (a) sin t 0.75 55. (a) sin t 0.25 t 0.25 or 2.89 t 4.0 or t 5.4 (b) cos t 0.25 (b) cos t 0.75 t 1.82 or 4.46 t 0.7 or t 5.6 57. yt 14 et cos 6t (a) t 0 1 4 1 2 3 4 1 y 0.25 0.0138 0.1501 0.0249 0.0883 (b) From the table feature of a graphing utility we see that y 0 when t 5 seconds. (c) As t increases, the displacement oscillates but decreases in amplitude. 58. y t 1 cos 6t 4 (a) y 0 (b) y (c) y 59. False. sint sin t means the function is odd, not that the sine of a negative angle is a negative number. 1 cos 0 0.2500 feet 4 For example: sin 4 4 cos 2 0.0177 feet 1 1 3 3 3 sin 1 1. 2 2 Even though the angle is negative, the sine value is positive. 2 4 cos 3 0.2475 feet 1 1 60. True. tan a tana 6 since the period of tan is . 61. (a) The points have y-axis symmetry. (b) sin t1 sin t1 since they have the same y-value. (c) cos t1 cos t1 since the x-values have the opposite signs. 62. cos x cos sec 63. 1 sec x 1 y 3x 2 2 So sec and cos are even. 1 x 3y 2 2 sin y sin y sin csc 1 y 2x 3y 2 y 2 2 x y 3 3 1 csc csc y (x, y) 2 2 2 f 1x x x 1 3 3 3 So sin and csc are odd. tan tan y x y tan x x cot y cot x cot y So tan and cot are odd. 1 f x 3x 2 2 θ −θ x (x, −y) Section 4.2 1 64. f x x3 1 4 y x2 4 x y2 4 x 2 y2 4 1 x y3 1 4 ± x2 4 y y x2 x4 x y2 y4 xy 4x y 2 xy y 4x 2 y3 x 1 yx 1 4x 2 3 4 3 4x 1 x f1 67. f x x2 x4 xy 4 y 2 f 1x x2 4, x ≥ 0 1 x 1 y3 4 y 66. f x f x x2 4, x ≥ 2 65. 1 y x3 1 4 4x 1 Trigonometric Functions: The Unit Circle 2x x3 y 22x 1 x1 f 1x 22x 1 x1 y 8 Intercept: 0, 0 6 Vertical asymptote: x 3 2 4 Horizontal asymptote: y 2 x −6 − 4 − 2 −2 2 4 6 8 10 2 4 6 −4 x 1 0 1 2 4 5 6 y 1 2 0 1 4 8 5 4 68. f x −6 −8 5x 5x , x 3, 2 x2 x 6 x 3x 2 y 8 Horizontal asymptote: x 0 6 4 Vertical asymptote: x 3, x 2 2 Intercept: 0, 0 −4 −2 −2 x 8 −4 x 6 4 2 y 5 4 10 3 5 2 69. f x 0 1 3 5 0 5 4 5 2 25 24 −6 −8 x 5x 2 x2 3x 10 x5 ,x2 2x2 8 2x 2x 2 2x 2 5 Intercepts: 5, 0, 0, 4 x Hole in the graph at 2, 7 8 4 0 1 4 3 4 3 1 0 2 5 4 1 3 1 4 5 2 1 Vertical asymptote: x 2 Horizontal asymptote: y 5 y 1 2 y 1 −6 −5 −2 −1 −1 −2 −3 −4 x 1 2 349 350 Chapter 4 70. f x Trigonometry x3 6x2 x 1 1 7 15x 4 x 2x2 5x 8 2 4 42x2 5x 8 Vertical asymptote: 2x 2 5x 8 0 x 5 ± 52 428 22 x 5 ± 89 ; x 1.11, x 3.61 4 4 3 32 y 15 4 17 5 155 32 1 0 2 9 1 8 3 2 3 4 7 5 29 4 1 y −6 −4 −2 −2 1 7 Slant asymptote: y x 2 4 y-intercept: x x 2 6 8 4 0, 18 x-intercept: 5.86, 0 Section 4.3 ■ Right Triangle Trigonometry You should know the right triangle definition of trigonometric functions. ■ (a) sin opp hyp (b) cos adj hyp (c) tan opp adj (d) csc hyp opp (e) sec hyp adj (f) cot adj opp You should know the following identities. (a) sin 1 csc (b) csc 1 sin (c) cos 1 sec (d) sec 1 cos (e) tan 1 cot (f) cot 1 tan (g) tan sin cos (h) cot cos sin (i) sin2 cos2 1 (j) 1 tan2 sec2 (k) 1 cot2 csc2 ■ You should know that two acute angles and are complementary if 90, and that cofunctions of complementary angles are equal. ■ You should know the trigonometric function values of 30, 45, and 60, or be able to construct triangles from which you can determine them. Vocabulary Check 1. (i) (iv) hypotenuse sec adjacent adjacent cos hypotenuse (v) (ii) adjacent cot opposite (iv) (iii) hypotenuse csc opposite (iii) (v) opposite sin hypotenuse (i) (vi) opposite tan adjacent 2. opposite; adjacent; hypotenuse 3. elevation; depression (vi) (ii) Section 4.3 Right Triangle Trigonometry 1. hyp 62 82 36 64 100 10 6 sin 6 3 opp hyp 10 5 csc hyp 10 5 opp 6 3 cos adj 8 4 hyp 10 5 sec hyp 10 5 adj 8 4 tan opp 6 3 adj 8 4 cot adj 8 4 opp 6 3 θ 8 adj 132 52 169 25 12 2. 13 5 θ sin 5 opp hyp 13 csc hyp 13 opp 5 cos 12 adj hyp 13 sec hyp 13 adj 12 tan 5 opp adj 12 cot 12 adj opp 5 b 3. adj 412 92 1681 81 1600 40 41 θ 9 sin opp 9 hyp 41 csc hyp 41 opp 9 cos 40 adj hyp 41 sec hyp 41 adj 40 tan opp 9 adj 40 cot adj 40 opp 9 hyp 42 42 32 42 4. 4 θ 4 sin 2 opp 4 1 hyp 42 2 2 csc hyp 42 2 opp 4 cos 2 adj 4 1 hyp 42 2 2 sec hyp 42 2 adj 4 tan opp 4 1 adj 4 cot adj 4 1 opp 4 5. adj 32 12 8 22 θ 3 1 sin opp 1 hyp 3 csc hyp 3 opp cos adj 22 hyp 3 sec hyp 3 32 adj 22 4 tan 2 1 opp adj 22 4 cot adj 22 opp adj 62 22 32 42 6 2 sin opp 2 1 hyp 6 3 csc hyp 6 3 opp 2 cos adj 42 22 hyp 6 3 sec hyp 6 3 32 adj 42 22 4 tan 2 opp 2 1 adj 42 22 4 cot adj 42 22 opp 2 θ The function values are the same since the triangles are similar and the corresponding sides are proportional. 351 352 Chapter 4 Trigonometry 6. 4 θ 8 7.5 θ hyp 7.52 42 15 hyp 152 82 289 17 sin opp 8 hyp 17 csc hyp 17 opp 8 cos 15 adj hyp 17 sec hyp 17 adj 15 tan 8 opp adj 15 cot 15 adj opp 8 17 2 sin opp 4 8 hyp 172 17 csc hyp 172 17 opp 4 8 cos 7.5 15 adj hyp 172 17 sec hyp 172 17 adj 7.5 15 tan 4 8 opp adj 7.5 15 cot 7.5 15 adj opp 4 8 The function values are the same because the triangles are similar, and corresponding sides are proportional. 7. opp 52 42 3 5 θ 4 sin opp 3 hyp 5 csc hyp 5 opp 3 cos adj 4 hyp 5 sec hyp 5 adj 4 tan opp 3 adj 4 cot adj 4 opp 3 opp 1.252 12 0.75 1.25 θ 1 sin opp 0.75 3 hyp 1.25 5 csc hyp 1.25 5 opp 0.75 3 cos adj 1 4 hyp 1.25 5 sec hyp 1.25 5 adj 1 4 tan opp 0.75 3 adj 1 4 cot 1 4 adj opp 0.75 3 The function values are the same since the triangles are similar and the corresponding sides are proportional. 8. θ 1 3 2 θ hyp 12 22 5 6 5 1 opp sin hyp 5 5 hyp 5 5 csc opp 1 adj 25 2 cos hyp 5 5 hyp 5 sec adj 2 tan opp 1 adj 2 cot adj 2 2 opp 1 hyp 32 62 35 sin 5 3 1 35 5 5 csc 35 5 3 cos 6 2 25 35 5 5 sec 35 5 6 2 tan 3 1 6 2 cot 6 2 3 The function values are the same because the triangles are similar, and corresponding sides are proportional. Section 4.3 9. Given: sin 3 opp 4 hyp 10. Given: cos 32 adj2 42 7 adj hyp 4 3 θ hyp 4 csc opp 3 csc hyp 7 76 opp 26 12 sec hyp 47 adj 7 sec hyp 7 adj 5 cot 7 adj opp 3 cot 5 adj 56 opp 26 12 11. Given: sec 2 opp 2 12 2 hyp 1 adj 12. Given: cot 22 hyp opp 3 sin opp 3 hyp 2 2 1 adj cos hyp 2 52 12 cos adj 5 526 hyp 26 26 tan opp 1 adj 5 csc hyp 23 opp 3 csc hyp 26 26 opp 1 cot 3 adj opp 3 sec hyp 26 adj 5 3 opp 1 adj 14. Given: sec 32 12 hyp2 hyp 10 sin opp 310 hyp 10 cos 10 adj hyp 10 hyp 10 csc opp 3 hyp 10 sec adj adj 1 cot opp 3 2 6 θ 5 θ 26 1 5 26 26 opp 1 hyp 26 26 θ 1 5 adj 1 opp sin 3 opp 3 tan adj 13. Given: tan 3 7 opp 26 sin hyp 7 opp 26 tan adj 5 7 opp 37 tan adj 7 6 hyp 1 adj opp 62 12 35 10 θ 1 3 353 adj 5 7 hyp opp 72 52 24 26 4 adj 7 cos Right Triangle Trigonometry opp 35 hyp 6 6 sin cos adj 1 hyp 6 θ tan opp 35 35 adj 1 csc 6 hyp 635 opp 35 35 cot 35 adj 1 opp 35 35 1 35 354 Chapter 4 Trigonometry 15. Given: cot 22 32 3 adj 2 opp 16. Given: csc hyp 2 13 hyp 13 adj 2 172 17 hyp 4 opp 17 273 42 273 sin 4 opp hyp 17 cos 273 adj hyp 17 tan opp 4 4273 273 adj 273 hyp 13 csc opp 2 sec 17 hyp 17273 273 adj 273 hyp 13 adj 3 cot 273 adj opp 4 sin cos θ 2 opp 213 hyp 13 13 3 adj hyp 13 3 313 13 opp 2 tan adj 3 sec 17. 30 30 180 6 radian sin 30 opp 1 hyp 2 60° 2 1 30° 18. 2 1 3 4 θ 45 45 180 4 radian cos 45 2 1 adj hyp 2 2 45° 1 180 60 3 3 19. π 6 2 tan 3 180 45 4 4 20. opp 3 3 3 adj 1 hyp 2 2 4 adj 1 2 sec 1 π 4 π 3 1 1 cot 21. 30° 3 3 60 3 1 3 adj opp csc 2 22. radian 3 2 45 45 1 hyp opp 180 4 radian 45° 60° 1 1 23. π 3 2 180 30 6 6 1 cos π 6 24. 2 3 adj 6 hyp 2 1 sin π 4 3 180 45 4 4 2 opp 1 4 hyp 2 2 1 cot 1 25. 45° 2 1 45° 1 1 adj 1 opp 45 45 radian 4 tan 26. 180 3 3 60° 2 1 30 30 30° 3 radian 6 1 3 180 opp adj Section 4.3 27. sin 60 3 2 (a) tan 60 3 1 28. sin 30 , tan 30 2 3 1 (a) csc 30 2 sin 30 1 2 , cos 60 sin 60 3 cos 60 (b) sin 30 cos 60 (c) cos 30 sin 60 1 2 (b) cot 60 tan90 60 tan 30 3 2 3 cos 60 1 (d) cot 60 3 sin 60 3 29. csc 13 2 , sec Right Triangle Trigonometry 13 3 1 3 3 sin 30 2 (c) cos 30 tan 30 23 2 3 3 3 33 1 3 (d) cot 30 tan 30 3 3 30. sec 5, tan 26 3 (a) sin 1 2 213 csc 13 13 (a) cos 1 1 sec 5 (b) cos 1 3 313 sec 13 13 (b) cot 6 1 1 tan 26 12 (c) tan 213 13 sin 2 cos 3 313 13 (d) sec90 csc 31. cos (c) cot90º tan 26 (d) sin tan cos 26 5 1 26 5 13 2 1 3 32. tan 5 1 3 cos (a) cot 1 1 tan 5 (b) sin2 cos2 1 (b) cos 1 1 sec 1 tan2 (a) sec 3 sin2 3 1 2 1 sin2 8 9 sin 22 3 1 2 cos 3 1 (c) cot sin 2 2 4 2 2 3 1 (d) sin90 cos 3 33. tan cot tan tan 1 1 1 1 52 1 26 (c) tan90º cot 26 26 1 1 tan 5 (d) csc 1 cot 2 1 15 1 251 2625 34. cos sec cos 2 cos1 1 26 5 355 356 Chapter 4 35. tan cos Trigonometry sin cos cos sin 37. 1 cos 1 cos 1 cos2 36. cot sin cos sin cos sin 38. 1 sin 1 sin 1 sin2 cos2 sin2 cos 2 cos 2 sin2 39. sec tan sec tan sec2 tan2 41. 1 tan2 tan2 sin2 1 sin2 1 2 sin2 1 sin cos sin2 cos2 cos sin sin cos 1 sin cos 1 sin 40. sin2 cos2 sin2 1 sin2 42. tan cot tan cot tan tan tan 1 1 cos cot 1 cot 1 cot2 csc2 csc sec 43. (a) sin 10 0.1736 44. (a) tan 23.5 0.4348 (b) cos 80 0.1736 Note: cos 80 sin90 80 sin 10 45. (a) sin 16.35 0.2815 47. (a) sec 42 12 sec 42.2 1 1.3499 cos 42.2 1 sin 48 1 0.4348 tan 66.5 18 0.9598 60 56 0.9609 (b) sin 73 56 sin 73 60 46. (a) cos 16 18 cos 16 1 3.5523 (b) csc 16.35 sin 16.35 (b) csc 48 7 (b) cot 66.5 7 60 48. (a) cos 4 50 15 cos 4 50 15 60 3600 0.9964 1.3432 (b) sec 4 50 15 1 cos 4 50 15 1.0036 49. (a) cot 11 15 1 5.0273 tan 11.25 (b) tan 11 15 tan 11.25 0.1989 51. (a) csc 32 40 3 1 1.8527 sin 32.6675 (b) tan 44 28 16 tan 44.4711 0.9817 8 10 1.7946 60 3600 10 8 0.5572 60 3600 50. (a) sec 56 8 10 sec 56 (b) cos 56 8 10 cos 56 52. (a) sec (b) cot 95 20 32 2.6695 95 30 32 0.0699 Section 4.3 53. (a) sin 1 ⇒ 30 2 6 54. (a) cos (b) csc 2 ⇒ 30 6 58. (a) cot tan 3 3 (b) cot 1 ⇒ 45 4 23 ⇒ 60 3 3 2 2 ⇒ 45 3 tan 30 30 x cos 60. sin 60 30° 1 2 2 2 ⇒ 45 y 18 y 18 sin 60 18 3 x 303 x 61. tan 60 32 x 3 32 x 32 62. sin 45 r 20 r 20 20 202 sin 45 22 3 x 32 x 60° x 32 3 323 3 x 45 (b) tan 64. (a) h Height of the building: y h 270 feet x 123 45 tan 82 443.2 meters 6 Distance between friends: 132 82° 45 45 cos 82 ⇒ y y cos 82 3 Not drawn to scale 45 m 323.34 meters 66. tan 65. 3000 ft 1500 ft θ 1500 1 3000 2 6 6 h 3 135 (c) 2135 h x 45 tan 82 0 4 23 9 1 30 3 x 30 sin 4 (b) sec 2 3 ⇒ 60 opp adj tan 54 357 55. (a) sec 2 ⇒ 60 4 (b) sin 3 59. 63. tan 82 csc 3 3 4 57. (a) 1 ⇒ 60 2 3 3 2 ⇒ 45 (b) tan 1 ⇒ 45º 56. (a) tan 3 ⇒ 60 (b) cos 2 Right Triangle Trigonometry w 100 w 100 tan 54 137.6 feet 358 Chapter 4 Trigonometry 67. x 150 ft 23° 5 ft y (a) sin 23 x 145 x (b) tan 23 145 371.1 feet sin 23 y 145 y (c) Moving down the line: 145sin 23 61.85 feet per second 6 145 341.6 feet tan 23 Dropping vertically: 145 24.17 feet per second 6 68. Let h the height of the mountain. 69. (x1, y1) Let x the horizontal distance from where the 9 angle of elevation is sighted to the point at that level directly below the mountain peak. Then tan 3.5 tan 9 h h and tan 9 . x 13 x h h ⇒ x x tan 9 Substitute x tan 3.5 30° sin 30 y1 56 1256 28 y1 sin 3056 h into the expression for tan 3.5. tan 9 tan 3.5 56 h h 13 tan 9 h tan 9 h 13 tan 9 cos 30 x1 56 x1 cos 3056 3 2 56 283 x1, y1 283, 28 (x2, y2) h tan 3.5 13 tan 9 tan 3.5 h tan 9 13 tan 9 tan 3.5 htan 9 tan 3.5 13 tan 9 tan 3.5 h tan 9 tan 3.5 1.2953 h 56 60° The mountain is about 1.3 miles high. sin 60 y2 56 y2 sin 60°56 cos 60° 2356 283 x2 56 x2 cos 60°56 x2, y2 28, 283 70. tan 3 x 15 x 15 tan 3 d 5 2x 5 215 tan 3 6.57 centimeters 1256 28 Section 4.3 (e) 71. (a) 20 Angle, Height (in meters) 80 19.7 70 18.8 60 17.3 h 85° 50 15.3 h (b) sin 85 20 40 12.9 (c) h 20 sin 85 19.9 meters 30 10.0 (d) The side of the triangle labeled h will become shorter. 20 6.8 10 3.5 72. x 9.4, y 3.4 Right Triangle Trigonometry 359 (f) The height of the balloon decreases as decreases. 20 h θ 73. True, sin 20 y 0.34 10 cot 20 x 2.75 y cos 20 x 0.94 10 sec 20 10 1.06 x tan 20 y 0.36 x csc 20 10 2.92 y csc x 74. True, sec 30 csc 60 because sec90 csc . 75. False, 76. True, cot2 10 csc2 10 1 because 77. False, 1 cot2 csc2 2 2 1 1 ⇒ sin 60 csc 60 sin 60 1 sin x sin 60 2 2 2 1 sin 60 cos 30 cot 30 1.7321; sin 30 sin 30 sin 2 0.0349 cot 2 csc2 1 cot 2 csc2 1. 78. False, tan5 2 tan25. 79. This is true because the corresponding sides of similar triangles are proportional. tan5 tan 25 0.4663 2 tan2 5 tan 5tan 5 0.0077 80. Yes. Given tan , sec can be found from the identity 1 tan2 sec2 . 81. (a) 82. (a) (b) In the interval 0, 0.5, > sin . 0.1 0.2 0.3 0.4 0.5 sin 0.0998 0.1987 0.2955 0.3894 0.4794 0 18 36 54 72 90 sin 0 0.3090 0.5878 0.8090 0.9511 1 cos 1 0.9511 0.8090 0.5878 0.3090 0 —CONTINUED— (c) As approaches 0, sin approaches . 360 Chapter 4 Trigonometry 82. —CONTINUED— (b) sin increases from 0 to 1 as increases from 0 to 90. (c) cos decreases from 1 to 0 as increases from 0 to 90. (d) As the angle increases the length of the side opposite the angle increases relative to the length of the hypotenuse and the length of the side adjacent to the angle decreases relative to the length of the hypotenuse. Thus the sine increases and the cosine decreases. 83. x2 6x x2 4x 12 x2 12x 36 xx 6 x2 36 x 6x 2 84. x , x ±6 x2 2t 2 5t 12 t 2 16 2t 2 5t 12 2 2 9 4t 4t 12t 9 9 4t 2 85. x 6x 6 x 6x 6 4t 2 12t 9 t 2 16 2t 3t 4 2t 32t 3 2t 3 2t 3 3 , t ± , 4 3 2t3 2t t 4t 4 t 4 4t 2 3 2 x 3x 2x 2 2x 22 xx 2 2 x 2 x 2 x 4x 4 x 2x 22 3x2 4 2x2 4x 4 x2 2x x 2x 22 2x2 10x 20 2x2 5x 10 x 2x 22 x 2x 22 12 x x 4 4x 12 x x 1 86. 12 x 4, x 0, 12 12 12 x 4x 1 x x 3 1 Section 4.4 ■ Trigonometric Functions of Any Angle Know the Definitions of Trigonometric Functions of Any Angle. If is in standard position, x, y a point on the terminal side and r x2 y2 0, then: sin y r r csc , y 0 y cos x r r sec , x 0 x y tan , x 0 x x cot , y 0 y ■ You should know the signs of the trigonometric functions in each quadrant. ■ You should know the trigonometric function values of the quadrant angles 0, ■ You should be able to find reference angles. ■ You should be able to evaluate trigonometric functions of any angle. (Use reference angles.) ■ You should know that the period of sine and cosine is 2. 3 , , and . 2 2 Section 4.4 Trigonometric Functions of Any Angle Vocabulary Check 1. sin 4. y r 2. csc r x 5. 3. tan x cos r y x 6. cot 7. reference 1. (a) x, y 4, 3 (b) x, y 8, 15 r 16 9 5 r 64 225 17 sin y 3 r 5 csc r 5 y 3 sin y 15 r 17 csc r 17 y 15 cos x 4 r 5 sec 5 r x 4 cos 8 x r 17 sec 17 r x 8 tan y 3 x 4 cot x 4 y 3 tan y 15 x 8 cot x 8 y 15 (b) x 1, y 1 2. (a) x 12, y 5 r 12 12 2 r 122 52 13 sin y 5 r 13 sin 2 y 1 2 r 2 cos x 12 r 13 cos 2 1 x 2 r 2 tan y 5 5 x 12 12 tan y 1 1 x 1 csc 13 13 r y 5 5 csc 2 r 2 y 1 sec r 13 13 x 12 12 r 2 sec 2 x 1 cot x 12 12 y 5 5 cot 3. (a) x, y 3, 1 x 1 1 y 1 (b) x, y 4, 1 r 16 1 17 r 3 1 2 sin y 1 r 2 csc r 2 y sin y 17 r 17 csc r 17 y cos 3 x r 2 sec r 23 x 3 cos x 417 r 17 sec 17 r x 4 tan y 3 x 3 cot x 3 y tan y 1 x 4 cot x 4 y 361 362 Chapter 4 Trigonometry 4. (a) x 3, y 1 (b) x 4, y 4 r 32 12 10 r 42 42 42 sin 10 1 y 10 r 10 sin 2 y 4 r 42 2 cos x 3 310 r 10 10 cos 2 4 x r 4 2 2 tan y 1 x 3 tan y 4 1 x 4 csc 10 r 10 y 1 csc r 42 2 y 4 sec 10 r x 3 sec 42 r 2 x 4 cot x 3 3 y 1 cot 4 x 1 y 4 5. x, y 7, 24 7. x, y 4, 10 6. x 8, y 15 r 49 576 25 r 16 100 229 r 82 152 17 sin y 24 r 25 sin y 15 r 17 sin y 529 r 29 cos x 7 r 25 cos x 8 r 17 cos 229 x r 29 tan y 24 x 7 tan y 15 x 8 tan y 5 x 2 csc r 25 y 24 csc r 17 y 15 csc 29 r y 5 sec r 25 x 7 sec r 17 x 8 sec 29 r x 2 cot x 7 y 24 cot x 8 y 15 cot x 2 y 5 8. x 5, y 2 9. x, y 3.5, 6.8 r 5 2 29 2 sin 2 y 2 229 29 r 29 r 12.25 46.24 5849 10 sin y 685849 0.9 r 5849 cos 355849 x 0.5 r 5849 tan y 68 1.9 x 35 29 29 r sec x 5 5 csc 5849 r 1.1 y 68 x 5 5 y 2 2 sec 5849 r 2.2 x 35 cot x 35 0.5 y 68 x 529 5 cos 29 r 29 y 2 2 tan x 5 5 29 29 r csc y 2 2 cot Section 4.4 Trigonometric Functions of Any Angle 3 31 1 7 10. x 3 , y 7 2 2 4 4 r 72 314 2 2 1157 4 314 y 311157 0.9 r 1157 11574 x 72 141157 cos 0.4 r 1157 11574 31 y 314 tan 2.2 x 72 14 sin 11. sin < 0 ⇒ lies in Quadrant III or in Quadrant IV. cos < 0 ⇒ lies in Quadrant II or in Quadrant III. sin < 0 and cos < 0 ⇒ lies in Quadrant III. csc 1157 11574 r 1.1 31 y 314 sec 11574 1157 r 2.4 x 72 14 cot 72 14 x 0.5 y 314 31 12. sin > 0 and cos > 0 y x > 0 and > 0 r r Quadrant I 13. sin > 0 ⇒ lies in Quadrant I or in Quadrant II. tan < 0 ⇒ lies in Quadrant II or in Quadrant IV. sin > 0 and tan < 0 ⇒ lies in Quadrant II. 14. sec > 0 and cot < 0 r x > 0 and < 0 x y Quadrant IV 15. sin y 3 ⇒ x2 25 9 16 r 5 y 3 r 5 x 4 cos r 5 y 3 tan x 4 17. tan x 4 ⇒ y 2 25 16 9 r 5 in Quadrant III ⇒ y 3 in Quadrant II ⇒ x 4 sin 16. cos 5 r y 3 5 r sec x 4 4 x cot y 3 csc y 15 x 8 18. cos tan < 0 ⇒ y 15 x 8, y 15, r 17 sin y 15 r 17 8 x cos r 17 y 15 tan x 8 r 17 y 15 r 17 sec x 8 x 8 cot y 15 csc csc 8 x ⇒ y 15 r 17 sin < 0 and tan < 0 ⇒ is in Quadrant IV ⇒ y < 0 and x > 0. sin 5 3 5 sec 4 4 cot 3 3 y r 5 4 x cos r 5 y 3 tan x 4 sin y 15 15 r 17 17 x 8 cos r 17 y 15 15 tan x 8 8 17 15 17 sec 8 8 cot 15 csc 363 364 Chapter 4 19. cot Trigonometry 3 3 x y 1 1 20. csc cos > 0 ⇒ is in Quadrant IV ⇒ x is positive; x 3, y 1, r 10 sin cos 10 y r 10 csc x 310 r 10 sec 1 y tan x 3 21. sec r 10 y 10 r x 3 x cot 3 y 2 r ⇒ y2 4 1 3 x 1 4 r ⇒ x 15 y 1 cot < 0 ⇒ x 15 sin y 1 r 4 csc 4 cos 15 x r 4 sec tan 15 y x 15 cot 15 22. sin 0 ⇒ 0 n sin > 0 ⇒ is in Quadrant II ⇒ y 3 sec 1 ⇒ 2n y 0, x r sin y 3 r 2 csc r 23 y 3 sin 0 cos x 1 r 2 sec r 2 x cos tan y 3 x cot 3 x y 3 tan 23. cot is undefined, 3 ≤ ≤ ⇒ y0 ⇒ 2 2 sin 0 csc is undefined cos 1 sec 1 tan 0 cot is undefined 25. To find a point on the terminal side of use any point on the line y x that lies in Quadrant II. 1, 1 is one such point. x 1, y 1, r 2 sin 1 2 cos 1 2 tan 1 csc 2 sec 2 2 2 csc r is undefined y x r 1 r r sec r 1 x y 0 x cot x is undefined y 24. tan is undefined ⇒ n 2 2 3 , x 0, y r 2 y r r 1 sin csc 1 r r y x 0 r cos 0 sec is undefined. r r x y x 0 tan is undefined. cot 0 x y y ≤ ≤ 2 ⇒ 26. Let x > 0. x, 3 x, Quadrant III 1 r x 2 1 2 10 x x 9 3 10 y 13x r 10 10 x3 310 x x cos r 10 10 x3 sin 2 415 15 tan csc cot 1 y 13 x 1 x x 3 r 10 x3 10 y 13 x sec r 10 x3 10 x x 3 cot x x 3 y 13 x Section 4.4 27. To find a point on the terminal side of , use any point on the line y 2x that lies in Quadrant III. 1, 2 is one such point. 4 4x 3y 0 ⇒ y x 3 x, 3 x, Quadrant IV 4 2 25 sin 5 5 r 5 1 cos 5 5 csc sec cot 2 2 1 5 2 5 1 32. sec since 2 16 2 5 x x 9 3 sin 4 y 43 x r 53 x 5 csc cos x 3 x r 53 x 5 sec tan y 43 x 4 x x 3 tan 5 4 5 3 3 4 1 1 2 2 30. csc y 0 r r 1 3 1 2 y 1 since r 1 1 x 1 31. x, y 0, 1, r 1 3 corresponds to 0, 1. 2 sec sin y 1 2 r 37. 203 x 0 0 2 y 1 since corresponds to 0, 1. 2 38. 309 203 180 23 360 309 51 y y 203° x 1 (undefined) y 0 since corresponds to 1, 0. 36. cot r 1 ⇒ undefined y 0 r 1 3 ⇒ undefined 2 x 0 34. cot 33. x, y 0, 1, r 1 35. x, y 1, 0, r 1 θ′ 2 5 3 corresponds to 1, 0. 2 csc x 5 29. x, y 1, 0, r 1 sin 365 28. Let x > 0. x 1, y 2, r 5 tan Trigonometric Functions of Any Angle 309° x x θ′ 366 Chapter 4 Trigonometry 245 39. 40. 145 is coterminal with 215. 360 245 115 coterminal angle 215 180 35 180 115 65 y y θ′ x θ′ x − 145° −245° 41. 2 3 42. y 2 3 3 2π 3 7 4 2 θ′ y 7 4 4 7π 4 x x θ′ 43. 3.5 11 is coterminal 3 5 with . 3 3.5 3.5 x θ′ 45. 225, 360 225 45, Quadrant III sin 225 sin 45 2 cos 225 cos 45 2 2 2 47. 750 is coterminal with 30. 5 3 3 x θ′ 46. 300, 360 300 60, Quadrant IV sin 300 sin 60 3 2 1 cos 300 cos 60 2 48. 405 is coterminal with 315. 360 315 45, Quadrant IV 30, Quadrant I 1 2 cos 750 cos 30 tan 750 tan 30 2 11π 3 tan 300 tan 60 3 tan 225 tan 45 1 sin 750 sin 30 y 44. y sin405 sin 45 3 2 3 3 cos405 cos 45 2 2 2 2 tan405 tan 45 1 Section 4.4 240 180 60, Quadrant III 210 180 30, Quadrant III 1 2 cos150 cos 30 tan150 tan 30 51. 3 sin840 sin 60 3 2 cos840 cos 60 2 3 52. , , Quadrant I 4 4 53. , , Quadrant IV 6 6 sin 3 4 sin 3 3 2 sin 2 4 2 sin cos 4 1 cos 3 3 2 cos 2 4 2 cos tan 4 tan 3 3 3 tan 1 4 tan 3 is coterminal with . 2 2 3 2 3 6 tan 6 3 3 11 is coterminal with . 4 4 56. 10 4 is coterminal with . 3 3 3 0 2 sin sin 3 10 sin 3 3 2 3 is undefined. 2 2 11 sin 4 4 2 cos 2 11 cos 4 4 2 cos 10 1 cos 3 3 2 tan 11 tan 1 4 4 tan 10 tan 3 3 3 2 tan 57. 3 is coterminal with , . 2 2 2 3 , Quadrant II 4 4 58. 3 sin 1 2 2 2 3 cos 0 2 2 sin 3 tan which is undefined. 2 2 cos cos 4 , Quadrant III 3 3 25 7 is coterminal with . 4 4 tan 6 cos 6 1 2 cos sin 3 1 2 cos tan 55. 6 sin 6 2 2 sin sin 1 2 tan840 tan 60 3 3 4 , , Quadrant III 3 3 54. 367 50. 840 is coterminal with 240. 49. 150 is coterminal with 210. sin150 sin 30 Trigonometric Functions of Any Angle 7 in Quadrant IV. 4 4 2 25 sin 4 4 2 2 25 cos 4 4 2 25 tan 1 4 4 tan 368 Chapter 4 Trigonometry sin 59. 3 5 60. cot 3 1 cot 2 csc 2 sin2 cos2 1 cos2 1 cos2 1 3 5 10 2 4 5 1 cot 2 csc 2 cot 2 csc 2 1 2 1 3 sec2 1 2 sec2 1 9 4 2 sec2 1 sin 5 8 64. sec cos 1 1 ⇒ sec sec cos sec 1 8 58 5 4 tan2 tan 2 66. sec 225 1 2.0000 sin330 1 3.2361 cos 72 1 0.2245 tan 1.35 77. sin0.65 0.6052 11 8 9 4 9 2 1 65 16 tan > 0 in Quadrant III. 65. sin 10 0.1736 79. cot 2 tan2 sec 2 1 tan 74. cot 1.35 13 1 tan2 sec 2 cot 3 71. sec 72 2 13 4 sec cot < 0 in Quadrant IV. 68. csc330 3 sec < 0 in Quadrant III. 10 1 1 sin csc 10 10 63. cos 62. csc 2 cot 2 csc 2 csc > 0 in Quadrant II. csc 16 25 cot 2 sec2 1 tan2 10 csc 9 25 cos > 0 in Quadrant IV. cos 3 2 1 32 csc 2 cos2 1 sin2 cos2 61. tan 4 67. cos110 0.3420 1 28.6363 tan 178 69. tan 304 1.4826 70. cot 178 72. tan188 0.1405 73. tan 4.5 4.6373 75. tan 0.3640 9 9 0.3640 76. tan 78. sec 0.29 1 0.4142 11 tan 8 1 1.4142 cos 225 65 80. csc 1 1.0436 cos 0.29 15 14 1 4.4940 15 sin 14 Section 4.4 81. (a) sin Trigonometric Functions of Any Angle 369 1 ⇒ reference angle is 30 or and is in Quadrant I or Quadrant II. 2 6 Values in degrees: 30, 150 Values in radians: (b) sin 5 , 6 6 1 ⇒ reference angle is 30 or and is in Quadrant III or Quadrant IV. 2 6 Values in degrees: 210, 330 Values in radians: 82. (a) cos 2 2 7 11 , 6 6 and is in Quadrant I or IV. 4 ⇒ reference angle is 45 or Values in degrees: 45, 315 Values in radians: (b) cos 2 2 7 , 4 4 ⇒ reference angle is 45 or and is in Quadrant II or III. 4 Values in degrees: 135, 225 Values in radians: 83. (a) csc 3 5 , 4 4 23 ⇒ reference angle is 60 or and is in Quadrant I or Quadrant II. 3 3 Values in degrees: 60, 120 Values in radians: 2 , 3 3 (b) cot 1 ⇒ reference angle is 45 or and is in Quadrant II or Quadrant IV. 4 Values in degrees: 135, 315 Values in radians: 3 7 , 4 4 84. (a) sec 2 ⇒ reference angle is 60 or Quadrant I or IV. and is in 3 5 , 3 3 (b) sec 2 ⇒ reference angle is 60 or in Quadrant II or III. Values in degrees: 120, 240 Values in radians: 2 4 , 3 3 Quadrant I or Quadrant III. and is in 4 Values in degrees: 45, 225 Values in degrees: 60, 300 Values in radians: 85. (a) tan 1 ⇒ reference angle is 45 or Values in radians: and is 3 5 , 4 4 (b) cot 3 ⇒ reference angle is 30 or is in Quadrant II or Quadrant IV. Values in degrees: 150, 330 Values in radians: 5 11 , 6 6 and 6 370 Chapter 4 Trigonometry ⇒ reference angle is 60 or and is 2 3 in Quadrant I or II. 86. (a) sin 3 87. (a) New York City: N 22.099 sin0.522t 2.219 55.008 Values in degrees: 60, 120 Values in radians: Fairbanks: 2 , 3 3 F 36.641 sin0.502t 1.831 25.610 (b) (b) sin ⇒ reference angle is 60 or and 2 3 is in Quadrant III or IV. 3 Month Values in degrees: 240, 300 Values in radians: 4 5 , 3 3 New York City Fairbanks February 34.6 1.4 March 41.6 13.9 May 63.4 48.6 June 72.5 59.5 August 75.5 55.6 September 68.6 41.7 November 46.8 6.5 (c) The periods are about the same for both models, approximately 12 months. 88. S 23.1 0.442t 4.3 cos t 6 89. yt 2 cos 6t (a) y0 2 cos 0 2 centimeters (a) For February 2006, t 2. S 23.1 0.4422 4.3 cos 2 26,134 units 6 (b) For February 2007, t 14. S 23.1 0.44214 4.3 cos 14 31,438 units 6 (b) y 4 2 cos2 0.14 centimeter (c) y 2 2 cos 3 1.98 centimeters 1 3 1 (c) For June 2006, t 6. S 23.1 0.4426 4.3 cos 6 21,452 units 6 (d) For June 2007, t 18. S 23.1 0.44218 4.3 cos 18 26,756 units 6 90. yt 2et cos 6t 91. I0.7 5e1.4 sin 0.7 0.79 ampere (a) t 0 y0 2e0 cos 0 2 centimeters 1 (b) t 4 y14 2e14 cos6 14 0.11 centimeters 1 (c) t 2 y2 2e12 cos6 1 I 5e2t sin t 12 1.2 centimeters Section 4.4 92. sin 6 6 ⇒ d d sin 371 93. False. In each of the four quadrants, the sign of the secant function and the cosine function will be the same since they are reciprocals of each other. (a) 30 d Trigonometric Functions of Any Angle 6 6 12 miles sin 30 12 (b) 90 d 6 6 6 miles sin 90º 1 (c) 120 d 6 6.9 miles sin 120 94. False. For example, if n 1 and 225, 0 ≤ 135 ≤ 360, but 360n 135 is not the reference angle. The reference angle would be 45. For in Quadrant II, 180 . For in Quadrant III, 180. For in Quadrant IV, 360 . 95. As increases from 0 to 90, x decreases from 12 cm to 0 cm and y increases from 0 cm to 12 cm. y x Therefore, sin increases from 0 to 1 and cos decreases from 1 to 0. Thus, 12 12 y tan increases without bound, and when 90 the tangent is undefined. x 96. Determine the trigonometric function of the reference angle and prefix the appropriate sign. 97. y x2 3x 4 x 4x 1 98. y 2x2 5x x2x 5 x-intercepts: 4, 0, 1, 0 y-intercept: 0, 4 8 No asymptotes 4 Domain: All real numbers x 5 x-intercepts: 0, 0, 2, 0 y y-intercepts: 0, 0 6 −8 −6 2 No asymptotes 2 (− 4, 0) y 1 (1, 0) −2 −2 2 −4 4 x 6 8 Domain: All real numbers x (52 , 0( (0, 0) −3 −2 −1 −1 1 2 3 4 x 5 −2 (0, −4) −3 −4 −8 99. f x x3 8 100. gx x4 2x2 3 x2 3x2 1 y x2 3x 1x 1 12 x-intercept: 2, 0 10 y-intercept: 0, 8 x-intercepts: 1, 0, 1, 0 (0, 8) 4 y-intercepts: 0, 3 No asymptotes Domain: All real numbers x y (− 2, 0) x −8 − 6 − 4 2 −4 4 6 3 2 No asymptotes 8 Domain: All real numbers x (−1, 0) 1 (1, 0) x −4 −3 −2 2 −3 −4 (0, − 3) 3 4 372 Chapter 4 101. f x Trigonometry x7 x7 x2 4x 4 x 22 y 4 x-intercept: 7, 0 2 (7, 0) x y-intercept: 7 0, 4 −8 −2 2 4 0, − 7 4 ( 6 8 ( Vertical asymptote: x 2 Horizontal asymptote: y 0 Domain: All real numbers except x 2 102. hx x2 1 x 1x 1 x5 x5 103. y 2x1 x-intercepts: 1, 0, 1, 0, To find the y-intercept, let x 0: y-intercept: 0, 21 y-intercept: 02 1 1 05 5 Horizontal asymptote: y 0 Domain: All real numbers x 0, 51 Vertical asymptote: x 5 To find the slant asymptote, use long division: x 1 0 1 2 3 y 1 4 1 2 1 2 4 1 24 x5 x5 x5 x2 y Slant asymptote: y x 5 5 Domain: All real numbers except x 5 4 3 y 8 2 ) ) 0, 1 1 2 (1, 0) x − 12 −8 (−1, 0) −8 4 1 0, − 5 −2 −1 ( ( x 1 2 3 −1 − 16 − 24 104. y 3x1 2 y 7 This is an exponential function (always positive) translated two units upward. There are no x-intercepts. 6 5 To find the y-intercept, let x 0: 3 y 301 2 3 2 5 y-intercepts: 0, 5 The horizontal asymptote is the horizontal asymptote of y 3x1 translated two units upward. Horizontal asymptote: y 2 Domain: All real numbers x (0, 5) 2 1 −5 −4 −3 −2 −1 x 1 2 3 4 Section 4.5 105. y ln x4 Graphs of Sine and Cosine Functions y 12 Domain: All real numbers except x 0 9 x-intercepts: ± 1, 0 6 Vertical asymptote: x 0 (−1, 0) − 12 − 9 − 6 − 3 (1, 0) x 3 6 9 12 106. y log10x 2 y To find the x-intercept, let y 0: 3 2 0 log10x 2 ⇒ 10 x 2 ⇒ x 1 x-intercepts: 1, 0 To find the y-intercept, let x 0: y log10x 2 log10 2 0.301 (−1, 0) −3 −1 x 1 2 3 −1 −2 −3 y-intercepts: 0, 0.301 The vertical asymptote is the horizontal asymptote of y log10 x translated two units to the left. Vertical asymptote: x 2 Domain: All real numbers x such that x > 2 Section 4.5 Graphs of Sine and Cosine Functions ■ You should be able to graph y a sinbx c and y a cosbx c. Assume b > 0. ■ Amplitude: a ■ 2 Period: b ■ Shift: Solve bx c 0 and bx c 2. ■ Key increments: 1 (period) 4 Vocabulary Check 1. cycle 2. amplitude 2 b 4. phase shift 3. 5. vertical shift 373 374 Chapter 4 Trigonometry 2. y 2 cos 3x 1. y 3 sin 2x Period: 2 2 Period: Amplitude: 3 3 4. y 3 sin Period: x 3 Amplitude: a 3 3 2 2 1 10. y 1 sin 8x 3 Period: Amplitude: a 13. y 1 3 1 sin 2x 4 Period: 2 1 2 Amplitude: Period: 1 1 4 4 16. f x cos x, gx cosx g is a horizontal shift of f units to the left. 6. y 2 6 3 Amplitude: 1 1 2 2 2x 3 2 2 3 b 23 2x 1 cos 2 3 14. y 1 1 2 2 gx cos 2x The period of f is twice that of g. 3 2 Period: 2 2 20 b 10 Amplitude: a 2 10 5 Amplitude: 3 3 5 x cos 2 4 2 2 8 b 14 Amplitude: a 5 2 15. f x sin x 2 3 17. f x cos 2x gx cos 2x The graph of g is a reflection in the x-axis of the graph of f. 19. f x cos x Period: 2 x cos 3 10 Period: 2 2 4 b 2 Amplitude: a 12. y 2 3 23 Amplitude: 5 5 2 2 9. y 3 sin 10x Period: 3 x cos 2 2 Period: Amplitude: a 1 1 11. y 2 2 b 8 4 2 4 12 Amplitude: x 1 sin 2 3 Period: Amplitude: 2 2 Period: 8. y cos 7. y 2 sin x Period: 2 2 b 3 x 5 cos 2 2 Amplitude: a 2 5. y 2 2 6 b 13 3. y 20. f x sin x, gx sin 3x The period of g is one-third the period of f. gx sinx The graph of g is a horizontal shift to the right units of the graph of f a phase shift. 18. f x sin 3x, gx sin3x g is a reflection of f about the y-axis. 21. f x sin 2x f x 3 sin 2x The graph of g is a vertical shift three units upward of the graph of f. Section 4.5 22. f x cos 4x, gx 2 cos 4x g is a vertical shift of f two units downward. Graphs of Sine and Cosine Functions 23. The graph of g has twice the amplitude as the graph of f. The period is the same. 25. The graph of g is a horizontal shift units to the right of the graph of f. 24. The period of g is one-third the period of f. 26. Shift the graph of f two units upward to obtain the graph of g. 27. f x 2 sin x Period: y 2 2 2 b 1 5 4 3 g f Amplitude: 2 −π 2 Symmetry: origin Key points: Intercept 0, 0 Minimum , 2 2 Intercept Maximum , 0 3 ,0 2 x 3π 2 Intercept −5 2, 0 Since gx 4 sin x 2 f x, generate key points for the graph of gx by multiplying the y-coordinate of each key point of f x by 2. 28. f x sin x Period: y 2 2 2 b 1 2 g f Amplitude: 1 6π Symmetry: origin Key points: Intercept 0, 0 Since gx sin Maximum ,1 2 Intercept Minimum Intercept , 0 3 , 1 2 2, 0 x −2 3x f 3x , the graph of gx is the graph of f x, but stretched horizontally by a factor of 3. Generate key points for the graph of gx by multiplying the x-coordinate of each key point of f x by 3. 29. f x cos x Period: y 2 2 2 b 1 g Amplitude: 1 Symmetry: y-axis Key points: Maximum 0, 1 375 π Intercept Minimum Intercept Maximum 2 , 0 , 1 32, 0 2, 1 −1 2π f Since gx 1 cosx f x 1, the graph of gx is the graph of f x, but translated upward by one unit. Generate key points for the graph of gx by adding 1 to the y-coordinate of each key point of f x. x 376 Chapter 4 Trigonometry 30. f x 2 cos 2x Period: y 2 2 b 2 2 f g Amplitude: 2 π Symmetry: y-axis Key points: Maximum 0, 2 Intercept Minimum Intercept Maximum 4 , 0 2 , 2 34, 0 , 2 x −2 Since gx cos 4x 2 f 2x, the graph of gx is the graph of f x, but 1 i) shrunk horizontally by a factor of 2, 1 ii) shrunk vertically by a factor of 2, and iii) reflected about the x-axis. Generate key points for the graph of gx by i) dividing the x-coordinate of each key point of f x by 2, and ii) dividing the y-coordinate of each key point of f x by 2. 1 x 31. f x sin 2 2 Period: y 5 2 2 4 b 12 g 4 3 Amplitude: 1 2 2 1 Symmetry: origin Key points: Intercept 0, 0 Minimum Intercept Maximum Intercept , 21 2, 0 3, 12 4, 0 f −π −1 3π x 1 x sin 3 f x, the graph of gx is the graph of f x, but translated upward by three units. 2 2 Generate key points for the graph of gx by adding 3 to the y-coordinate of each key point of f x. Since gx 3 32. f x 4 sin x Period: y 2 2 2 b 4 f 2 1 x Amplitude: 4 g Symmetry: origin Key points: Intercept 0, 0 Maximum 1 ,2 2 Intercept 1, 0 Minimum 3 , 2 2 Intercept −8 2, 0 Since gx 4 sin x 3 f x 3, the graph of gx is the graph of f x, but translated downward by three units. Generate key points for the graph of gx by subtracting 3 from the y-coordinate of each key point of f x. Section 4.5 Graphs of Sine and Cosine Functions 33. f x 2 cos x Period: 377 y 2 2 2 b 1 3 f Amplitude: 2 π Symmetry: y-axis x 2π g Key points: Maximum Intercept Minimum Intercept Maximum , 2 3 ,0 2 2, 2 ,0 2 0, 2 −3 Since gx 2 cosx f x , the graph of gx is the graph of f x, but with a phase shift (horizontal translation) of . Generate key points for the graph of gx by shifting each key point of f x units to the left. 34. f x cos x Period: y 2 2 2 b 1 2 g Amplitude: 1 π Symmetry: y-axis x 2π f Key points: Minimum Intercept Maximum Intercept Minimum , 1 3 ,0 2 2, 1 ,0 2 0, 1 −2 Since gx cosx f x , the graph of gx is the graph of f x, but with a phase shift (horizontal translation) of . Generate key points for the graph of gx by shifting each key point of f x units to the right. 35. y 3 sin x Period: 2 4 Amplitude: 3 2 3 − π 2 π 2 −3 2 −3 0, 3, 2 , 0, , 3, 1 1 − 2π −π π 2π x −1 1 2 , 4, , 0, −2 −1 − 43 1 y Period: 2 2 3 1 3 3 1 , 0 , 2, 2 3 1 4 38. y 4 cos x 1 Key points: 0, 0, y 2 Key points: 4 3 1 x 3π 2 3 Period: 2 1 y 2 , 4, 2, 0 1 cos x 3 Amplitude: −π 2 −4 3 , 3 , 2, 0 2 37. y Amplitude: 1 0, 0, , 3 , , 0, 2 1 sin x 4 Period: 2 3 Key points: 36. y y 4 Amplitude: 4 π 2 π 2π x Key points: 0, 4, 2 , 0, , 4, 3 , 0 , 2, 4 2 − 2π −π π −2 −4 2π x 378 Chapter 4 Trigonometry x 2 39. y cos 40. y sin 4x y 2 y 2 2 Period: 4 Period: Amplitude: 1 Amplitude: 1 Key points: − 2π 2π 0, 1, , 0, 2, 1, −1 3, 0, 4, 1 −2 4π 1 x x π 4 Key points: 0, 0, 8 , 1, 4 , 0, 3 −2 8 , 1, 2 , 0 41. y cos 2x Period: 42. y sin y 2 1 2 2 1 x 1 Period: 2 2 1 Amplitude: 1 −6 0, 0, 2, 1, 4, 0, −2 −2 6, 1, 8, 0 2 2x ; a 1, b , c0 3 3 44. y 10 cos 2 3 23 Period: x 6 2 12 6 Amplitude: 10 Amplitude: 1 Key points: 0, 0, Key points: 4, 1, 2, 0, 4, 1, 3, 0 3 3 9 0, 10, 3, 0, 6, 10, 9, 0, 12, 10 y y 3 12 2 8 4 x −1 2 3 − 12 x −4 4 8 12 −2 −3 − 12 45. y sin x ; a 1, b 1, c 4 4 y Period: 2 3 Amplitude: 1 1 2 Shift: Set x 0 and x 2 4 4 x 4 Key points: −2 Key points: 1 1 3 0, 1, , 0 , , 1 , , 0 4 2 4 43. y sin y 2 8 Period: 4 Amplitude: 1 Key points: x 4 3 −π −2 9 x 4 5 7 π −3 9 4 , 0, 4 , 1, 4 , 0, 4 , 1, 4 , 0 x x 2 6 Section 4.5 46. y sinx Graphs of Sine and Cosine Functions 47. y 3 cosx Period: 2 Period: 2 Amplitude: 1 Amplitude: 3 Shift: Set x 0 and x Key points: , 0, Shift: Set x 0 x 2 x 2 and x x 3 x Key points: , 3, , 0 , 0, 3, , 0 , , 3 2 2 32, 1, 2, 0, 52, 1, 3, 0 y y 6 2 4 2 −π 2 −1 x 3π 2 −π x π −4 −2 −6 48. y 4 cos x 4 49. y 2 sin 2x 3 Period: 2 Period: 3 Amplitude: 4 Amplitude: 1 Shift: Set x 0 4 x and 4 x 2 4 x Key points: 0, 2, 7 4 34, 1, 32, 2, 94, 3, 3, 2 y 5 3 5 7 Key points: , 4 , , 0 , , 4 , ,0 , ,4 4 4 4 4 4 4 2 y 1 6 x –3 –2 –1 1 2 −1 2 −π −2 π 2π x −4 −6 50. y 3 5 cos Period: 379 t 12 y 16 12 8 4 2 24 12 Amplitude: 5 Key points: 0, 2, 6, 3, 12, 8, 18, 3, 24, 2 t − 12 4 −8 − 12 − 16 − 20 − 24 12 3 380 Chapter 4 51. y 2 Period: Trigonometry 1 cos 60x 10 52. y 2 cos x 3 Period: 2 1 2 60 30 Amplitude: 2 1 Amplitude: 10 Key points: Vertical shift two units upward 0, 1, Key points: 1 1 1 1 0, 2.1, ,2 , , 1.9 , ,2 , , 2.1 120 60 40 30 2 , 3, , 5, 32, 3, 2, 1 y 1 −π y π 2π 2.2 −4 −5 −6 −7 1.8 x − 0.1 0 0.1 0.2 53. y 3 cosx 3 y Period: 2 4 2 Amplitude: 3 Shift: Set x 0 x 2 and x π −8 4 4 y 10 Period: 2 6 Amplitude: 4 Shift: Set x 0 4 x Key points: 55. y and x 4 4 2 4 x 2 − 2π − π 7 4 2π 3π x 4 , 8, 4 , 4, 34, 0, 54, 4, 74, 8 y 4 3 Period: 4 2 2 Amplitude: 3 1 −1 x x 0 and 2 2 4 2 4 x Key points: π −4 2 x 2 1 cos ; a , b ,c 3 2 4 3 2 4 Shift: x x Key points: , 0, , 3 , 0, 6, , 3 , , 0 2 2 54. y 4 cos x 2π 2 2 x 3 −2 −3 −4 9 2 5 2 7 9 2 2 , 3 , 2 , 0, 2 , 3 , 2 , 0, 2 , 3 π 4π x x Section 4.5 Graphs of Sine and Cosine Functions 56. y 3 cos6x Period: y 2 6 3 3 2 Amplitude: 3 Shift: Set 6x 0 x Key points: and 6 x x π 6x 2 6 6 , 3, 12 , 0, 0, 3, 12 , 0, 6 , 3 57. y 2 sin4x 58. y 4 sin 4 3x 3 2 59. y cos 2x 8 −6 6 1 2 3 − 12 12 −3 −4 60. y 3 cos 3 −8 x 2 2 2 61. y 0.1 sin −1 x 10 62. y 1 sin 120 t 100 0.12 2 −6 0.02 6 − 20 − 0.03 − 0.12 −6 63. f x a cos x d Amplitude: 20 1 3 1 2 ⇒ a 2 2 − 0.02 64. f x a cos x d Amplitude: 1 3 2 2 Vertical shift one unit upward of 1 2 cos 0 d gx 2 cos x ⇒ d 1 d 1 2 1 Thus, f x 2 cos x 1. a 2, d 1 65. f x a cos x d Amplitude: 1 8 0 4 2 Since f x is the graph of gx 4 cos x reflected in the x-axis and shifted vertically four units upward, we have a 4 and d 4. Thus, f x 4 cos x 4. 0.03 66. f x a cos x d Amplitude: 2 4 1 2 Reflected in the x-axis: a 1 4 1 cos 0 d d 3 a 1, d 3 381 382 Chapter 4 Trigonometry 68. y a sinbx c 67. y a sinbx c Amplitude: a 3 Amplitude: 2 ⇒ a 2 Since the graph is reflected in the x-axis, we have a 3. Period: 4 2 1 4 ⇒ b b 2 2 ⇒ b2 Period: b Phase shift: c 0 Phase shift: c 0 1 a 2, b , c 0 2 Thus, y 3 sin 2x. 69. y a sinbx c 70. y a sinbx c Amplitude: a 2 Amplitude: 2 ⇒ a 2 Period: 2 ⇒ b 1 Period: 2 Phase shift: bx c 0 when x 1 4 c0 ⇒ Thus, y 2 sin x 4 c 4 . 4 71. y1 sin x y2 2 4 ⇒ b b 2 c 1 ⇒ c b 2 a 2, b ,c 2 2 72. y1 cos x 2 1 2 Phase shift: −2 2 y2 1 2 −2 y1 y2 when x , In the interval 2, 2, −2 −2 5 7 11 1 sin x when x , , , . 2 6 6 6 6 73. y 0.85 sin t 3 v 1.00 (a) Time for one cycle 2 6 sec 3 0.75 0.50 0.25 t 60 10 cycles per min (b) Cycles per min 6 2 − 0.25 (c) Amplitude: 0.85; Period: 6 4 8 10 − 1.00 3 9 Key points: 0, 0, , 0.85 , 3, 0, , 0.85 , 6, 0 2 2 74. v 1.75 sin t 2 (a) Period (b) 1 cycle 4 seconds 2 4 seconds 2 2 60 seconds 15 cycles per minute 1 minute (c) v 3 2 1 t 1 −2 −3 3 5 7 Section 4.5 (b) f 2 1 seconds 880 440 (a) Period: 1 440 cycles per second p (b) 1 1 77. (a) a high low 83.5 29.6 26.95 2 2 (c) 383 5 t 3 76. P 100 20 cos 75. y 0.001 sin 880t (a) Period: Graphs of Sine and Cosine Functions 2 6 seconds 53 5 1 heartbeat 65 seconds 60 seconds 50 heartbeats per minute 1 minute 100 p 2high time low time 27 1 12 b 2 2 p 12 6 0 c 7 ⇒ c7 3.67 b 6 The model is a good fit. (d) Tallahassee average maximum: 77.90 1 1 d high low 83.5 29.6 56.55 2 2 Ct 56.55 26.95 cos (b) 12 0 Chicago average maximum: 56.55 The constant term, d, gives the average maximum temperature. 6t 3.67 100 (e) The period for both models is 2 12 months. 6 This is as we expected since one full period is one year. 0 12 (f) Chicago has the greater variability in temperature throughout the year. The amplitude, a, determines this 1 variability since it is 2high temp low temp. 0 The model is a good fit. 78. (a) and (c) (b) Vertical shift: y Amplitude: Percent of moon’s face illuminated 1.0 1 1 ⇒ d 2 2 1 1 ⇒ a 2 2 0.8 0.6 Period: 0.4 0.2 x 10 20 30 40 88768 7.4 (average length of interval in data) 5 2 47.4 29.6 b Day of the year Reasonably good fit (d) Period is 29.6 days. (e) March 12 ⇒ x 71. y 0.44 44% The Naval observatory says that 50% of the moon’s face will be illuminated on March 12, 2007. b 2 0.21 29.6 Horizontal shift: 0.213 7.4 C 0 C 0.92 y 1 1 sin0.21x 0.92 2 2 384 Chapter 4 Trigonometry 79. C 30.3 21.6 sin Period (a) 2 t 365 10.9 80. (a) Period 2 365 2 365 2 12 minutes 6 The wheel takes 12 minutes to revolve once. (b) Amplitude: 50 feet Yes, this is what is expected because there are 365 days in a year. The radius of the wheel is 50 feet. (b) The average daily fuel consumption is given by the amount of the vertical shift (from 0) which is given by the constant 30.3. (c) 60 (c) 110 0 20 0 0 365 0 The consumption exceeds 40 gallons per day when 124 < x < 252. 81. False. The graph of sin(x 2) is the graph of sin(x) translated to the left by one period, and the graphs are indeed identical. 1 82. False. y 2 cos 2x has an amplitude that is half that of y cos x. For y a cos bx, the amplitude is a . 83. True. Since cos x sin x , y cos x sin x , and so is a reflection in the x-axis of y sin x . 2 2 2 84. Answers will vary. 85. Since the graphs are the same, the conjecture is that y 2 sinx cos x f=g 1 − 3π 2 π 2 3π 2 . 2 x −2 86. f x sin x, gx cos x x 2 0 1 0 1 0 sin x cos x 2 2 y 2 Conjecture: sin x cos x f=g 1 3 2 2 0 1 0 0 1 0 − 3π 2 π 2 −2 3π 2 x 2 Section 4.5 87. (a) Graphs of Sine and Cosine Functions 385 x3 x5 x7 3! 5! 7! x4 x6 x2 cos x 1 2! 4! 6! (c) sin x x 2 −2 2 2 2 −2 The graphs are nearly the same for (b) < x < . 2 2 −2 2 −2 2 −2 2 −2 −2 The graphs now agree over a wider range, 2 3 3 < x < . 4 4 −2 The graphs are nearly the same for 88. (a) sin sin < x < . 2 2 1 1 12 3 125 0.4794 2 2 3! 5! 1 0.4794 (by calculator) 2 63 65 0.5000 (c) sin 1 6 3! 5! sin (b) sin 1 1 0.5 (by calculator) 6 (e) cos 1 1 1 1 0.8417 3! 5! sin 1 0.8415 (by calculator) 0.5 2 0.54 0.8776 2! 4! cos0.5 0.8776 (by calculator) (d) cos0.5 1 (f) cos 1 1 0.5417 2! 4! cos cos 1 0.5403 (by calculator) 42 42 1 0.7074 4 2! 4! 0.7071 (by calculator) 4 The error in the approximation is not the same in each case. The error appears to increase as x moves farther away from 0. 89. log10 x 2 log10x 212 12 log10x 2 90. log2x2x 3 log2 x2 log2x 3 2 log2 x log2x 3 91. ln t3 ln t 3 lnt 1 3 ln t lnt 1 t1 z 92. ln 2 z 1 z 1 ln 2 ln z lnz2 1 1 2 z 1 2 93. 1 2 log10 x log10 y 12 log10xy log10 xy 94. 2 log2 x log2xy log2 x2 log2xy log2 x2(xy) log2 x3y 95. ln 3x 4 ln y ln 3x ln y4 ln 3xy 4 1 1 ln z lnz2 1 2 2 386 96. Chapter 4 Trigonometry 97. Answers will vary. 1 1 ln 2x 2 ln x 3 ln x ln 2x ln x2 ln x3 2 2 1 2x ln 2 ln x3 2 x 2xx ln x 2x lnx x ln 3 2 3 2 lnx22x Section 4.6 ■ ■ Graphs of Other Trigonometric Functions You should be able to graph y a tanbx c y a cotbx c y a secbx c y a cscbx c When graphing y a secbx c or y a cscbx c you should first graph y a cosbx c or y a sinbx c because (a) The x-intercepts of sine and cosine are the vertical asymptotes of cosecant and secant. (b) The maximums of sine and cosine are the local minimums of cosecant and secant. (c) The minimums of sine and cosine are the local maximums of cosecant and secant. ■ You should be able to graph using a damping factor. Vocabulary Check 1. vertical 2. reciprocal 3. damping 5. x n 6. , 1 1, 7. 2 1. y sec 2x Period: 2 2 Matches graph (e). 2. y tan Period: x 2 2 b 12 Asymptotes: x , x 4. 3. y 1 cot x 2 Period: 1 Matches graph (a). Matches graph (c). 4. y csc x Period: 2 Matches graph (d). 5. y 1 x sec 2 2 Period: 2 2 4 b 2 Asymptotes: x 1, x 1 Matches graph (f). 6. y 2 sec Period: x 2 2 2 4 b 2 Asymptotes: x 1, x 1 Reflected in x-axis Matches graph (b). Section 4.6 7. y 1 tan x 3 Graphs of Other Trigonometric Functions 8. y y 1 tan x 4 y Period: 3 2 Period: Two consecutive asymptotes: 1 Two consecutive asymptotes: x and x 2 2 −π x π 3 2 x ,x 2 2 4 0 4 x 4 0 4 y 1 3 0 1 3 y 1 4 0 1 4 Period: 4 2 1 Two consecutive asymptotes: 3x ⇒ x 2 6 3x Period: 3 −π 3 x π 3 x 12 0 12 y 1 0 1 1 11. y sec x 2 y 1 −π x π 3 y 1 0 1 2 x 1 −3 1 2 5 6 y 2 1 2 Period: 2 −1 0 1 4 0 3 y 3 3 2 0 3 1 4 1 2 y 2 4 2 8 6 4 Two consecutive asymptotes: x 0, x x 2π 14. y 3 csc 4x 3 −2 1 4 −8 Two consecutive asymptotes: 1 2 Two consecutive asymptotes: −4 1 sec x 4 y 1 1 6 3 1 4 x y y x 0, x 1 x 2 2 Period: x 3 13. y csc x x 2 x ,x 2 2 x ,x 2 2 x 1 Period: 2 2 Two consecutive asymptotes: y Two consecutive asymptotes: 12. y 3 Period: 2 −3 1 1 x ,x 2 2 ⇒ x 2 6 x π 10. y 3 tan x y 3 1 −π x 9. y tan 3x 387 2 −π 4 4 −4 x 24 8 5 24 y 6 3 6 −2 π 4 x 388 Chapter 4 Trigonometry 15. y sec x 1 Period: 16. y 2 sec 4x 2 y 2 2 −3 −2 −1 x 1 −1 2 3 y 1 17. y csc Period: 1 3 6 4 Two consecutive asymptotes: 2 x ,x 8 8 1 1 x ,x 2 2 x 2 4 2 Period: Two consecutive asymptotes: y 0 1 3 x 0 1 y 2 x 2 6 2 4 12 2 Two consecutive asymptotes: x π 12 0 2 6 4 2 Two consecutive asymptotes: x 3 5 3 x 2 3 2 5 2 y 2 1 2 y 2 1 2 Period: x 2 y 20. y 3 cot 3 2 12 2 Period: 1 Two consecutive asymptotes: − 2π x 2π x 0 ⇒ x0 2 x y 2 1 1 sec 2x 2 Period: x y 1 3 2 0 1 y 6 2 2 4 2 Two consecutive asymptotes: x −2 2 3 0 6 1 2 1 −π x 1 2 1 3 2 y 3 0 3 1 22. y tan x 2 y 2 2 6 x 2 x 0, x 2 x ⇒ x 2 2 21. y x π 2π x 0, x 3 x 0, x 2 19. y cot x π 2 y 2 6 13 Period: 4 0 π 4 x 3 18. y csc y 12 −π 4 y 3 Period: π x 2 Two consecutive asymptotes: −π x ,x 2 2 x y 1 2 4 0 4 0 1 2 π x Section 4.6 23. y tan x 4 y 24. y tanx 6 Period: 4 4 4 2 x −4 Two consecutive asymptotes: 4 x 1 0 1 y 1 0 1 25. y csc x Period: 2 4 Two consecutive asymptotes: 2 1 − 3π 2 x 6 2 5 6 y 2 1 2 Two consecutive asymptotes: 2 1 x ,x 2 2 y 29. y 4 −π π 2π 3π x 2 Two consecutive asymptotes: y 1 2 4 y 1 0 1 4 7 12 1 4 1 2 y 2 2 Two consecutive asymptotes: 2 π π 2 −1 3π 2 2π 3 4 x −2 x 12 4 5 12 y 2 1 2 y 2 2 3 2 Two consecutive asymptotes: 1 x 1 x 1 3 0 1 3 y 1 0 1 2 2 y Period: 2π 3 x ,x 4 4 12 0 1 4 30. y 2 cot x y Period: 2 x 4 x π 1 1 x ,x 2 2 3 2 1 csc x 4 4 −1 1 x Period: 3 0 2 3 28. y sec x 1 y 4 3 Two consecutive asymptotes: x 0, x Period: 2 x π 2 27. y 2 secx x 4 Period: 3 x 0, x Period: 26. y csc2x y 389 y x ,x 2 2 x ⇒ x 2 4 2 x ⇒ x2 4 2 Graphs of Other Trigonometric Functions x Two consecutive asymptotes: − 3π 2 x ,x 2 2 x y 2 4 3π 2 −2 0 4 0 2 x 390 Chapter 4 31. y tan Trigonometry x 3 32. y tan 2x 3 5 −5 4 − 3 4 5 3 4 − 2 2 −3 −5 34. y sec x ⇒ y 1 cos x 35. y tan x −4 4 36. y 2 1 cot x 4 2 4 tan x 3 − 3 2 3 2 −2 1 3 −3 2 cos 4x 33. y 2 sec 4x 2 3 −3 − 3 2 3 2 −3 37. y csc4x y 38. y 2 sec(2x ) ⇒ 3 1 sin4x − 2 y 2 4 2 cos2x − −3 39. y 0.1 tan 4x 4 −4 40. y 0.6 −6 1 x sec 3 2 2 ⇒ y 1 x 3 cos 2 2 6 2 −0.6 −6 6 −2 41. tan x 1 43. cot x 42. tan x 3 7 3 5 x , , , 4 4 4 4 5 2 4 x , , , 3 3 3 3 y 3 3 4 2 5 , x , , 3 3 3 3 y 2 y 2 3 2 1 1 π 2π x π 2π x −3 π 2 3π 2 x Section 4.6 Graphs of Other Trigonometric Functions 46. sec x 2 45. sec x 2 44. cot x 1 x 7 3 5 , , , 4 4 4 4 x± 2 4 ,± 3 3 x 5 5 , , , 3 3 3 3 y y 391 y 3 2 − π2 − 3π 2 1 x π 2 −2π 3π 2 −π 1 π 2π x −2π −π π 2π x −3 47. csc x 2 x 48. csc x y 7 5 3 , , , 4 4 4 4 −π 2 2 1 cos x π 2 2 1 x 3π 2 − 3π 2 y −π 3π 2 y tanx tan x 3 x π 2 2 f x tan x 50. 4 f x secx 3 2 1 cosx −1 3 2 4 5 x , , , 3 3 3 3 2 − 3π f x sec x y 3 1 49. 23 3 −π π 2π x Thus, the function is odd and the graph of y tan x is symmetric about the origin. 1 cos x − 3π 2 − π2 π 2 −3 f x Thus, f x sec x is an even function and the graph has y-axis symmetry. 51. f x 2 sin x gx (a) 52. f x tan 1 csc x 2 x 1 x , gx sec 2 2 2 (a) 3 g y −1 1 f 3 1 −1 −3 f 2 g π 4 π 2 3π 4 π (b) f > g on the interval, x 5 < x < 6 6 (c) As x → , f x 2 sin x → 0 and gx 12 csc x → ± since g x is the reciprocal of f x. (b) 1 The interval in which f < g is 1, 3 . (c) 1 The interval in which 2f < 2g is 1, 3 , which is the same interval as part (b). x 3π 2 392 Chapter 4 Trigonometry 54. y1 sin x sec x, y2 tan x 53. y1 sin x csc x and y2 1 2 4 −3 −2 3 2 −2 −4 sin x csc x sin x sin x 1, sin x 0 1 sin x sec x sin x The expressions are equivalent except when sin x 0 and y1 is undefined. 55. y1 The expressions are equivalent. cos x 1 and y2 cot x sin x tan x cot x 1 sin x tan x cos x cos x 56. y1 sec 2 x 1, y2 tan2 x 3 1 tan2 x sec2 x cos x sin x 4 The expressions are equivalent. tan2 x sec2 x 1 −2 2 − 3 2 3 2 The expressions are equivalent. −1 −4 58. f x x sin x 57. f x x cos x Matches graph (a) as x → 0, f x → 0. As x → 0, f x → 0 and f x > 0. Matches graph (d). 59. gx x sin x 60. gx x cos x As x → 0, gx → 0 and gx is odd. Matches graph (c) as x → 0, gx → 0. Matches graph (b). 61. f x sin x cos x 2 3 gx 0 1 −3 −2 −1 2 y 4 gx 2 sin x 2 f x gx The graph is the line y 0. 62. f x sin x cos x y x 1 2 3 −1 sin x cos x −2 2 It appears that f x gx. That is, −π 2 sin x. 2 x π −4 −3 63. f x sin2 x 64. f x cos2 y 1 gx 1 cos 2x 2 x 2 y 3 3 1 gx 1 cos x 2 2 f x gx −π π –1 x 2 It appears that f x gx. That is, x 1 1 cos x. cos2 2 2 −6 x −3 3 −1 6 Section 4.6 65. gx ex 2 sin x 2 Graphs of Other Trigonometric Functions 1 ex 2 ≤ gx ≤ ex 2 2 2 −8 8 The damping factor is y ex 2. 2 As x → , gx →0. −1 67. f x 2x4 cos x 66. f x ex cos x 68. hx 2x 4 sin x 2 2x4 ≤ f x ≤ 2x4 Damping factor: ex Damping factor: 2x 4 2 Damping factor: y 2x4. 3 1 6 −3 −8 6 −9 8 9 −3 As x → −1 , f x → 0. −6 As x → , h x → 0. As x→ , f x → 0. 69. y 6 cos x, x > 0 x 70. y 6 4 sin 2x, x > 0 x 71. gx 2 6 8 0 −6 6 0 −2 6 −1 −2 As x → 0, y → 72. f (x) . As x → 0, y → 1 cos x x 73. f x sin As x → 0, gx → 1. . 1 x 74. h(x) x sin 2 1 −6 − 6 sin x x 1 x 2 − −2 −1 d −1 As x → 0, f x oscillates between 1 and 1. As x → 0, f (x) → 0. 75. tan x 7 d As x → 0, h(x) oscillates. 76. cos x 7 7 cot x tan x d 27 d 27 27 sec x, < x < cos x 2 2 d d 14 80 6 2 −2 −6 π 4 π 2 3π 4 π x 60 Distance Ground distance 10 40 20 −π 2 − 10 − 14 Angle of elevation −π 4 0 π 4 Angle of camera π 2 x 393 394 Chapter 4 Trigonometry 77. C 5000 2000 sin (a) t t , R 25,000 15,000 cos 12 12 50,000 R C 0 100 0 (b) As the predator population increases, the number of prey decreases. When the number of prey is small, the number of predators decreases. (c) The period for both C and R is: p 2 24 months 12 When the prey population is highest, the predator population is increasing most rapidly. When the prey population is lowest, the predator population is decreasing most rapidly. When the predator population is lowest, the prey population is increasing most rapidly. When the predator population is highest, the prey population is decreasing most rapidly. In addition, weather, food sources for the prey, hunting, all affect the populations of both the predator and the prey. t 6 S Lawn mower sales (in thousands of units) 78. S 74 3t 40 cos 150 135 120 105 90 75 60 45 30 15 t 2 4 6 8 10 12 Month (1 ↔ January) 79. Ht 54.33 20.38 cos t t 15.69 sin 6 6 Lt 39.36 15.70 cos t t 14.16 sin 6 6 (a) Period of cos t 2 : 12 6 6 Period of sin t 2 : 12 6 6 (b) From the graph, it appears that the greatest difference between high and low temperatures occurs in summer. The smallest difference occurs in winter. (c) The highest high and low temperatures appear to occur around the middle of July, roughly one month after the time when the sun is northernmost in the sky. Period of Ht : 12 months Period of Lt : 12 months 80. (a) 1 y et4 cos 4t 2 0.6 0 81. True. Since y csc x 1 , sin x 4 −0.6 (b) The displacement is a damped sine wave. y → 0 as t increases. for a given value of x, the y-coordinate of csc x is the reciprocal of the y-coordinate of sin x. Section 4.6 Graphs of Other Trigonometric Functions 83. As x → 82. True. y sec x 1 cos x As x → 395 from the left, f x tan x → . 2 from the right, f x tan x → . 2 If the reciprocal of y sin x is translated 2 units to the left, we have y 1 sin x 2 1 sec x. cos x 84. As x → from the left, f (x) csc x → . As x → from the right, f (x) csc x → . 85. f x x cos x (a) (b) xn cosxn1 2 x0 1 −3 3 x1 cos 1 0.5403 x2 cos 0.5403 0.8576 −2 The zero between 0 and 1 occurs at x 0.7391. x3 cos 0.8576 0.6543 x4 cos 0.6543 0.7935 x5 cos 0.7935 0.7014 x6 cos 0.7014 0.7640 x7 cos 0.7640 0.7221 x8 cos 0.7221 0.7504 x9 cos 0.7504 0.7314 This sequence appears to be approaching the zero of f : x 0.7391. 86. y tan x yx 2x3 16x5 3! 5! − 3 2 The graphs are nearly the same for 1.1 < x < 1.1. 88. (a) y1 87. y1 sec x 6 4 1 sin x sin 3x 3 y2 1 3 2 y2 2 4 1 1 sin x sin 3x sin 5x 3 5 2 −3 3 −3 3 y2 y1 −2 —CONTINUED— x2 5x4 2! 4! − 3 2 The graph appears to coincide on the interval 1.1 ≤ x ≤ 1.1. −6 6 −2 3 2 −6 396 Chapter 4 Trigonometry 88. —CONTINUED— (b) y3 4 1 1 1 sin x sin 3x sin 5x sin 7x 3 5 7 2 −3 3 −2 (c) y4 4 1 1 1 1 sin x sin 3x sin 5x sin 7x sin 9x 3 5 7 9 89. e2x 54 90. 83x 98 2x ln 54 91. 3x log8 98 ln 54 x 1.994 2 x 300 100 1 e x 300 1 e x 100 ln 98 0.735 3 ln 8 3 1 e x 2 e x ln 2 x x ln 2 0.693 92. 1 0.15 365 365t 93. ln3x 2 73 5 3x 2 e73 0.15 1 1.00041096 365 3x 2 e73 1.00041096365t 5 x 365t log1.00041096 5 t 2 e73 3 1.684 1031 1 log10 5 10.732 365 log10 1.00041096 94. ln(14 2x) 68 95. lnx2 1 3.2 14 2x e68 x2 1 e3.2 14 e68 2x x 14 e68 1.702 1029 2 96. lnx 4 5 1 2 lnx x2 e3.2 1 4 5 lnx 4 10 x 4 e10 x e10 4 22,022.466 x ± e3.2 1 ± 4.851 97. log8 x log8x 1 13 98. log6 x log6x2 1 log664x log8xx 1 13 log6x(x2 1 log664x xx 1 813 x2 x 2 x2 x 2 0 x 2x 1 0 x 2, 1 x 1 is extraneous (not in the domain of log8 x) so only x 2 is a solution. xx2 1 64x x2 1 64 x ± 65 Since 65 is not in the domain of log6 x, the only solution is x 65 8.062. Section 4.7 Section 4.7 ■ ■ Inverse Trigonometric Functions 397 Inverse Trigonometric Functions You should know the definitions, domains, and ranges of y arcsin x, y arccos x, and y arctan x. Function Domain Range y arcsin x ⇒ x sin y 1 ≤ x ≤ 1 y arccos x ⇒ x cos y 1 ≤ x ≤ 1 0 ≤ y ≤ y arctan x ⇒ x tan y < x < ≤ y ≤ 2 2 < x < 2 2 You should know the inverse properties of the inverse trigonometric functions. sinarcsin x x and arcsinsin y y, ≤ y ≤ 2 2 cosarccos x x and arccoscos y y, 0 ≤ y ≤ tanarctan x x and arctantan y y, ■ < y < 2 2 You should be able to use the triangle technique to convert trigonometric functions of inverse trigonometric functions into algebraic expressions. Vocabulary Check Function Alternative Notation Domain 1. y arcsin x y sin1 x 1 ≤ x ≤ 1 2. y arccos x y cos1 x 1 ≤ x ≤ 1 3. y arctan x y tan1 x < x < Range ≤ y ≤ 2 2 0 ≤ y ≤ < y < 2 2 1. y arcsin 1 1 ⇒ sin y for ≤ y ≤ ⇒ y 2 2 2 2 6 2. y arcsin 0 ⇒ sin y 0 for 3. y arccos 1 1 ⇒ cos y for 0 ≤ y ≤ ⇒ y 2 2 3 4. y arccos 0 ⇒ cos y 0 for 0 ≤ y ≤ ⇒ y 5. y arctan 3 3 ⇒ tan y ⇒ y < y < 2 2 6 3 3 for ≤ y ≤ ⇒ y0 2 2 6. y arctan1 ⇒ tan y 1 for < y < ⇒ y 2 2 4 2 398 Chapter 4 7. y arccos Trigonometry 3 2 ⇒ cos y 23 for 0 ≤ y ≤ ⇒ y 5 6 9. y arctan 3 ⇒ tan y 3 for 2 ⇒ cos y 2 for 1 1 0 ≤ y ≤ ⇒ y 13. y arcsin 3 2 3 2 < y < ⇒ y0 2 2 2 ⇒ sin y 3 3 ⇒ tan y 33 for < y < ⇒ y 2 2 6 2 − 2 Graph y2 tan1 x. g f −2 19. arccos 0.28 cos1 0.28 1.29 20. arcsin 0.45 0.47 21. arcsin0.75 sin10.75 0.85 22. arccos0.7 2.35 24. arctan 15 1.50 2 g Graph y3 x. −1 23. arctan3 tan13 1.25 for ≤ y ≤ ⇒ y 2 2 4 Graph y1 tan x. 1.5 f 2 16. y arccos 1 ⇒ cos y 1 for 0 ≤ y ≤ ⇒ y 0 gx arcsin x −1.5 2 18. f x tan x and gx arctan x 1 yx 2 ≤ y ≤ ⇒ y 2 2 3 17. f x sin x < y < ⇒ y 2 2 3 14. y arctan for 15. y arctan 0 ⇒ tan y 0 for ⇒ sin y 22 for ≤ y ≤ ⇒ y 2 2 4 12. y arcsin 2 3 ⇒ sin y 2 10. y arctan3 ⇒ tan y 3 for < y < ⇒ y 2 2 3 11. y arccos 2 8. y arcsin 25. arcsin 0.31 sin1 0.31 0.32 27. arccos0.41 cos10.41 1.99 26. arccos 0.26 1.31 28. arcsin0.125 0.13 29. arctan 0.92 tan1 0.92 0.74 30. arctan 2.8 1.23 31. arcsin4 sin10.75 0.85 32. arccos 3 1.91 33. arctan72 tan13.5 1.29 3 34. arctan 95 1.50 7 1 35. This is the graph of y arctan x. The coordinates are 3, 3 , 3 , 6 , and 1, 4 . 3 Section 4.7 36. arccos1 2 3 2 arccos 37. tan 1 Inverse Trigonometric Functions x 4 tan arctan 3 cos 6 2 x x 4 θ 38. cos 4 4 x 39. sin arccos 399 4 x x2 5 sin arcsin x2 5 5 x+2 θ 40. tan x1 10 41. cos x1 arctan 10 x3 2x arccos x3 2x 2x θ x+3 42. tan x1 1 2 x 1 x1 arctan 43. sinarcsin 0.3 0.3 44. tanarctan 25 25 46. sinarcsin0.2 0.2 47. arcsinsin 3 arcsin0 0 1 x1 x1 45. cosarccos0.1 0.1 Note: 3 is not in the range of the arcsine function. 48. arccos cos Note: 7 arccos 0 2 2 3 49. Let y arctan , 4 7 is not in the range of the arccosine function. 2 y 3 tan y , 0 < y < , 4 2 3 and sin y . 5 5 3 y x 4 4 50. Let u arcsin , 5 4 sin u , 0 < u < , 5 2 4 5 sec arcsin sec u . 5 3 51. Let y arctan 2, 5 4 u 3 y 2 tan y 2 , 0 < y < , 1 2 and cos y 1 5 5 5 . 5 2 y 1 x 400 Chapter 4 Trigonometry 5 52. Let u arccos cos u 5 5 sin arccos 5 53. Let y arcsin , ,0 < u < 5 5 , 2 sin u 5 sin y 2 5 25 5 5 , 13 5 12 , 0 < y < , and cos y . 13 2 13 y . 2 13 u 5 y 1 x 12 54. Let u arctan tan u 5 , 5 , 12 55. Let y arctan 34 3 tan y , < y < 0, and sec y . 5 2 5 5 , < u < 0, 12 2 csc arctan 5 12 3 csc u 5 . 13 y 5 x y −3 34 12 u 13 −5 3 , 4 , 56. Let u arcsin 3 57. Let y arccos 3 sin u , < u < 0, 4 2 3 tan arcsin 4 5 2 cos y , < y < , and sin y . 3 2 3 y 37 3 . tan u 7 7 5 3 y 7 −2 u 4 2 −3 x Section 4.7 5 58. Let u arctan , 8 401 59. Let y arctan x, 5 tan u , 0 < u < , 8 2 cot arctan Inverse Trigonometric Functions 89 u x 1 and cot y . x 8 5 8 cot u . 8 5 x2 + 1 x tan y x , 1 5 x 60. Let u arctan x, tan u x , 1 x sinarctan x sin u . x2 1 y 1 61. Let y arcsin2x, sin y 2x 2x , 1 1 2x and cos y 1 4x2. y 1 − 4x 2 x2 + 1 x u 1 62. Let u arctan 3x, tan u 3x 63. Let y arccos x, 3x , 1 x cos y x , 1 9x 2 + 1 secarctan 3x sec u 9x2 1. 3x 1 and sin y 1 x2. 1 − x2 y x u 1 64. Let u arcsinx 1, sin u x 1 65. Let y arccos x1 , 1 secarcsinx 1 sec u 1 3 , x x cos y , 3 1 2x x2 . 3 and tan y 9 x2 x . 9 − x2 y x x −1 u 2x − x 2 1 66. Let u arctan , x 67. Let y arctan x2 + 1 1 tan u , x 1 1 cot u x. cot arctan x tan y x 2 and csc y u x x 2 , x2 + 2 , x x2 2 x . y 2 402 Chapter 4 68. Let u arcsin sin u Trigonometry xh , r xh , r cos arcsin r xh cos u r r 2 x−h x h . r 2 u r 2 − (x − h) 2 69. f x sinarctan 2x, gx 2x 1 4x2 2 They are equal. Let y arctan 2x, tan y 2x −2 1 4x2 . 1 + 4x 2 2x 2x f x gx 1 4x2 y The graph has horizontal asymptotes at y ± 1. x 2 70. f x tan arccos gx 1 2 −3 4 x 2 9 71. Let y arctan . x 3 x −2 x Let u arccos . 2 2 x tan u 2 4 x 2 x 4 − x2 arcsin y arcsin , x > 0; , x < 0. 9 gx y 6 x u, 36 x 2 36 x2 6 9 x2 81 x 36 x 2 then sin u 9 x2 81 x 2 + 81 u Thus, f x gx. 72. If arcsin 9 9 9 and sin y , x > 0; , x < 0. x2 81 x2 81 x arcsin y These are equal because: f x tan arccos tan y Thus, Asymptote: x 0 3 2x , 1 2x and sin y −3 6 73. Let y arccos , 6 x arccos . 6 cos y 36 − x 2 3 x2 2x 10 and sin y u x 3 x2 2x 10 x 1 x 12 9 Thus, y arcsin . x 1 x − 1 3 3 x 12 9 x2 2x 10 (x − 1) 2 + 9 y . Then, . Section 4.7 74. If arccos x2 u, 2 then cos u arccos Inverse Trigonometric Functions 403 75. y 2 arccos x Domain: 1 ≤ x ≤ 1 x2 , 2 Range: 0 ≤ y ≤ 2 x2 arctan 2 4x x 2 x2 This is the graph of f x arccos x with a factor of 2. . y 2π 2 4x − x 2 π u −2 x−2 76. y arcsin x 2 Range: ≤ y ≤ 2 2 This is the graph of f x arcsin x with a horizontal stretch of a factor of 2. Range: x −2 1 π ≤ y ≤ 2 2 π Range: This is the graph of y arccos t shifted two units to the left. arctan x 2 2 4 π < y < 2 2 y 81. hv tanarccos v Domain: all real numbers 3 y −4 This is the graph of gx arctanx with a horizontal stretch of a factor of 2. t −1 2 −π Domain: all real numbers Range: 0 ≤ y ≤ −2 1 79. f x arctan 2x y −3 x −1 This is the graph of gx arcsinx shifted one unit to the right. Domain: 3 ≤ t ≤ 1 x −2 −π 1 v2 v Domain: 1 ≤ v ≤ 1, v 0 π Range: 0 < y ≤ This is the graph of y arctan x shifted upward 2 units. y 2 −π −4 2 Domain: 0 ≤ x ≤ 2 π 78. gt arccost 2 80. f x 1 77. f x arcsinx 1 y Domain: 2 ≤ x ≤ 2 x −1 Range: all real numbers −4 −2 y x 2 4 π 1 1 − v2 y v −2 v 1 2 404 Chapter 4 82. f x arccos Trigonometry x 4 y 83. f x 2 arccos2x 2 Domain: 4 ≤ x ≤ 4 π Range: 0 ≤ y ≤ −4 x −2 2 −1 4 1 0 85. f x arctan2x 3 84. f x arcsin4x 2 86. f x 3 arctan x 2 −4 −0.5 −2 0.5 − −2 87. f x arcsin 4 4 −2 23 2.412 88. f x 1 arccos 2.82 2 4 4 −4 −4 5 5 −2 −2 89. f t 3 cos 2t 3 sin 2t 32 32 sin 2t arctan 3 3 32 sin2t arctan 1 32 sin 2t 4 6 −2 2 −6 The graph implies that the identity is true. 90. f t 4 cos t 3 sin t 91. (a) sin 42 32 sin t arctan 5 sin t arctan 4 3 4 3 sin arcsin 6 −6 6 −6 The graph implies that A cos t B sin t A2 B2 sin t arctan is true. 5 s A B 5 s (b) s 40: arcsin 5 0.13 40 s 20: arcsin 5 0.25 20 Section 4.7 92. (a) tan s 750 Inverse Trigonometric Functions 93. arctan arctan (a) s 750 3x x2 4 1.5 (b) When s 300, arctan 0 300 0.38 21.8. 750 6 − 0.5 When s 1200, (b) is maximum when x 2 feet. 1200 arctan 1.01 58.0. 750 (c) The graph has a horizontal asymptote at 0. As x increases, decreases. 94. (a) tan 11 17 95. arctan 11 0.5743 32.9 17 1 (b) r 40 20 2 tan 20 ft θ 41 ft (a) tan h h r 20 h 20 tan 20 20 41 arctan 11 12.94 feet 17 (b) tan 26 41 26.0 20 h 50 h 50 tan 26 24.39 feet 96. (a) tan 6 x 97. (a) tan arctan 6 x x 20 arctan x 20 (b) x 7 miles 6 arctan 0.71 40.6 7 12 31.0 20 6 1.41 80.5 1 99. False. 98. False. 5 is not in the range of arcsinx. 6 arcsin 5 14.0 20 x 12: arctan x 1 mile arctan (b) x 5: arctan 1 2 6 5 is not in the range of the arctangent function. 4 arctan 1 100. False. arctan x is defined for all real x, but arcsin x and arccos x require 1 ≤ x ≤ 1. Also, for example, arctan 1 Since arctan 1 arcsin 1 . arccos 1 2 arcsin 1 undefined. , but 4 arccos 1 0 4 405 406 Chapter 4 Trigonometry 101. y arccot x if and only if cot y x. Domain: < x < 102. y arcsec x if and only if sec y x where x ≤ 1 x ≥ 1 and 0 ≤ y < and < y ≤ . 2 2 The domain of y arcsec x is , 1 1, y Range: 0 < x < π 2 2 , . and the range is 0, π 2 y −2 −1 x 1 2 π π 2 −2 103. y arccsc x if and only if csc y x. x 1 2 y Domain: , 1 1, Range: −1 π 2 2 , 0 0, 2 −2 x −1 1 2 −π 2 104. (a) y arcsec 2 ⇒ sec y 2 and 0 ≤ y < (b) y arcsec 1 ⇒ sec y 1 and 0 ≤ y < < y ≤ ⇒ y 2 2 4 < y ≤ ⇒ y0 2 2 (c) y arccot 3 ⇒ cot y 3 and 0 < y < ⇒ y (d) y arccsc 2 ⇒ csc y 2 and 5 6 ≤ y < 00 < y ≤ ⇒ y 2 2 6 105. Area arctan b arctan a 106. f x x (a) a 0, b 1 gx 6 arctan x Area arctan 1 arctan 0 0 4 4 12 g (b) a 1, b 1 Area arctan 1 arctan1 4 4 2 (c) a 0, b 3 Area arctan 3 arctan 0 1.25 0 1.25 (d) a 1, b 3 Area arctan 3 arctan1 4 2.03 1.25 f 0 6 0 As x increases to infinity, g approaches 3, but f has no maximum. Using the solve feature of the graphing utility, you find a 87.54. Section 4.7 Inverse Trigonometric Functions 407 107. f x sinx, f 1x arcsinx (a) f f 1 sinarcsin x f 1 f arcsinsin x 2 2 − − −2 −2 (b) The graphs coincide with the graph of y x only for certain values of x. f f 1 x over its entire domain, 1 ≤ x ≤ 1. f 1 f x over the region 2 ≤ x ≤ , corresponding to the region where sin x is 2 one-to-one and thus has an inverse. (b) Let y arctanx. Then, 108. (a) Let y arcsinx. Then, sin y x tan y x, sin y x tan y x siny x tany x, y arcsin x y arcsin x. < y < 2 2 arctantany arctan x Therefore, arcsinx arcsin x. y arctan x (c) Let y2 y1. 2 arctan x arctan < y < 2 2 y arctan x Thus, arctanx arctanx. 1 y1 y2 x y1 2 y 2 1 y2 (d) Let arcsin x and arccos x, then sin x and cos x. Thus, sin cos which implies that and are complementary angles and we have 2 arcsin x arccos x . 2 1 y1 x (e) arcsin x arcsin x x arctan 1 x2 1 y2 1 x y1 1 − x2 109. 8.23.4 1279.284 110. 10142 10 10 0.051 142 196 408 Chapter 4 Trigonometry 111. 1.150 117.391 3 opp 4 hyp sin 113. 112. 162 114. tan 2 adj2 32 42 adj2 9 16 adj2 7 4 3 θ adj 7 cos tan cot 4 3 37 7 3 csc 4 3 cos 5 adj 6 hyp opp2 25 36 opp 11 2 opp 11 cot sec sin 2 5 cot 1 2 2 θ 1 116. sec 3 opp2 52 62 tan 1 5 1 csc 5 2 4 47 7 7 sin cos sec 5 7 sec 115. hyp 12 22 5 7 7 1 2.718 108 162 6 opp 32 12 8 θ 5 cos 1 3 sin 22 3 11 6 11 5 5 511 11 11 6 5 6 611 csc 11 11 3 22 θ 1 tan 22 sec 3 csc cot 3 22 32 4 2 1 4 22 Section 4.7 Inverse Trigonometric Functions 117. Let x the number of people presently in the group. Each person’s share is now 250,000x. If two more join the group, each person’s share would then be 250,000x 2. Share per person with Original share 6250 two more people per person 250,000 250,000 6250 x2 x 250,000x 250,000x 2 6250xx 2 250,000x 250,000x 500,000 6250x2 12500x 6250x2 12500x 500,000 0 6250x2 2x 80 0 6250x 10x 8 0 x 10 or x 8 x 10 is not possible. There were 8 people in the original group. 118. Rate downstream: 18 x Rate upstream: 18 x rate time distance ⇒ t d r Time to go upstream Time to go downstream 4 35 35 4 18 x 18 x 3518 x 3518 x 418 x18 x 630 35x 630 35x 4324 x2 1260 4324 x2 315 324 x2 x2 9 x ±3 The speed of the current is 3 miles per hour. 0.035 4 0.035 12 0.035 365 119. (a) A 15,000 1 (b) A 15,000 1 (c) A 15,000 1 (d) A 15,000e0.03510 410 $21,253.63 1210 36510 $21,275.17 $21,285.66 $21,286.01 120. Data: 2, 742,000, 4, 632,000 To find: 8, y P P0 Assume: ert 742,000 P0er 2 632,000 P0er 4 Then: er 2 P0er 4 632 P0er 2 742 y P0er 8 P0er 4 632,000 er 22 632,000 742 458,504.31 632 2 er 4 409 410 Chapter 4 Section 4.8 Trigonometry Applications and Models ■ You should be able to solve right triangles. ■ You should be able to solve right triangle applications. ■ You should be able to solve applications of simple harmonic motion. Vocabulary Check 1. elevation; depression 2. bearing 3. harmonic motion 2. Given: B 54, c 15 1. Given: A 20, b 10 tan A a ⇒ a b tan A 10 tan 20 3.64 b cos A b 10 b ⇒ c 10.64 c cos A cos 20 B 90 20 70 A 90 B sin B b ⇒ b c sin B c b = 10 3. Given: B 71, b 24 tan B b b 24 ⇒ a 8.26 a tan B tan 71 sin B b b 24 ⇒ c 25.38 c sin B sin 71 A cos B 4. Given: A 8.4, a 40.5 B 90 A 90 8.4 81.6 tan A a a ⇒ b b tan A B C c sin A A b = 24 40.5 274.27 tan 8.4 a a 40.5 ⇒ c 277.24 c sin A sin 8.4 B c a = 40.5 C 5. Given: a 6, b 10 8.4° 234 11.66 6 3 a ⇒ A arctan 30.96º tan A b 10 5 B 90 30.96 59.04 600 24.49 sin A c cos B A c = 35 a = 25 352 252 C b A a a ⇒ A arcsin c c arcsin a=6 B b c2 a2 B b = 10 A b 6. Given: a 25, c 35 c2 a2 b2 ⇒ c 36 100 C A b a ⇒ a c cos B 15 cos 54 8.82 c A 90 71 19 a 71° C c = 15 15 sin 54 12.14 c 20° C 54° a 90 54 36 B a B 25 45.58 35 a 25 a ⇒ B arccos arccos 44.42 c c 35 Section 4.8 7. Given: b 16, c 52 a 522 a 16 cos A 52 c = 52 a c2 b2 87.5601 9.36 1.32 b b 81.97 cos A ⇒ A arccos arccos c c 9.45 sin B A arccos 16 72.08º 52 C 411 8. Given: b 1.32, c 9.45 B 162 2448 1217 49.48 Applications and Models b b ⇒ B arcsin c c arcsin b = 16 A B 90 72.08 17.92 1.32 9.45 B a c = 9.45 8.03 C A b = 1.32 9. Given: A 12 15, c 430.5 B 90 12 15 77 45 sin 12 15 a 430.5 a 430.5 sin 12 15 91.34 cos 12 15 10. Given: B 65 12, a 14.2 A 90 B 90 65 12 24 48 cos B a 14.2 a ⇒ c 33.85 c cos B cos 65 12 tan B b ⇒ b a tan B 14.2 tan 65 12 30.73 a b 430.5 B b 430.5 cos 12 15 420.70 a = 14.2 B 12°15′ b C 11. tan 1 h 4 tan 52 2.56 inches 2 12. tan h 1 ⇒ h b tan 12b 2 1 h 10 tan 18 1.62 meters 2 h h θ θ θ θ 1 b 2 1 b 2 1 b 2 b b 13. tan A b A h 1 ⇒ h b tan 12b 2 1 b 2 c C c = 430.5 a 65°12′ h 1 ⇒ h b tan 12b 2 1 h 46 tan 41 19.99 inches 2 14. tan h 1 ⇒ h b tan 12b 2 1 h 11 tan 27 2.80 feet 2 h θ θ 1 b 2 1 b 2 b h θ θ 1 b 2 1 b 2 b 412 Chapter 4 15. tan 25 Trigonometry 50 x 16. tan 20 50 50 x tan 25 x 25° x 107.2 feet 17. 16 sin 80 600 x 600 600 tan 20 20° x 1648.5 feet h 20 18. tan 33 20 sin 80 h h 125 h 125 tan 33 20 ft 16 si 74 h 19.7 feet h h 81.2 feet 33° 125 80° 19. (a) 20. tan 51 h h 100 h 100 tan 51 y 123.5 feet x h 47° 40′ 50 ft 35° (b) Let the height of the church x and the height of the church and steeple y. Then, x y tan 35 and tan 47 40 50 50 51° 100 x 50 tan 35 and y 50 tan 47 40 h y x 50tan 47 40 tan 35. (c) h 19.9 feet 21. sin 34 x 4000 22. tan x 4000 sin 34º 75 50 23. (a) arctan 2236.8 feet 3 56.3 2 1 12 2 ft θ 1 17 3 ft 34° (b) tan x 4000 75 ft 24. 12,500 4000 16,500 14.03 Angle of depression 90 14.03 75.97 θ α i 4000 16,500 16,500 mi 0m 4,00 arcsin 1212 35.8 1713 The angle of elevation of the sum is 35.8. 50 ft 4000 16,500 1713 (c) arctan θ sin 1212 Not drawn to scale Section 4.8 25. 1200 feet 150 feet 400 feet 950 feet 5 miles 5 miles Not drawn to scale feet 26,400 feet 5280 1 mile θ 950 feet θ 5 miles 950 tan 26,400 arctan Applications and Models 26,400 2.06 950 (b) sin 18 26. (a) Since the airplane speed is 275sec60 min 16,500 min, ft sec ft s after one minute its distance travelled is 16,500 feet. a sin 18 16,500 10,000 275(sin 18) 117.7 seconds 275s a 16,500 sin 18 5099 ft 16500 10,000 275s 10,000 feet 18° a 18° x 4 27. sin 10.5 4 10.5° x x 4 sin 10.5 0.73 mile 28. θ 29. The plane has traveled 1.5600 900 miles. 100x 12x = y 4 miles = 21 ,120 feet Angle of grade: tan 12x 100x sin 38 a ⇒ a 554 miles north 900 cos 38 b ⇒ b 709 miles east 900 arctan 0.12 6.8 N Change in elevation: y sin 21,120 900 52° a 38° b W y 21,120 sin 21,120 sinarctan 0.12 2516.3 feet S 30. (a) Reno is 2472 sin 10 429 miles N of Miami. N Reno is 2472 cos 10 2434 miles W of Miami. (b) The return heading is 280. 100° Reno W 2472 mi 80° N 10° W S 10° E 280° S E Miami E 413 414 Chapter 4 31. Trigonometry 32. N N W E 1.4° W E 29° 120 428 a 88.6° 20 b S (a) cos 29 sin 29 a ⇒ a 104.95 nautical miles south 120 (a) t b ⇒ b 58.18 nautical miles west 120 (b) After 12 hours, the yacht will have traveled 240 nautical miles. 428 21.4 hours 20 240 sin 1.4 5.9 miles E 78.18 20 b ⇒ 36.7 a 104.95 (b) tan Not drawn to scale S 240 cos 1.4 239.9 miles S Bearing: S 36.7 W (c) Bearing from N is 178.6. Distance: d 104.952 78.182 130.9 nautical miles from port 33. 32, 68 (a) 90 32 58 32 (b) 90 22 Bearing from A to C: N 58º E C 54 N B tan C d φ C γ β α 50 β θ W A E d ⇒ tan 54 50 d ⇒ d 68.82 meters 50 S 34. tan 14 tan 34 d ⇒ x d cot 14 x 35. tan d d y 30 x cot 34 d 30 d cot 14 30 d cot 14 d 45 ⇒ 56.3 30 36. Bearing 180 arctan 208.0 or 528 W Bearing: N 56.3 W N N 45 Port Plane θ W 30 160 E Ship d cot 34 30 d cot 14 d W 30 cot 34 cot 14 tan 4 350 ⇒ d 3071.91 ft d 350 ⇒ D 5005.23 ft D 6.5° 4° 350 S1 d S2 D Not drawn to scale Distance between ships: D d 1933.3 ft 85 Airport S S 5.46 kilometers 37. tan 6.5 85 160 E Section 4.8 38. cot 55 d ⇒ d 7 kilometers 10 a ⇒ x a cot 57 x a tan 16 x 556 Distance between towns: tan 16 D d 18.8 7 11.8 kilometers cot 16 28° 55° T1 55° d 28° P2 a 57° 16° H 550 60 x a a cot 57 556 a cot 57 556 a a cot 16 a cot 57 T2 415 P1 39. tan 57 D cot 28 ⇒ D 18.8 kilometers 10 10 km Applications and Models D 55 ⇒ a 3.23 miles 6 17,054 ft 40. tan 2.5 h x h 2.5° 17 h x tan 2.5 tan 9 x x 9° x − 17 41. L1: 3x 2y 5 ⇒ y 3 3 5 ⇒ m1 x 2 2 2 L2: 3x y 1 ⇒ y x 1 ⇒ m2 1 Not drawn to scale tan h x 17 1 32 52 5 1 132 12 arctan 5 78.7 h 17 tan 9 h h 17 tan 2.5 tan 9 h 17 1.025 miles 1 1 tan 2.5 tan 9 5410 feet 42. L1 2x y 8 ⇒ m1 2 L2 x 5y 4 ⇒ m2 tan m2 m1 1 m2m1 arctan 1 5 tan m2 m1 15 2 arctan 1 152 1 m2m1 9 52.1 arctan 7 44. tan 43. The diagonal of the base has a length of a2 a2 2a. Now, we have a 2a arctan 1 2 a 1 2 θ 35.3. a2 2 a 45. sin 36 2a d ⇒ d 14.69 25 Length of side: 2d 29.4 inches arctan 2 54.7º a 2 d θ a 36° 25 416 Chapter 4 Trigonometry 46. 47. a 48. r 2 c b 30° r 30° 25 a 15° y c x sin 30 b Length of side 2a 212.5 10 10 10 10 a 2 3r 50. tan 35° 10 12 18 arctan b 10 cos b 10 tan 35 7 cos 35 3r y 2b 2 b tan 35 Distance 2a 9.06 centimeters 23r 25 inches a 35° 10 4.53 b r cos 30 a 25 sin 30 12.5 49. a c a c sin 15 17.5 sin 15 sin 15 b cos 30 r a 25 a 10 a 35 17.5 2 2 0.588 rad 33.7 3 18 a f a 18 21.6 feet cos θ 6 c φ b 6 36 21.6 10.8 feet f 2 10 12.2 cos 35 90 33.7 56.3 sin b 6 b 6 7.2 feet sin c 10.82 7.22 13 feet 51. d 0 when t 0, a 4, period 2 Use d a sin t since d 0 when t 0. Thus, d 4 sin t. d 3 sin 53. d 3 when t 0, a 3, period 1.5 Use d a cos t since d 3 when t 0. t 3 54. Displacement at t 0 is 2 ⇒ d a cos t. Amplitude: a 2 2 Period: 10 ⇒ 5 2 4 1.5 ⇒ 3 4 Amplitude: a 3 2 6 ⇒ Period: 3 2 2 ⇒ Thus, d 3 cos 52. Displacement at t 0 is 0 ⇒ d a sin t. 4t 3 t 3 cos 3 . d 2 cos t 5 9 Section 4.8 55. d 4 cos 8t 56. d (a) Maximum displacement amplitude 4 (b) Frequency 8 2 2 1 cos 20 t 2 (a) Maximum displacement: a (b) Frequency: 4 cycles per unit of time (c) d 4 cos 40 4 (d) 8 t Applications and Models 1 1 2 2 20 10 cycles per unit of time 2 2 (c) t 5 ⇒ d 1 ⇒ t 2 16 417 1 1 cos 100 2 2 (d) Least positive value for t for which d 0 1 cos 20 t 0 2 cos 20 t 0 20 t arccos 0 57. d 1 sin 120t 16 58. d (a) Maximum displacement amplitude 120 (b) Frequency 2 2 60 cycles per unit of time 1 (c) d sin 600 0 16 (d) 120t ⇒ t 1 120 1 16 20 t 2 t 2 1 1 20 40 1 sin 792t 64 (a) Maximum displacement: a (b) Frequency: 1 1 64 64 792 396 cycles per unit of time 2 2 (c) t 5 ⇒ d 1 sin3960 0 64 (d) Least positive value for t for which d 0 1 sin 792 t 0 64 sin 792 t 0 792 t arcsin 0 792 t t 59. d a sin t Frequency 2 264 2 1 792 792 60. At t 0, buoy is at its high point ⇒ d a cos t. Distance from high to low 2 a 3.5 7 a 4 Returns to high point every 10 seconds: 2264 528 Period: d 2 10 ⇒ 5 7 t cos 4 5 418 Chapter 4 61. y Trigonometry 1 cos 16t, t > 0 4 (a) y (b) Period: 1 (c) π 8 π 4 3π 8 π 2 2 16 8 1 ⇒ t cos 16t 0 when 16t 4 2 32 t −1 62. (a) (b) L1 L2 L1 L2 0.1 2 sin 0.1 3 cos 0.1 23.0 0.5 0.2 2 sin 0.2 3 cos 0.2 13.1 0.6 0.3 2 sin 0.3 3 cos 0.3 9.9 0.4 2 sin 0.4 3 cos 0.4 8.4 L1 L2 L1 L2 2 sin 0.5 2 sin 0.6 3 cos 0.5 3 cos 0.6 0.7 2 sin 0.7 3 cos 0.7 7.0 0.8 2 sin 0.8 3 cos 0.8 7.1 7.6 7.2 The minimum length of the elevator is 7.0 meters. (c) 2 3 L L1 L2 sin cos (d) 12 −2 2 −12 From the graph, it appears that the minimum length is 7.0 meters, which agrees with the estimate of part (b). (c) A 8 8 16 cos 63. (a) and (b) Base 1 Base 2 Altitude Area 8 8 16 cos 10º 8 sin 10º 22.1 8 8 16 cos 20º 8 sin 20º 42.5 8 8 16 cos 30º 8 sin 30º 59.7 8 8 16 cos 40º 8 sin 40º 72.7 8 8 16 cos 50º 8 sin 50º 80.5 8 8 16 cos 60º 8 sin 60º 83.1 8 8 16 cos 70º 8 sin 70º 80.7 8 sin 2 16 16 cos 4 sin 641 cos sin (d) 100 0 90 0 The maximum occurs when 60 and is approximately 83.1 square feet. The maximum of 83.1 square feet occurs when 60. 3 64. (a) (b) Average sales (in millions of dollars) S 15 1 a 14.3 1.7 6.3 2 2 12 ⇒ b b 6 12 9 Shift: d 14.3 6.3 8 6 3 S d a cos bt t 2 4 6 8 10 12 Month (1 ↔ January) (c) Period: Applications and Models S 8 6.3 cos 2 12 6 15 12 9 6 3 t 2 4 6 8 10 12 Month (1 ↔ January) t 6 Note: Another model is S 8 6.3 sin This corresponds to the 12 months in a year. Since the sales of outerwear is seasonal, this is reasonable. 419 S Average sales (in millions of dollars) Section 4.8 6t 2 . The model is a good fit. (d) The amplitude represents the maximum displacement from average sales of 8 million dollars. Sales are greatest in December (cold weather Christmas) and least in June. 65. False. Since the tower is not exactly vertical, a right triangle with sides 191 feet and d is not formed. 66. False. One period is the time for one complete cycle of the motion. 67. No. N 24 E means 24 east of north. 68. Aeronautical bearings are always taken clockwise from North (rather than the acute angle from a north-south line). 69. m 4, passes through 1, 2 1 70. Linear equation m 2 through 13, 0 y y 2 4x 1 y 12x b 7 6 b y 2 4x 4 5 0 y 4x 6 3 0 16 b 2 y 12 13 3 2 1 b 16 1 x −4 −3 −2 −1 −1 1 2 3 4 y −3 1 2x −2 1 6 −1 x 2 3 −1 −2 −3 71. Passes through 2, 6 and 3, 2 m 26 4 3 2 5 4 y 6 x 2 5 4 8 y6 x 5 5 22 4 y x 5 5 72. Linear equation through y m 7 6 4 3 2 −2 −1 −1 13 23 12 14 x 1 2 3 4 5 y y 3 1 34 1 14, 32 and 21, 13 2 1 4 3 −3 2 4 1 x 3 3 4 1 4 y x 3 3 −2 −1 x 2 −1 −2 −3 3 420 Chapter 4 Trigonometry Review Exercises for Chapter 4 1. 0.5 radian 3. 2. 4.5 radians 11 4 (a) 4. y 2 9 (a) 5. 4 3 (a) y y 11π 4 2π 9 x x x − (b) The angle lies in Quadrant II. (c) 11 3 2 4 4 23 3 (a) (a) 4 2 2 3 3 4 10 2 3 3 8. 280 7. 70 y − (c) Coterminal angles: 2 20 2 9 9 16 2 2 9 9 5 3 2 4 4 6. (b) The angle lies in Quadrant II. (b) Quadrant I (c) Coterminal angles: 4π 3 (a) y 23π 3 y 280° 70° x x x (b) Quadrant I (c) 23 8 3 3 17 23 2 3 3 (b) The angle lies in Quadrant I. (b) Quadrant IV (c) Coterminal angles: (c) 280 360 640 70 360 430 70 360 290 280 360 80 Review Exercises for Chapter 4 10. 405 9. 110 (a) (a) y 11. 480 480 y rad 180 8 radians 3 8.378 radians − 405° x x − 110° (b) Quadrant IV (b) The angle lies in Quadrant III. (c) 405 720 315 (c) Coterminal angles: 405 360 45 110 360 250 110 360 470 12. 127.5 2.225 180 13. 33 45 33.75 33.75 77 180 3.443 60 14. 196 77 196 16. 11 6 18. 5.7 180 15. 3 radian 0.589 radian 16 5 rad 5 rad 7 7 180 rad 128.571 17. 3.5 rad 3.5 rad 330.000 180 326.586 19. 138 138 23 radians 180 30 s r 20 rad 180 180 200.535 rad 20. 60 2330 48.17 inches 60 radians 180 s r 11 60 180 11 meters 3 s 11.52 meters 21. (a) Angular speed 3313 2 radians 22. linear speed angular speed radius 1 minute 5 rads 13.5 inches 66 23 radians per minute (b) Linear speed 666 23 inches 1 minute 67.5 inches per second 212.1 inches per second 400 inches per minute 23. 120 1 5 1 6.5 2 24. A r 2 2 2 6 120 2 radians 180 3 1 2 1 339.29 square inches A r 2 182 2 2 3 25. t 12.05 miles per hour 2 1 3 corresponds to the point , . 3 2 2 A 55.31 square millimeters 26. t 2 2 3 , x, y , 4 2 2 421 422 Chapter 4 Trigonometry 27. t 3 1 5 corresponds to the point , . 6 2 2 29. t 3 1 7 , . corresponds to the point 6 2 2 28. t 30. t 4 1 3 , x, y , 3 2 2 2 2 , . corresponds to the point 4 2 2 sin 1 7 y 6 2 csc 7 1 2 6 y sin 2 y 4 2 csc 1 2 4 y cos 3 7 x 6 2 sec 7 1 23 6 x 3 cos 2 x 4 2 sec 1 2 4 x tan 3 7 y 1 6 x 3 3 cot 7 x 3 6 y tan y 1 4 x cot x 1 4 y 31. t 3 2 y 3 2 sin 2 1 23 3 y 3 2 1 3 . corresponds to the point , 3 2 2 csc 32. t 2 corresponds to the point 1, 0. sin 2 y 0 csc 2 1 is undefined. y 2 1 x cos 3 2 2 1 sec 2 3 x cos 2 x 1 sec 2 1 1 x 2 y tan 3 3 x 2 x 3 cot 3 y 3 tan 2 y 0 x cot 2 x is undefined. y 33. sin 11 3 2 sin 4 4 2 36. cos 39. sec 13 1 5 cos 3 3 2 12 5 37. tan 33 75.3130 38. csc 10.5 1 1.1368 sin 10.5 9 0.3420 1 3.2361 12 cos 5 40. sin opp 4 441 hyp 41 41 17 5 1 sin 6 6 2 35. sin 41. opp 4, adj 5, hyp 42 52 41 sin 34. cos 4 cos 0 1 csc hyp 41 opp 4 adj 5 541 cos hyp 41 41 41 hyp sec adj 5 opp 4 tan adj 5 adj 5 cot opp 4 43. adj 4, hyp 8, opp 82 42 48 43 sin opp 43 3 hyp 8 2 csc hyp 8 23 opp 43 3 cos 4 1 adj hyp 8 2 sec hyp 8 2 adj 4 tan opp 43 3 adj 4 cot 3 adj 4 opp 43 3 42. adj 6, opp 6 hyp 62 62 62 sin 2 opp 6 hyp 62 2 csc hyp 62 2 opp 6 cos 2 adj 6 hyp 62 2 sec hyp 62 2 adj 6 tan opp 6 1 adj 6 cot adj 6 1 opp 6 Review Exercises for Chapter 4 44. opp 5, hyp 9 45. sin adj 92 52 214 1 3 (a) csc opp 5 sin hyp 9 1 3 sin (b) sin2 cos2 1 adj 214 cos hyp 9 13 2 cos2 1 tan opp 5 514 adj 28 214 csc hyp 9 opp 5 cos2 sec hyp 9 914 adj 28 214 8 9 cos cot 214 adj opp 5 89 cos 22 3 46. tan 4 (a) cot (c) sec 1 3 32 cos 22 4 (d) tan 2 sin 13 1 cos 223 22 4 47. csc 4 1 1 tan 4 (a) sin (b) sec 1 tan2 1 16 17 (c) cos 1 9 cos2 1 (d) csc 1 cot2 (b) sin2 cos2 1 14 2 17 1 1 sec 17 17 1 161 1 1 csc 4 cos2 1 17 cos2 1 4 1 16 cos2 15 16 cos 1516 cos 15 4 1 4 415 cos 15 15 15 sin 14 1 (d) tan cos 154 15 15 (c) sec 48. csc 5 (a) sin 1 1 csc 5 (c) tan (b) cot csc2 1 25 1 26 49. tan 33 0.6494 50. csc 11 6 1 1 cot 26 12 (d) sec90 csc 5 1 5.2408 sin 11 51. sin 34.2 0.5621 423 424 Chapter 4 52. sec 79.3 Trigonometry 1 5.3860 cos 79.3 53. cot 15 14 1 tan15 14 60 54. cos 78 11 58 cos 78 3.6722 55. sin 1 10 x 3.5 0.2045 x 3.5 sin 1 10 0.07 kilometer or 71.3 meters 25 x 52° x Not drawn to scale 58. x, y 3, 4 57. x 12, y 16, r 144 256 400 20 sin 25 x 25 19.5 feet x tan 52 tan 52 56. m 3.5 k 1°10' y 4 r 5 csc r 32 42 5 5 r y 4 x 3 cos r 5 r 5 sec x 3 y 4 tan x 3 x 3 cot y 4 sin y 4 r 5 csc r 5 y 4 cos x 3 r 5 sec r 5 x 3 tan y 4 x 3 cot x 3 y 4 2 5 59. x , y 3 2 r 23 52 2 2 241 6 sin y 15241 52 15 r 241 2416 241 csc 2416 r 2241 241 y 52 30 15 cos x 4241 23 4 r 241 2416 241 sec 2416 241 r x 23 4 tan y 52 15 x 23 4 cot x 23 4 y 52 15 60. x, y r 10 2 , 3 3 103 32 2 2 11 58 60 3600 226 3 sin y 26 23 r 26 2263 csc r 2263 26 y 23 cos x 103 526 r 26 2 26 3 sec r 2263 26 x 103 5 tan y 23 1 x 103 5 cot x 103 5 y 23 Review Exercises for Chapter 4 61. x 0.5, y 4.5 r 0.52 4.52 20.5 82 2 sin 4.5 y 982 r 82 822 csc 822 82 r y 4.5 9 cos x 0.5 82 r 82 822 sec 822 r 82 x 0.5 tan 4.5 y 9 x 0.5 cot 1 x 0.5 y 4.5 9 62. x, y 0.3, 0.4 r 0.32 0.42 0.5 sin y 0.4 4 0.8 r 0.5 5 csc 0.5 5 r 1.25 y 0.4 4 cos x 0.3 3 0.6 r 0.5 5 sec r 0.5 5 1.67 x 0.3 3 tan y 0.4 4 1.33 x 0.3 3 cot x 0.3 3 0.75 y 0.4 4 63. x, 4x, x > 0 x x, y 4x r x2 4x2 17 x sin y 4x 417 r 17 17 x csc 17 x 17 r y 4x 4 cos 17 x x r 17 17 x sec 17 x r 17 x x tan y 4x 4 x x cot x x 1 y 4x 4 64. x, y 2x, 3x, x > 0 r 2x2 3x2 13x 6 65. sec , tan < 0 ⇒ is in Quadrant IV. 5 r 6, x 5, y 36 25 11 sin y 313 3x r 13 13x sin 11 y r 6 cos 213 x 2x r 13 13x cos x 5 r 6 tan y 3x 3 x 2x 2 tan 11 y x 5 csc 13x 13 r y 3x 3 csc r 611 y 11 sec 13x 13 r x 2x 2 sec 6 5 cot x 2x 2 y 3x 3 cot 511 11 425 426 Chapter 4 Trigonometry 3 66. csc , cos < 0 2 3 67. sin , cos < 0 ⇒ is in Quadrant II. 8 y 3, r 8, x 55 is in Quadrant II. sin 1 2 csc 3 cos 1 sin2 5 3 sin y 3 r 8 cos 55 x r 8 tan sin 25 cos 5 tan 3 y 355 55 x 55 sec 1 35 cos 5 csc 8 3 sec 5 1 cot tan 2 cot 5 68. tan , cos < 0 4 69. cos 855 55 55 3 x 2 ⇒ y2 21 r 5 sin > 0 ⇒ is in Quadrant II ⇒ y 21 is in Quadrant III. sec 1 tan2 cos 8 55 1 2516 41 4 1 441 sec 41 sin 1 cos2 1 4116 5 4141 sin y 21 r 5 tan 21 y x 2 csc 5 r 521 y 21 21 csc 41 1 sin 5 sec r 5 5 x 2 2 cot 1 4 tan 5 cot x 2 221 y 21 21 2 1 70. sin , cos > 0 4 2 71. 264 264 180 84 is in Quadrant IV. y csc 1 2 sin cos 1 sin2 1 23 sec cos 3 tan 3 sin cos 3 cot 1 3 tan 1 14 3 264° 2 x θ′ Review Exercises for Chapter 4 73. 72. 635 720 85 85 y 6 5 74. 17 18 3 3 3 6 4 2 5 5 6 635° 4 5 5 y 3 3 y x θ′ 17π 3 θ′ x − 3 3 2 75. sin x θ′ 6π 5 76. sin 2 4 2 77. sin 3 7 sin 3 3 2 7 1 cos 3 3 2 7 tan 3 3 3 cos 1 3 2 cos 2 4 2 cos tan 3 3 tan 2 1 4 22 tan 5 2 sin 4 4 2 2 5 cos 4 4 2 5 tan 4 4 78. sin cos tan 79. sin 495 sin 45 2 22 2 1 2 cos150 2 tan150 3 2 3 12 3 32 1 81. sin240 sin 60 3 82. sin315 2 cos240 cos 60 1 2 cos315 tan240 tan 60 3 83. sin 4 0.7568 86. cot4.8 2 cos 495 cos 45 80. sin150 tan 495 tan 45 1 2 1 0.0878 tan4.8 tan315 84. tan 3 0.1425 12 5 87. sec 1 3.2361 12 cos 5 2 2 2 2 22 1 22 85. sin3.2 0.0584 88. tan 257 4.3813 427 428 Chapter 4 Trigonometry 89. y sin x 91. f x 5 sin 90. y cos x Amplitude: 1 Amplitude: 1 Period: 2 Period: 2 Amplitude: 5 Period: y y 2 5 25 y 2 2 2x 5 6 1 4 x π 2 − 3π 2 −π 2π π x 2 −1 6π −2 x −2 −2 −6 4x 92. f x 8 cos Shift the graph of y sin x two units upward. Amplitude: 8 Period: 94. y 4 cos x 93. y 2 sin x 2 8 14 Amplitude: 1 Period: y 2 2 4 y y 3 8 −3 2 −2 −1 x 1 2 3 −1 −2 4π 8π −π x π −1 2π x −3 −2 −4 −5 −6 −6 −8 95. gt 5 sint 2 96. gt 3 cost (a) a 2, Amplitude: 3 5 2 Amplitude: 97. y a sin bx Period: 2 Period: 2 2 1 ⇒ b 528 b 264 y y 2 sin528x 4 y 3 4 2 3 1 π 1 −1 (b) f π t 1 1264 264 cycles per second. t −3 −2 −4 −3 −4 98. (a) St 18.09 1.41 sin 6t 4.60 (b) Period 2 26 12 6 12 months 1 year, so this is expected. 22 (c) Amplitude: 1.41 0 12 14 The amplitude represents the maximum change in the time of sunset from the average time d 18.09. Review Exercises for Chapter 4 100. f t tan t 99. f x tan x y 4 429 101. f x cot x y y 4 4 3 3 2 2 1 1 3 2 1 x π −π 2 t π 2 −π x π −3 −4 102. gt 2 cot 2t 104. ht sec t 103. f x sec x Graph y cos x first. y 3 4 y y 2 1 1 −π π t t π −π −1 π x −2 −3 −4 106. f t 3 csc 2t 105. f x csc x Graph y sin x first. 4 y y 4 2 3 2 π 1 − 3π 2 π 2 t x −3 −4 107. f x x cos x 108. gx x 4 cos x 6 Graph y x and y x first. As x → , f x → . 300 Damping factor: x 4 −9 9 As x → , f x → . −2 −300 −6 21 arcsin 21 6 2 109. arcsin 110. arcsin1 112. arcsin0.213 0.21 radian 113. sin 10.44 0.46 radian 115. arccos 3 2 6 116. arccos 2 22 4 111. arcsin 0.4 0.41 radian 114. sin10.89 1.10 radians 117. cos 11 430 Chapter 4 118. cos1 Trigonometry 23 6 120. arccos0.888 2.66 radians 119. arccos 0.324 1.24 radians 121. tan1 1.5 0.98 radian 122. tan18.2 1.45 radians 123. f x 2 arcsin x 2 sin1x 124. y 3 arccos x 3 −1.5 1.5 −1.5 − 125. f x arctan 1.5 0 2x tan 2x 126. f x arcsin 2x 1 2 2 −1.5 −4 1.5 4 − 2 − 2 127. cosarctan 34 45 128. Let u arccos 35. Use a right triangle. Let arctan 34 then tan and cos 45 . tanarccos 35 tan u 43 5 3 4 5 3 θ 4 u 4 3 13 129. secarctan 12 5 5 130. Let u arcsin 12 13 . Use a right triangle. Let arctan 13 then tan 12 5 and sec 5 . 12 5 13 12 cot arcsin 12 13 5 cot u 5 12 u θ 13 −12 5 131. Let y arccos cos y 2x . Then 132. secarcsinx 1 x x and tan y tan arccos 2 2 4 x 2 x . arcsinx 1 ⇒ ≤ ≤ 2 2 sin x 1 cos 12 x 12 x2 x 2 sec 4 − x2 1 x2 x 1 y x θ 12 − (x − 1)2 x −1 Review Exercises for Chapter 4 133. tan 70 30 arctan d1 ⇒ d1 483 650 cos 25 d2 ⇒ d2 734 810 d cos 48 3 ⇒ d3 435 650 tan d4 ⇒ d4 342 810 h h 25 tan 21 9.6 feet 135. sin 48 sin 25 h 25 134. tan 21 70 66.8 30 431 21° 25 N d1 d2 1217 48° W d3 A d3 d4 93 B 25° 48° 65° 810 650 D θ d1 d2 E d4 C S 93 ⇒ 4.4 1217 sec 4.4 D ⇒ D 1217 sec 4.4 1221 1217 The distance is 1221 miles and the bearing is 85.6. 136. Amplitude: 1.5 0.75 inches 2 Period: 3 seconds 137. False. The sine or cosine functions are often useful for modeling simple harmonic motion. 138. True. The inverse sine, y arcsin x, is defined where 1 ≤ x ≤ 1 and ≤ y ≤ . 2 2 140. False. The range of arctan is , , so arctan1 . 2 2 4 141. y 3 sin x d a cos bt a 0.75 b 2 3 d 0.75 cos 23 t 139. False. For each there corresponds exactly one value of y. Amplitude: 3 Period: 2 Matches graph d 142. y 3 sin x matches graph (a). 143. y 2 sin x Period: 2 Amplitude: 2 Amplitude: 3 Period: 2 144. y 2 sin Period: 4 Amplitude: 2 Matches graph b 145. f sec is undefined at the zeros of g cos since sec x matches graph (c). 2 1 . cos 432 Chapter 4 146. (a) Trigonometry tan 2 cot (b) tan 0.1 0.4 0.7 1.0 1.3 9.9666 2.3652 1.1872 0.6421 0.2776 9.9666 2.3652 1.1872 0.6421 0.2776 cot 2 1 148. y Aekt cos bt 5 et10 cos 6t 147. The ranges for the other four trigonometric functions are not bounded. For y tan x and y cot x, the range is , . For y sec x and y csc x, the range is , 1 1, . 1 1 1 1 (a) A is changed from 5 to 3 : The displacement is increased. (b) k is changed from 10 to 3 : The friction damps the oscillations more rapidly. (c) b is changed from 6 to 9: The frequency of oscillation is increased. 1 149. A 2 r 2, s r 1 (a) A 2 r 20.8 0.4r 2, r > 0 (b) A 12 102 50, > 0 s r0.8 0.8r, r > 0 s 10, > 0 As r increases, the area function increases more rapidly. 30 A 4 A s s 3 0 0 0 6 0 150. Answers will vary. Problem Solving for Chapter 4 1. (a) 8:57 6:45 2 hours 12 minutes 132 minutes 2. Gear 1: 132 11 revolutions 48 4 1142 112 radians or 990 Gear 2: 24 360 332.308 5.80 radians 26 Gear 3: 24 360 392.727 6.85 radians 22 Gear 4: 40 5 360 450 radians 32 2 Gear 5: 24 360 454.737 7.94 radians 19 (b) s r 47.255.5 816.42 feet 3. (a) sin 39 d (b) tan 39 x 3000 d 3000 4767 feet sin 39 3000 x 3000 3705 feet tan 39 (c) tan 63 24 3 360 270 radians 32 2 w 3705 3000 3000 tan 63 w 3705 w 3000 tan 63 3705 2183 feet Problem Solving for Chapter 4 433 4. (a) ABC, ADE, and AFG are all similar triangles since they all have the same angles. A is part of all three triangles and C E G 90. Thus, B D F. (b) Since the triangles are similar, the ratios of corresponding sides are equal. BC DE FG AB AD AF opp BC DE FG sin A it does not matter which triangle is used to calculate sin A. hyp AB AD AF Any triangle similar to these three triangles could be used to find sin A. The value of sin A would not change. (c) Since the ratios: (d) Since the values of all six trigonometric functions can be found by taking the ratios of the sides of a right triangle, similar triangles would yield the same values. 5. (a) hx cos2 x 6. Given: f is an even function and g is an odd function. hx f x2 (a) 3 hx f x2 −2 f x2 since f is even 2 hx −1 Thus, h is an even function. h is even. (b) hx hx gx2 (b) sin2 x hx gx2 3 gx2 since g is odd −2 gx2 2 hx −1 Thus, h is an even function. h is even. Conjecture: The square of either an even function or an odd function is an even function. 7. If we alter the model so that h 1 when t 0, we can use either a sine or a cosine model. 8. P 100 20 cos (a) 1 1 a max min 101 1 50 2 2 83t 130 1 1 d max min 101 1 51 2 2 0 b 8 5 70 For the cosine model we have: h 51 50 cos8 t For the sine model we have: h 51 50 sin 8 t 2 Notice that we needed the horizontal shift so that the sine value was one when t 0 . Another model would be: h 51 50 sin 8 t Here we wanted the sine value to be 1 when t 0. 3 2 (b) Period 2 6 3 sec 83 8 4 This is the time between heartbeats. (c) Amplitude: 20 The blood pressure ranges between 100 20 80 and 100 20 120. (d) Pulse rate (e) Period 64 60 secmin 80 beatsmin 3 4 secbeat 60 15 sec 64 16 64 60 ⇒ b 2b 60 32 2 15 434 Chapter 4 Trigonometry 9. Physical (23 days): P sin 2 t ,t ≥ 0 23 Emotional (28 days): E sin 2 t ,t ≥ 0 28 Intellectual (33 days): I sin 2 t ,t ≥ 0 33 (a) 2 E P I 7300 7380 −2 leap years remaining July days August days 31 1 11 20 years t 365 20 5 (b) Number of days since birth until September 1, 2006: day in September t 7348 2 7349 I 7379 P E −2 All three drop early in the month, then peak toward the middle of the month, and drop again toward the latter part of the month. (c) For September 22, 2006, use t 7369. P 0.631 E 0.901 I 0.945 10. f x 2 cos 2x 3 sin 3x gx 2 cos 2x 3 sin 4 x (a) 11. (a) Both graphs have a period of 2 and intersect when x 5.35. They should also intersect when x 5.35 2 3.35 and x 5.35 2 7.35. (b) The graphs intersect when x 5.35 32 0.65. 6 g − f −6 (b) The period of f x is 2. The period of gx is . (c) hx A cos x B sin x is periodic since the sine and cosine functions are periodic. (c) Since 13.35 5.35 42 and 4.65 5.35 52 the graphs will intersect again at these values. Therefore f 13.35 g4.65. Problem Solving for Chapter 4 12. (a) f t 2c f t is true since this is a two period horizontal shift. 13. θ1 1 1 (b) f t c f t is not true. 2 2 x (a) 1 f t is a doubling of the period of f t. 2 d y θ2 2 ft 1 f t c is a horizontal translation of f t. 2 435 sin 1 1.333 sin 2 sin 2 1 1 (c) f t c f t is not true. 2 2 sin 1 sin 60 0.6497 1.333 1.333 2 40.52 1 1 1 f t c f t c is a horizontal 2 2 2 1 translation of f t by half a period. 2 (b) tan 2 tan 1 12 2 sin12. For example, sin x ⇒ x 2 tan 40.52 1.71 feet 2 y ⇒ y 2 tan 60 3.46 feet 2 (c) d y x 3.46 1.71 1.75 feet (d) As you more closer to the rock, 1 decreases, which causes y to decrease, which in turn causes d to decrease. 14. arctan x x (a) x3 x5 x7 3 5 7 (b) 2 − 2 2 −2 The graphs are nearly the same for 1 < x < 1. 2 − 2 2 −2 The accuracy of the approximation improved slightly by adding the next term x99. 436 Chapter 4 Chapter 4 Trigonometry Practice Test 1. Express 350° in radian measure. 2. Express 59 in degree measure. 3. Convert 135 14 12 to decimal form. 4. Convert 22.569 to D M S form. 5. If cos 23, use the trigonometric identities to find tan . 6. Find given sin 0.9063. 7. Solve for x in the figure below. 8. Find the reference angle for 65. 35 20° x 10. Find sec given that lies in Quadrant III and tan 6. 9. Evaluate csc 3.92. x 11. Graph y 3 sin . 2 12. Graph y 2 cosx . 13. Graph y tan 2x. 14. Graph y csc x 15. Graph y 2x sin x, using a graphing calculator. 16. Graph y 3x cos x, using a graphing calculator. 17. Evaluate arcsin 1. 18. Evaluate arctan3. 19. Evaluate sin arccos . 4 4 . 35 20. Write an algebraic expression for cos arcsin For Exercises 21–23, solve the right triangle. B c A b 21. A 40, c 12 a C 22. B 6.84, a 21.3 23. a 5, b 9 24. A 20-foot ladder leans against the side of a barn. Find the height of the top of the ladder if the angle of elevation of the ladder is 67°. 25. An observer in a lighthouse 250 feet above sea level spots a ship off the shore. If the angle of depression to the ship is 5°, how far out is the ship? x . 4