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C H A P T E R Trigonometry 6 Section 6.1 Angles and Their Measure . . . . . . . . . . . . . . . . . 532 Section 6.2 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 541 Section 6.3 Trigonometric Functions of Any Angle Section 6.4 Graphs of Sine and Cosine Functions . . . . . . . . . . . 567 Section 6.5 Graphs of Other Trigonometric Functions . . . . . . . . . 579 Section 6.6 Inverse Trigonometric Functions . . . . . . . . . . . . . . 591 Section 6.7 Applications and Models . . . . . . . . . . . . . . . . . . 604 . . . . . . . . . . 552 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 C H A P T E R Trigonometry Section 6.1 6 Angles and Their 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 180º. (d) Straight: Measure 180. ■ Two positive angles, 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. ■ Speed distancetime ■ Angular speed t srt Vocabulary Check 1. Trigonometry 2. angle 3. coterminal 4. degree 5. acute; obtuse 6. complementary; supplementary 7. radian 8. linear 1 10. A 2r 2 9. angular 1. 3. The angle shown is approximately 210. The angle shown is approximately 60. 5. (a) Since 90 < 130 < 180; 130 lies in Quadrant II. (b) Since 270 < 285 < 360; 285 lies in Quadrant IV. 532 2. 4. The angle shown is approximately 120. The angle shown is approximately 330. 6. (a) Since 0 < 8.3 < 90; 8.3 lies in Quadrant I. (b) Since 180 < 257 30 < 270; 257 30 lies in Quadrant III. Section 6.1 7. (a) Since 180 < 13250 < 90; 13250 lies in Quadrant III. 8. (a) Since 270 < 260 < 180; 260 lies in Quadrant II. (b) Since 90 < 3.4 < 0; 3.4 lies in Quadrant IV. (b) Since 360 < 336 < 270; 336 lies in Quadrant I. 9. (a) y Angles and Their Measure (b) y 150° 30° x 10. (a) 270 x (b) 120 y y x x −270° 11. (a) − 120° (b) y y 480° 405° x x 12. (a) 750 (b) 600 y y − 600° x x −750° 13. (a) Coterminal angles for 45 45 360 405 45 360 315 (b) Coterminal angles for 36 14. (a) 120 360 480 120 360 240 (b) 420 720 300 420 360 60 15. (a) Coterminal angles for 300 300 360 660 300 360 60 (b) Coterminal angles for 740 36 360 324 740° 2360° 20° 36 360 396 20 360 340 533 534 Chapter 6 Trigonometry 16. (a) 520 720 200 45 17. (a) 5445 54 60 54.75 30 (b) 12830 128 60 128.5 520 360 160 (b) 230 360 590 230 360 130 10 18. (a) 24510 245 60 30 19. (a) 8518 30 85 18 60 3600 85.308 25 (b) 33025 330 3600 330.007 245 0.167 245.167 12 (b) 212 2 60 2 0.2 2.2 36 20. (a) 13536 135 3600 21. (a) 240.6 240 0.660 24036 (b) 145.8 145 0.860 14548 135 0.01 135.01 16 20 (b) 40816 20 408 60 3600 408 0.2667 0.0056 408.272 22. (a) 345.12 345 0.1260 23. (a) 2.5 2 0.5(60) 230 (b) 3.58 [3 0.58(60)] 345 7 0.260 [334.8] 345 7 12 [334 0.8(60)] (b) 0.45 0 0.4560 0 27 33448 027 24. (a) 0.355 0 0.35560 25. (a) Complement: 90 24 66 Supplement: 180 24 156 0 21 0.360 0 21 18 (b) Complement: Not possible. 126 is greater than 90. Supplement: 180 126 54 02118 (b) 0.7865 0 0.786560 0 47 0.1960 0 47 11.4 04711.4 26. (a) Complement: 90 87 3 27. Supplement: 180 87 93 The angle shown is approximately 2 radians. (b) Complement: Not possible. 166 > 90 Supplement: 180 166 14 28. The angle shown is approximately 5.5 radians. 29. The angle shown is approximately 3 radians. Section 6.1 30. The angle shown is approximately 4 radians. 31. (a) Since 0 < (b) Since < 32. (a) Since (b) Since 34. (a) Since 35. (a) < < 0; lies in Quadrant IV. 2 12 12 33. (a) Since 3 11 11 < < ; lies in Quadrant II. 2 9 9 3 < 6.02 < 2 ; 6.02 lies in Quadrant IV. 2 5 4 y Angles and Their Measure lies in Quadrant I. < ; 5 2 5 7 3 7 lies in Quadrant III. < ; 5 2 5 < 1 < 0; 1 lies in Quadrant IV. 2 (b) Since < 2 < ; 2 lies in Quadrant III. 2 (b) Since (b) 2 3 < 2.25 < , 2.25 lies in Quadrant II. 2 y 5π 4 x x − 36. (a) 7 4 y (b) 5 2 2π 3 y 5π 2 x x − 37. (a) 11 6 7π 4 y (b) 3 y 11π 6 x x −3 38. (a) 4 (b) 7 y y 7π 4 x x 535 536 Chapter 6 Trigonometry 39. (a) Coterminal angles for 6 40. (a) 13 2 6 6 11 2 6 6 5 6 41. (a) Coterminal angles for 7 5 2 6 6 9 7 4 4 4 11 2 6 6 9 2 4 4 (b) (b) Coterminal angles for 42. (a) 7 19 2 6 6 23 11 2 6 6 5 17 2 6 6 2 28 2 15 15 5 7 2 6 6 2 32 2 15 15 26 8 2 9 9 Supplement: 98 8 2 45 45 Supplement: Supplement: (b) 144 144 (b) > 2 180 5 6 7 4 2 3 180 180 9 4 (b) 240 240 180 3 47. (a) 20 20 3 180 2 180 49. (a) 3 3 180 270 2 2 (b) 7 7 180 210 6 6 4 5 180 20 9 9 11 12 12 180 6 (b) 150 150 7 7 180 105 12 12 53. 115 115 11 is greater than 12 2 3 4 4 48. (a) 270 270 50. (a) 3 4 180 (b) 120 120 45. (a) 30 30 2 3 3 (b) Complement: none 46. (a) 315 315 Supplement: 2 3 6 44. (a) Complement: 11 12 12 (b) Complement: Not possible; 82 8 2 45 45 2 15 5 2 12 12 43. (a) Complement: 10 8 2 9 9 (b) (b) Coterminal angles for 9 4 180 2.007 radians 51. (a) 7 7 180 420 3 3 (b) 11 180 11 66 30 30 54. 87.4 87.4 52. (a) 11 11 180 330 6 6 (b) 34 34 180 408 15 15 180 1.525 radians Section 6.1 55. 216.35 216.35 57. 0.83 0.83 59. 180 3.776 radians 62. 4.8 4.8 56. 48.27 48.27 180 0.014 radian 180 25.714 7 7 864.000 180 65. s r 58. 0.54 0.54 60. 6 5 180 s r radians 29 10 15 15 180 337.500 8 8 61. 114.592 64. 0.57 0.57 67. s r 29 10 6 5 180 0.842 radians radians 68. 32 7 32 7 537 180 0.009 radians 5 5 180 81.818 11 11 63. 2 2 66. Angles and Their Measure 32.659 180 s r 60 75 radians 4 60 75 5 radian Because the angle represented is clockwise, this angle is 45 radian. 69. s r 70. r 14 feet, s 8 feet 6 27 6 2 radian 27 9 s 8 4 radian r 14 7 73. s r, in radians s 15180 180 15 inches 3s 3(1) 3 meters 1 77. A r 2 2 s 160 kilometers 25 14.5 3 25 50 radians 14.5 29 74. r 9 feet, 60 s r 9 47.12 inches 75. 3s r, in radians 72. r 80 kilometers, 71. 3s r 1 79. A r 2 2 3 76. r 20 centimeters, s r 20 4 4 5 centimeters 15.71 centimeters 4 1 1 A r 2 122 2 2 4 8.38 square inches 18 mm2 56.5 mm2 80. r 1.4 miles, 330 1 12.27 square feet A 2.52225 2 180 1 330 A 1.42 2 180 s 160 2 radians r 80 3 3 feet 9.42 feet 78. r 12 mm, 8 1 A 42 square inches 2 3 3 21.56 5.6 square miles 12 538 Chapter 6 Trigonometry 81. 411550 3247 39 8.46972 0.14782 radian s r 40000.14782 591 miles 82. 473718 3747 36 94942 9.82833 0.17154 radian r 4000 miles s r 40000.17154 686 miles 450 s 0.071 radian 4.04° r 6378 83. 84. r 6378 kilometers 400 s 0.062716 radian r 6378 0.062716 3.59 180 The difference in latitude is about 3.59. 85. s 2.5 25 5 radian r 6 60 12 87. (a) 65 miles per hour 65528060 5720 feet per minute The circumference of the tire is C 2.5 feet. The number of revolutions per minute is r 57202.5 728.3 rev/minute. 88. (a) 2-inch diameter pulley 86. s 24 180 4.8 radians 4.8 275 r 5 (b) The angular speed is t. 5720 (2) 4576 radians 2.5 Angular speed 89. (a) Angular speed 1700 rpm 1700 2 radiansminute (b) Linear speed 1 ft in. 52002 feet 7.25 2 12 in. On the 4-inch diameter pulley: 1 minute 2 3141 feet per minute 3 r2 s 10681.4 2 52002 radians 1 minute 10,400 radians per minute 10681.4 radiansminute Since r 1, the belt moves 10681.4 inchesminute. 4576 radians 4576 radiansminute 1 minute 164.5 feet per second 10681.4 5340.7 2 This pulley is turning at 5340.7 radiansminute. (b) 5340.7 850 rpm 2 90. (a) 4 rpm 42 radiansminute 8 25.13274 radiansminute (b) r 25 ft r 200 ftminute t Linear speed 2525.13274 ftminute 628.32 ftminute Section 6.1 Angles and Their Measure 91. (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 92. A R 2 r 2 2 1 93. A r 2 2 R 25 14 39 25 125° r 14 A r 1 125 2 180 14 1 352140 2 180 140° 35 476.39 square meters 39 2 14 2 460.069 1496.62 square meters 1445 in. 2 94. (a) Arc length of larger sprocket in feet: s r 2 1 feet s 2 3 3 Therefore, the chain moves 2 feet as does the smaller rear sprocket. 3 Thus, the angle of the smaller sprocket is 2 ft s 3 4 r 2 ft 12 r 2 inches 12 feet 2 and the arc length of the tire in feet is: s r s 4 12 14 14 feet 3 14 s 14 3 Speed feet per second t 1 sec 3 3600 seconds 1 mile 14 feet 10 miles per hour 3 seconds 1 hour 5280 feet (b) Since the arc length of the tire is Distance 14 feet and the cyclist is pedaling at a rate of one revolution per second, we have: 3 1 mile feet n revolutions 143 revolutions 5280 feet 7 n miles 7920 (c) Distance Rate Time 1 mile t seconds 143 feetsecond5280 feet 7 t miles 7920 (d) The functions are both linear. 539 540 Chapter 6 Trigonometry 95. False. A measurement of 4 radians corresponds to two complete revolutions from the initial to the terminal side of an angle. 96. True. If and are coterminal angles, then n360 or n2, where n is an integer. The difference between and is n360, or n2 if expressed in radians. 97. False. The terminal side of 1260 lies on the negative x-axis. 98. (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. (c) Two angles with the same initial and terminal sides are coterminal. (d) An obtuse angle measures between 90° and 180°. 100. 1 radian 99. The speed increases, since the linear speed is proportional to the radius. 180 57.3, so one radian is much larger than one degree. 101. Since the arc length s is given by s r, if the central angle is fixed while the radius r increases, then s increases in proportion to r . 102. Area of circle r 2 θ Area of sector Measure of central angle of sector Area of circle Measure of central angle of circle r Area of sector r2 2 Area of sector r 2 103. 4 4 42 42 105. 23 2 6 2 2 2 21 r 2 42 2 8 2 36 212 2 2 104. 2 2 107. 22 62 4 36 40 4 2 3 2 3 55 5 210 2 3 3 23 3 105 25 12 2 5 2 2 106. 10 210 108. 182 122 324 144 468 36 109. 182 62 324 36 288 144 111. Horizontal shift two units to the right y 2 2 52 4 13 613 110. 172 92 289 81 208 2 122 16 13 413 112. f (x) x5 4 y = x5 3 y 4 Vertical shift four units downward 2 2 1 −2 x 2 3 −1 −3 x −2 4 y = x5 1 3 −2 −2 −3 2 y = x5 − 4 y = (x − 2)5 −6 Section 6.2 113. Reflection in the x-axis and a vertical shift two units upward 114. f x x 35 Reflection in the x-axis and a horizontal shift three units to the left y 6 5 4 Right Triangle Trigonometry y = x5 3 1 −3 −2 y = −(x + 3)5 2 1 2 x 1 −1 3 −2 y = 2 − x5 −3 −3 Section 6.2 ■ Right Triangle Trigonometry (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 θ Adjacent side (k) 1 cot2 csc2 ■ You should know that two acute angles and are complementary if 90, and that the 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 hypotenuse sec adjacent (e) 2. opposite; adjacent; hypotenuse 3. elevation; depression (ii) adjacent cot opposite (f) (iii) hypotenuse csc opposite (d) (iv) adjacent cos hypotenuse (b) (v) opposite sin hypotenuse (a) (vi) opposite tan adjacent (c) Opposite side opp adj us (c) tan en adj hyp ot (b) cos yp opp hyp H (a) sin e You should know the right triangle definition of trigonometric functions. ■ 1. (i) y = x5 1 −5 −4 −3 −2 y 3 x −1 541 2 542 Chapter 6 Trigonometry 1. hyp 62 82 36 64 100 10 6 θ 8 sin opp 6 3 hyp 10 5 csc hyp 10 5 opp 6 3 cos 8 4 adj hyp 10 5 sec hyp 10 5 adj 8 4 tan opp 6 3 adj 8 4 cot 8 4 adj opp 6 3 2. adj 132 52 169 25 12 13 sin 5 opp hyp 13 csc hyp 13 opp 5 cos adj 12 hyp 13 sec hyp 13 adj 12 tan opp 5 adj 12 cot adj 12 opp 5 5 θ b 3. adj 412 92 1681 81 1600 40 41 9 θ sin opp 9 hyp 41 csc hyp 41 opp 9 cos adj 40 hyp 41 sec hyp 41 adj 40 tan opp 9 adj 40 cot adj 40 opp 9 4. hyp 42 42 32 42 sin 2 opp 4 1 hyp 42 2 2 csc hyp 42 2 opp 4 cos 2 4 adj 1 hyp 42 2 2 sec hyp 42 2 adj 4 tan opp 4 1 adj 4 cot adj 4 1 opp 4 4 θ 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 opp 1 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 6 hyp 3 32 adj 42 22 4 tan opp 2 2 1 5 adj42 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. Section 6.2 Right Triangle Trigonometry 543 6. θ 7.5 8 4 θ 15 hyp 7.52 42 hyp 152 82 289 17 sin opp 8 hyp 17 csc hyp 17 opp 8 15 adj cos hyp 17 hyp 17 sec adj 15 8 opp tan adj 15 15 adj cot opp 8 17 2 sin opp 4 8 hyp 172 17 csc hyp 172 17 opp 4 8 cos 7.5 adj 15 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 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 csc hyp 1.25 5 opp 0.75 3 sec hyp 1.25 5 adj 1 4 cot adj 1 4 opp 0.75 3 4 opp 1.252 12 0.75 opp 0.75 3 sin hyp 1.25 5 1.25 θ 1 adj 1 4 cos hyp 1.25 5 opp 0.75 3 tan adj 1 4 The function values are the same since the triangles are similar and the corresponding sides are proportional. 8. θ 1 3 2 hyp 12 22 θ 5 6 5 opp 1 sin hyp 5 5 hyp 5 5 csc opp 1 adj 2 25 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 opp 1 3 hyp 35 5 5 csc hyp 35 5 opp 3 cos adj 6 2 25 hyp 35 5 5 sec hyp 35 5 adj 6 2 tan opp 3 1 adj 6 2 cot adj 6 2 opp 3 The function values are the same because the triangles are similar, and corresponding sides are proportional. 544 Chapter 6 9. Given: sin Trigonometry 3 opp 4 hyp 10. opp 72 52 24 26 32 adj2 42 4 adj 7 7 adj cos hyp 4 opp 26 hyp 7 tan opp 26 adj 5 csc 7 hyp 76 opp 26 12 3 θ 7 opp 37 tan adj 7 cot hyp 47 sec adj 7 2 6 θ 5 adj 5 56 opp 26 12 hyp 4 opp 3 11. Given: sec 2 2 hyp 1 adj 12. hyp 52 12 26 opp2 12 22 sin opp 3 2 opp 3 sin hyp 2 3 1 26 opp 1 hyp 26 26 cos adj 5 526 hyp 26 26 tan opp 1 adj 5 θ adj 1 cos hyp 2 tan opp 3 adj csc hyp 26 26 opp 1 cot 3 adj opp 3 sec hyp 26 adj 5 csc hyp 23 opp 3 13. Given: tan 3 3 opp 1 adj hyp 10 opp 310 hyp 10 10 adj cos hyp 10 θ 26 5 14. opp 62 12 35 32 12 hyp2 sin 7 hyp 7 sec adj 5 7 adj cot opp 3 csc sin sin 10 opp 35 hyp 6 3 6 adj 1 cos hyp 6 θ 1 tan opp 35 35 adj 1 cot adj 1 opp 3 csc hyp 6 635 opp 35 35 sec hyp 10 adj cot 35 adj 1 opp 35 35 csc hyp 10 opp 3 θ 1 35 1 Section 6.2 15. Given: cot adj 3 2 opp Right Triangle Trigonometry 16. adj 172 42 273 22 32 hyp2 13 hyp 13 2 θ 2 opp 213 sin hyp 13 13 3 sin opp 4 hyp 17 17 cos 273 adj hyp 17 273 adj 3 313 hyp 13 13 tan 4 opp 4273 273 adj 273 tan opp 2 adj 3 sec hyp 17 17273 adj 273 273 csc hyp 13 opp 2 cot 273 adj opp 4 sec hyp 13 adj 3 30 30 180 6 radian sin 30 opp 1 hyp 2 60° 2 1 30° cos 45 180 60 3 3 19. π 6 20. degree opp 3 3 tan 3 adj 1 3 radian value 4 2 cos 45 3 2 18. degree 2 1 2 2 45° 2 1 radian value 4 2 sec 45 sec π 3 1 2 4 θ cos 17. 545 2 1 π 4 2 2 4 1 1 1 cot 21. 30° 3 3 1 3 adj opp 60 radian 3 3 22. degree radian value 4 2 csc 45 csc 45 60° 2 2 1 45° 1 1 1 23. π 3 2 180 30 6 6 1 cos π 6 3 adj 6 hyp 2 3 cot 1 45° 1 45° 1 1 adj 1 opp 45 45 180 4 radian value 4 2 sin 45 sin 25. 2 24. degree 1 radian value 6 1 3 tan 30 1 tan 30 3 1 π 4 2 1 4 2 2 26. degree 2 2 60° 2 1 30° 3 546 Chapter 6 27. sin 60 3 2 (a) tan 60 Trigonometry 1 2 , cos 60 sin 60 3 cos 60 (b) sin 30 cos 60 (c) cos 30 sin 60 13 2 , sec (a) csc 30 1 2 sin 30 (b) cot 60 tan90 60 tan 30 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 2 13 3 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 sin 21313 2 cos 31313 3 13 (d) sec90 csc 2 (c) tan 1 3 (a) sec (b) sin2 cos2 1 sin2 (c) cot90º tan 26 (d) sin tan cos 26 5 1 26 5 32. tan 5 1 3 cos 3 1 2 8 9 sin 22 3 1 2 cos 3 1 (c) cot sin 4 22 22 3 1 (d) sin90 cos 3 33. tan cot tan 35. tan cos (a) cot (b) cos 1 sin2 tan 1 sin 1 cos cos sin 3 30. sec 5, tan 26 (a) sin 31. cos 3 1 2 3 cos 60 1 (d) cot 60 3 sin 60 3 29. csc 3 1 28. sin 30 , tan 30 2 3 1 1 tan 5 1 1 tan2 (c) tan90º cot 1 1 52 1 26 26 26 1 5 (d) csc 1 cot 2 1 15 1 251 2625 34. cos sec cos 36. cot sin 2 1 1 cos cos sin cos sin 26 5 Section 6.2 37. 1 cos 1 cos 1 cos2 Right Triangle Trigonometry 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) tan 23.5 0.4348 (b) cot 66.5 44. (a) sin 16.35 0.2815 1 0.4348 tan 66.5 18 0.9598 60 56 (b) sin 7356 sin 73 0.9609 60 45. (a) cos 1618 cos 16 47. Make sure that your calculator is in radian mode. (a) cot (b) 16 tan 1 tan 16 5.0273 1 3.5523 sin 16.35 46. (a) sec 4212 sec 42.2 (b) csc 487 48. (a) sec 0.75 1 1.3499 cos 42.2 1 7 sin 48 60 1.3432 1 1.3667 cos 0.75 (Note: 0.75 is in radians) (b) cos 0.75 0.7317 0.1989 16 49. Make sure that your calculator is in radian mode. (a) csc 1 (b) (b) csc 16.35 1 1.1884 sin 1 (b) cot 1 tan 0.5463 2 51. (a) sin 50. (a) sec 1 ⇒ 30 2 6 (b) csc 2 ⇒ 30 6 2 1 2 2 52. (a) cos 1 2 2 1 1.1884 cos 1 2 1 0.5463 1 tan 2 2 ⇒ 45 (b) tan 1 ⇒ 45º 4 4 547 548 Chapter 6 Trigonometry 53. (a) sec 2 ⇒ 60 3 (b) cot 1 ⇒ 45 4 54. (a) tan 3 ⇒ 60 (b) cos 23 ⇒ 60 3 3 55. (a) csc 56. (a) cot ⇒ 45 (b) sin 2 4 2 tan 3 1 ⇒ 60 2 3 3 3 3 3 ⇒ 60 3 3 (b) sec 2 cos tan 30 57. 1 30 3 30 ° 32 58. sin 60 30 x tan 60 32 x 3 32 x x x tan 82 61. x 82° 45 m 2 ⇒ 45 60. sin 45 r 3 2 20 r 20 20 202 2 sin 45 2 323 32 3 3 x 45 (b) tan 62. (a) h 270 feet Distance between friends: 45 45 cos 82 ⇒ y y cos 82 6 132 3 Not drawn to scale 323.34 meters sin 3000 ft θ 1500 1 3000 2 1500 ft 30 6 6 h 3 135 (c) 2135 h h Height of the building: 123 45 tan 82 443.2 meters 63. 4 y 18 x 45 tan 82 y 2 93 3 x 32 60° y 18 sin 60 18 x 303 x 59. 30 x 1 2 64. tan opp adj tan 54 w 100 w 100 tan 54 137.6 feet Section 6.2 65. (a) sin 23 x (b) tan 23 y 150 x x 150 383.9 feet sin 23 5 ft 1 23° y 150 y 150 353.4 feet tan 23 (c) Moving down the line: Dropping vertically: 150sin 23 63.98 feet per second 6 150 25 feet per second 6 66. Let h the height of the mountain. 67. (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 Right Triangle Trigonometry 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 1256 28 549 550 Chapter 6 68. tan 3 Trigonometry x 15 x 15 tan 3 d 5 2x 5 215 tan 3 6.57 centimeters 69. (a) (e) 20 Angle, h 85° 80 19.7 70 18.8 60 17.3 50 15.3 h (b) sin 85 20 40 12.9 (c) h 20 sin 85 19.9 meters 30 10.0 20 6.8 10 3.5 (d) The side of the triangle labeled h will become shorter. (f) The height of the balloon decreases. Height (in meters) 20 h θ 70. x 9.397, y 3.420 sin 20 y 0.34 10 cos 20 x 0.94 10 tan 20 y 0.36 x cot 20 x 2.75 y sec 20 10 1.06 x csc 20 10 2.92 y 71. sin 60 csc 60 1 True, csc x 1 1 ⇒ sin 60 csc 60 sin 60 1 sin x sin 60 72. True, because sec90 csc . 73. sin 45 sin 45 1 False, 74. True, because 75. 1 cot 2 csc 2 cot 2 csc 2 1 cot 2 csc 2 1. 2 2 2 2 2 1 sin 60 sin 2 sin 30 False, sin 60 cos 30 cot 30 1.7321 sin 30 sin 30 sin 2 0.0349 Section 6.2 76. tan0.8 2 tan20.8 Right Triangle Trigonometry 77. This is true because the corresponding sides of similar triangles are proportional. False. tan0.82 tan 0.64 0.745 tan 20.8 tan 0.8 2 1.060 78. Yes. Given tan , sec can be found from the identity 1 tan2 sec2 . 79. (a) 80. (a) 0.1 0.2 0.3 0.4 0.5 (b) In the interval 0, 0.5, > sin . sin 0.0998 0.1987 0.2955 0.3894 0.4794 (c) As → 0, sin → 0, and 0 0.3 0.6 0.9 1.2 1.5 sin 0 0.2955 0.5646 0.7833 0.9320 0.9975 cos 1 0.9553 0.8253 0.6216 0.3624 0.0707 (b) On 0, 1.5, sin is an increasing function. (c) On 0, 1.5, cos is a decreasing function. (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. 81. x2 6x x2 4x 12 x2 12x 36 xx 6 x2 36 x 6x 2 82. 83. x 6x 6 x 6x 6 x , x ±6 x2 2t 2 5t 12 2t 2 5t 12 t 2 16 2 2 9 4t 2 9 4t 4t 12t 9 4t 2 12t 9 t 2 16 2t 3t 4 2t 32t 3 2t 3 3 2t3 2t t 4t 4 t 4 2t 3 3 , t ± , 4 4t 2 3 2 x 3x 2x 2 2x 22 xx 2 x 2 x 2 x2 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 84. 12 x 4, 12 12 x 4x 1 x x 3 1 x 0, 12 → 1. sin 551 552 Chapter 6 Trigonometry Section 6.3 ■ 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. ■ 3 You should know the trigonometric function values of the quadrant angles 0, , , and . 2 2 ■ 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. ■ You should know which trigonometric functions are odd and even. Even: cos x and sec x Odd: sin x, tan x, cot x, csc x Vocabulary Check 1. reference 2. periodic 3. period 4. even; odd 1. (a) x, y 4, 3 r 16 9 5 sin y 3 r 5 csc r 5 y 3 cos x 4 r 5 sec r 5 x 4 tan y 3 x 4 cot x 4 y 3 (b) x, y 8, 15 r 64 225 17 sin y 15 r 17 csc r 17 y 15 cos x 8 r 17 sec r 17 x 8 tan y 15 x 8 cot x 8 y 15 Section 6.3 r 12 12 2 r 122 52 13 y 5 r 13 12 x cos r 13 r 13 y 5 13 r sec x 12 sin csc y 5 5 x 12 12 cot x 12 12 y 5 5 3. (a) x, y 3, 1 sin 2 y 1 2 r 2 csc 2 r 2 y 1 cos 2 x 1 2 r 2 sec 2 r 2 x 1 tan 1 y 1 x 1 cot 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 4. (a) x 3, y 1 r 32 12 (b) x 4, y 4 10 r 42 42 42 sin 10 1 y 10 r 10 sin 2 y 4 r 4 2 2 cos x 3 310 10 r 10 cos 2 x 4 r 42 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 r 42 2 x 4 cot x 3 3 y 1 cot x 4 1 y 4 5. x, y 7, 24 6. x 8, y 15 r 82 152 17 r 49 576 25 y 24 r 25 x 7 cos r 25 y 24 tan x 7 sin 553 (b) x 1, y 1 2. (a) x 12, y 5 tan Trigonometric Functions of Any Angle r 25 y 24 r 25 sec x 7 x 7 cot y 24 csc sin y 15 r 17 csc r 17 y 15 cos x 8 r 17 sec r 17 x 8 tan y 15 x 8 cot x 8 y 15 554 Chapter 6 Trigonometry 7. x, y 4, 10 8. x 5, y 2 r 52 22 29 r 16 100 229 sin y 529 r 29 sin 2 y 229 29 r 29 cos 229 x r 29 cos 5 x 529 29 r 29 tan 5 y x 2 tan y 2 2 x 5 5 csc 29 r y 5 csc 29 29 r y 2 2 sec 29 r x 2 sec 29 29 r x 5 5 cot x 2 y 5 cot x 5 5 y 2 2 9. x, y 3.5, 6.8 r 3.52 6.82 58.49 y 6.8 6.858.49 r 58.49 58.49 x 3.5 3.558.49 cos r 58.49 58.49 y 6.8 tan x 3.5 sin csc 58.49 r y 6.8 sec 58.49 r x 3.5 cot x 3.5 y 6.8 1 7 3 31 10. x 3 , y 7 2 2 4 4 r 72 314 2 2 1157 4 31 311157 y 4 sin r 1157 1157 4 7 141157 x 2 cos r 1157 1157 4 31 y 4 31 3 tan 2 x 7 14 14 2 1157 r csc y 1157 4 31 31 4 1157 r sec x x cot y 11. sin < 0 ⇒ lies in Quadrant III or in Quadrant IV. 4 7 2 1157 14 7 2 14 31 31 4 12. sin > 0 ⇒ lies in Quadrant I or in Quadrant II. cos < 0 ⇒ lies in Quadrant II or in Quadrant III. cos > 0 ⇒ lies in Quadrant I or in Quadrant IV. sin < 0 and cos < 0 ⇒ lies in Quadrant III. sin > 0 and cos > 0 ⇒ lies in Quadrant I. 13. sin > 0 ⇒ lies in Quadrant I or in Quadrant II. 14. sec > 0 ⇒ lies in Quadrant I or in Quadrant IV. tan < 0 ⇒ lies in Quadrant II or in Quadrant IV. cot < 0 ⇒ lies in Quadrant II or in Quadrant IV. sin > 0 and tan < 0 ⇒ lies in Quadrant II. sec > 0 and cot < 0 ⇒ lies in Quadrant IV. Section 6.3 15. sin y 3 ⇒ x2 25 9 16 r 5 in Quadrant II ⇒ x 4 y 3 5 r sin csc r 5 y 3 4 x 5 r cos sec r 5 x 4 3 y 4 x tan cot x 4 y 3 17. sin < 0 and tan < 0 ⇒ is in Quadrant IV ⇒ y < 0 and x > 0. 15 y tan ⇒ r 17 x 8 y 15 r 17 x 8 cos r 17 y 15 tan x 8 csc sin 19. cot r 17 y 15 17 r x 8 x 8 cot y 15 sec 3 3 x y 1 1 x 3, y 1, r 10 10 y sin r 10 r csc 10 y x 310 r 10 1 y tan x 3 10 r x 3 x cot 3 y 21. sec sec r 2 ⇒ y2 4 1 3 x 1 sin > 0 ⇒ is in Quadrant II ⇒ y 3 sin y 3 r 2 csc x 1 cos r 2 tan r sec 2 x y 3 x 23. cot is undefined, sin 0 r 23 y 3 cot 3 x y 3 3 ≤ ≤ ⇒ y0 ⇒ 2 2 csc is undefined. cos 1 sec 1 tan 0 cot is undefined. x 4 ⇒ y 3 r 5 16. cos in Quadrant III ⇒ y 3 sin 3 y r 5 csc 5 3 cos 4 x r 5 sec 5 4 tan y 3 x 4 cot 18. cos 4 3 8 x ⇒ y 15 r 17 tan < 0 ⇒ y 15 sin y 15 15 r 17 17 csc cos 8 x r 17 sec tan y 15 15 x 8 8 cot 20. csc cos > 0 ⇒ is in Quadrant IV ⇒ x is positive; cos Trigonometric Functions of Any Angle 17 15 17 8 8 15 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 15 415 15 22. sin 0 ⇒ 0 n sec 1 ⇒ 2n y 0, x r sin 0 cos tan x r 1 r r y 0 r r csc is undefined. sec r r 1 x r cot is undefined. 555 556 Chapter 6 Trigonometry 24. tan is undefined ⇒ n 2 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. 3 , x 0, y r 2 r y r csc 1 1 sin y r r sec is undefined. x 0 cos 0 x 0 r r cot 0 y y tan is undefined. ≤ ≤ 2 ⇒ x 1, y 1, r 2 sin 1 2 cos 1 2 2 csc 2 2 2 sec 2 2 tan 1 cot 1 26. Let x > 0. x, 3 x, Quadrant III 1 r x 2 1 2 10 x x 9 3 10x 3 x 1 10 y r 10 10x 3 310 x x cos r 10 10x 3 1 x y 3 1 tan x x 3 sin csc r y 3 10 1 x 3 10x sec 10 r 3 x x 3 cot x y x 3 1 x 3 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. 28. Let x > 0. x 1, y 2, r 5 2 25 sin 5 5 5 1 cos 5 5 2 tan 2 1 4 4x 3y 0 ⇒ y x 3 x, 3 x, Quadrant IV 4 csc 5 5 2 2 5 sec 5 1 1 1 cot 2 2 r x 2 16 2 5 x x 9 3 3 x 4 sin y r cos x x 3 r 5 5 x 3 5 x 3 3 x 4 5 csc sec 5 4 5 3 4 y tan x 29. x, y 1, 0, r 1 sin y 0 0 r 1 30. x, y 0, 1, r 1 csc 1 3 1 2 1 x 4 3 cot 3 4 31. x, y 0, 1, r 1 sec r 1 3 ⇒ undefined 2 x 0 Section 6.3 r 1 1 x 1 sin 35. x, y 1, 0, r 1 csc y 1 1 2 r 1 cot 1 undefined. 0 37. 203 36. x, y 0, 1 x 0 cot 0 2 y 1 1 r ⇒ undefined y 0 557 34. x, y 1, 0, r 1 33. x, y 0, 1, r 1 32. x, y 1, 0, r 1 sec Trigonometric Functions of Any Angle 203 180 23 y 203° x θ′ 38. 309 245 39. 360 245 115 coterminal angle 360 309 51 180 115 65 y y 309° θ′ x θ′ x − 245° 145 40. 41. y 360 145 215 2 3 (coterminal angle) y 2π 3 2 3 3 θ′ x 215 180 35 42. 7 4 2 x θ′ −145° y y 43. 3.5 7 4 4 3.5 7π 4 3.5 x θ′ θ′ x 558 Chapter 6 44. Trigonometry 11 5 is coterminal with . 3 3 2 5 3 3 45. 225, 45, Quadrant III y sin 225 sin 45 11π 3 θ′ sin 750 sin 30 2 1 2 tan 750 tan 30 48. 405, 405 360 45 in Quadrant IV. 3 2 1 2 cos(150) cos 30 2 tan(150) tan 30 50. 840 is coterminal with 240. 51. 240 180 60 in Quadrant III. 3 2 1 cos(840) cos 60 2 tan(840) tan 60 3 , in Quadrant I. 4 4 3 sin(150) sin 30 2 2 sin(840) sin 60 2 3 49. 150, 30, Quadrant III tan405 tan 45 1 52. 1 2 cos 750 cos 30 tan 300 tan 60 3 cos405 cos 45 2 47. 750, 30, Quadrant I 3 sin405 sin 45 2 tan 225 tan 45 1 46. 300, 360 300 60 in Quadrant IV. cos 300 cos 60 2 cos 225 cos 45 x sin 300 sin 60 2 sin cos 2 4 2 cos tan 1 4 tan 3 3 4 sin 3 3 2 cos 4 1 cos 3 3 2 tan 4 tan 3 3 3 6 cos 6 3 sin 6 sin 6 2 2 4 2 2 4 , , Quadrant III 3 3 53. , , Quadrant IV 6 6 sin 3 1 3 54. 3 is coterminal with . 2 2 3 1 2 3 0 2 3 is undefined. 2 2 sin sin 2 cos cos 2 3 6 tan 6 3 2 tan tan Section 6.3 55. 11 , , Quadrant II 4 4 Trigonometric Functions of Any Angle 56. 10 4 is coterminal with . 3 3 4 in Quadrant III. 3 3 sin 2 11 sin 4 4 2 cos 2 11 cos 4 4 2 sin 3 10 sin 3 3 2 tan 11 tan 1 4 4 cos 10 1 cos 3 3 2 tan 10 tan 3 3 3 57. 3 , 2 2 58. 3 sin 1 2 2 2 3 cos 0 2 2 sin 3 tan which is undefined. 2 2 cos sin cos tan sin 59. 3 5 60. 2 25 sin 4 4 2 2 25 cos 4 4 2 25 tan 1 4 4 5 cos2 1 9 25 16 cos2 25 cos > 0 in Quadrant IV. 4 5 csc 2 1 cot 2 csc 2 cot 2 csc 2 1 cot 2 22 1 cot 2 3 cot < 0 in Quadrant IV. cot 3 cot 3 61. tan 3 3 2 sec2 1 tan2 1 32 csc 2 cos2 1 sin2 cos 1 cot 2 csc 2 sin2 cos2 1 cos2 1 7 in Quadrant IV. 4 4 tan 62. 25 7 is coterminal with . 4 4 10 2 csc 2 csc > 0 in Quadrant II. 10 csc csc 5 8 1 1 ⇒ sec sec cos sec 1 5 8 8 5 sec2 1 9 4 3 2 13 4 sec < 0 in Quadrant III. sec 64. cos 2 sec2 1 sin 10 1 1 sin csc 10 10 63. cos sec2 1 13 2 sec 9 4 1 tan2 sec 2 tan2 sec 2 1 4 tan2 tan 2 9 2 1 65 16 tan > 0 in Quadrant III. tan 65 4 559 560 Chapter 6 Trigonometry 65. sin 10 0.1736 68. csc330 71. sec 72 66. sec 225 1 2.0000 sin330 1 3.2361 cos 72 70. cot 178 28.6363 72. tan188 0.1405 73. tan 4.5 4.6373 75. cos 80. csc 1 ⇒ reference angle is 30 or and is in 2 6 Quadrant I or Quadrant II. 81. (a) sin 5 , 6 6 1 ⇒ reference angle is 30 or and is 2 6 in Quadrant III or Quadrant IV. 7 11 , Values in radians: 6 6 23 ⇒ reference angle is 60 or and 3 3 is in Quadrant I or Quadrant II. 2 7 , 4 4 ⇒ reference angle is 45 or and 2 4 is in Quadrant II or III. (b) cos 2 3 5 , 4 4 84. (a) sec 2 ⇒ reference angle is 60 or and is in 3 Quadrant I or IV. Values in degrees: 60, 300 Values in degrees: 60, 120 2 , 3 3 3 7 Values in radians: , 4 4 ⇒ reference angle is 45 or and is 2 4 in Quadrant I or IV. 82. (a) cos Values in radians: 83. (a) csc Values in degrees: 135, 315 Values in degrees: 135, 225 Values in degrees: 210, 330 in Quadrant II or Quadrant IV. 15 1 4.4940 14 sin1514 Values in radians: (b) sin (b) cot 1 ⇒ reference angle is 45 or 1 1.0436 cos 0.29 Values in degrees: 45, 315 Values in degrees: 30, 150 Values in radians: 76. tan 78. sec 0.29 11 1 0.4142 8 tan118 Values in radian: 9 0.3640 9 0.9397 77. sin0.65 0.6052 79. cot 67. cos110 0.3420 69. tan 304 1.4826 1 0.2245 tan 1.35 74. cot 1.35 1 1.4142 cos 225 Values in radians: and is 4 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 and is 3 Section 6.3 Trigonometric Functions of Any Angle Quadrant I or Quadrant III. ⇒ reference angle is 60 or and is 2 3 in Quadrant I or II. Values in degrees: 45, 225 Values in degrees: 60, 120 85. (a) tan 1 ⇒ reference angle is 45 or Values in radians: and is in 4 86. (a) sin 5 , 4 4 and 6 89. 91. 3 Values in degrees: 240, 300 Values in degrees: 150, 330 5 11 , 6 6 Values in radians: 22, 22 corresponds to t 4 on the unit circle. 2 , 3 3 ⇒ reference angle is 60 or and 2 3 is in Quadrant III or IV. (b) sin is in Quadrant II or Quadrant IV. Values in radians: 3 Values in radians: (b) cot 3 ⇒ reference angle is 30 or 87. 561 4 5 , 3 3 1 3 , x, y , 3 2 2 88. t sin 2 4 2 since sin t y sin cos 2 4 2 3 since sin t y. 3 2 since cos t x cos tan 1 4 1 since cos t x. 3 2 since tan t y x 3 y 2 tan 3 since tan t . 3 1 x 2 23, 12 corresponds to t 56 on the unit circle. 90. t 2 2 5 , x, y , 4 2 2 sin 5 1 6 2 since sin t y sin 2 5 since sin t y. 4 2 cos 3 5 6 2 since cos t x cos 2 5 since cos t x. 4 2 tan 3 5 6 3 since tan t y x 21, 23 corresponds to t 43 on the unit circle. 5 tan 4 92. t 2 2 2 y 1 since tan t . x 2 3 1 11 , x, y , 6 2 2 sin 3 4 3 2 since sin t y. sin 1 11 since sin t y. 6 2 cos 1 4 3 2 since cos t x. cos 11 3 since cos t x. 6 2 tan 4 3 3 since y tan t . x 1 2 11 tan 6 3 2 1 3 3 3 y since tan t . x 562 Chapter 6 Trigonometry 93. 0, 1 corresponds to t 3 on the unit circle. 2 3 1 2 since sin t y 3 0 cos 2 since cos t x since tan t sin tan 3 is undefined 2 95. (a) sin 5 1 94. t , x, y 1, 0 sin 0 since sin t y. cos 1 since cos t x. tan 0 y 0 since tan t . 1 x y x 96. (a) sin 0.75 y 0.7 (b) cos 2 0.4 97. (a) sin t 0.25 98. (a) sin t 0.75 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 0.72 or t 5.56 99. (a) New York City: N 22.1 sin0.52t 2.22 55.01 Fairbanks: F 36.6 sin0.50t 1.83 25.61 (b) Month New York City Fairbanks February 35 1 March 41 14 May 63 48 June 72 59 August 76 56 September 69 42 November 47 7 (c) The periods are about the same for both models, approximately 12 months. 100. S 23.1 0.442t 4.3 cos t 6 (a) t 2; S 23.1 0.4422 4.3 cos 2 26,134 snowboards 6 (b) t 14; S 23.1 0.44214 4.3 cos (c) t 6; S 23.1 0.4426 4.3 cos 6 21,452 snowboards 6 (d) t 18; S 23.1 0.44218 4.3 cos 101. yt 2 cos 6t (a) y0 2 cos 0 2 centimeters 1 3 (b) y4 2 cos2 0.14 centimeter 1 (c) y2 2 cos 3 1.98 centimeters 14 31,438 snowboards 6 18 26,756 snowboards 6 (b) cos 2.5 x 0.8 Section 6.3 Trigonometric Functions of Any Angle 563 103. I 5e2t sin t 102. yt 2et cos 6t I0.7 5e1.4 sin 0.7 0.79 (a) t 0 y0 2e0 cos 0 2 centimeters 1 (b) t 4 y14 2e14 cos6 14 0.11 centimeter 1 (c) t 2 y 1 2 2e12 cos6 12 1.2 centimeters 104. sin 6 6 ⇒ d d sin d (c) 120 (b) 90 (a) 30 6 6 6 miles d sin 90º 1 6 sin 30 d 6 12 miles 12 105. 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. 6 sin 120 6 32 6.9 miles 106. False. The reference angle is always acute by definition. For in Quadrant II, 180 . For in Quadrant III, 180. For in Quadrant IV, 360 . 107. ht) f tgt 108. As increases from 0 to 90, x decreases from 12 cm to y 0 cm, y increases from 0 cm to 12 cm, sin 12 x increases from 0 to 1, cos decreases from 1 to 0, 12 y and tan increases without bound (and is undefined x at 90). ht f tgt f tgt ht Therefore, ht is odd. 109. (a) 0 20 40 60 80 sin 0 0.342 0.643 0.866 0.985 sin180 0 0.342 0.643 0.866 0.985 (b) Conjecture: sin sin180 110. (a) cos 32 sin (b) cos 0 0.3 0.6 0.9 1.2 1.5 0 0.2955 0.5646 0.7833 0.9320 0.9975 0 0.2955 0.5646 0.7833 0.9320 0.9975 32 sin 564 Chapter 6 Trigonometry 111. If is obtuse, then 90 < < 180. The reference angle is 180 and we have the following: sin sin csc csc cos cos sec sec tan tan cot cot 112. Domain: All real numbers x Domain: All real numbers x 3 Range: 1, 1 Range: 1, 1 − Period: 2 2 Zeros: n Period: 2 2 The function is even. The function is odd. Domain: All real numbers x except x n Range: , − Zeros: n −3 2 −3 Range: , 1 1, 3 3 Period: 2 − − 2 −3 Domain: All real numbers x except x n Range: , 1 1, 2 3 Period: 2 − 2 Zeros: none The function is even. −3 Domain: All real numbers x except x n Range: , 3 Period: 2 − 2 −3 The function is odd. The secant function is similar to the tangent function because they both have vertical asymptotes at x n 2 Zeros: none The function is odd. The function is odd. Zeros: n 2 Domain: All real numbers x except x n Period: Zeros: n 3 . 2 The cotangent function and the cosecant function both have vertical asymptotes at x n. A maximum point on the sine curve corresponds to a relative minimum on the cosecant curve. The maximum points of sine and cosine are interchanged with the minimum points of cosecant and secant. The x-intercepts of the sine and cosine functions become vertical asymptotes of the cosecant and secant functions. The graphs of sine and cosine may be translated left or right (respectively) to 2 to coincide with each other. −3 Section 6.3 2 4x 113. f x Trigonometric Functions of Any Angle 114. gx 1 x2 4 Vertical asymptote: x 4 Vertical asymptotes: x 2, x 2 Horizontal asymptote: y 0 Horizontal asymptotes: y 0 x 0 1 2 3 5 6 x y 1 2 2 3 1 2 2 1 gx 565 ±3 ± 2.25 ± 1.75 ±1 0 0.2 0.94 1.07 0.33 0.25 y y 8 4 6 3 4 2 2 1 x −2 2 4 8 10 12 14 −4 −3 x −1 −4 1 2 3 4 −2 −6 −3 −8 −4 115. hx 2x2 12x 2xx 6 2x, x 6 x6 x6 y 12 No asymptotes 9 6 x 2 0 2 4 8 y 4 0 4 8 16 3 6 9 12 − 12 x5 x5 x2 2x 15 x 5x 3 3 −9 The graph has a hole at 2, 12. 116. f x x −9 − 6 − 3 117. y 2x1 y 5 x5 x 1 0 1 2 3 undefined, x 5 y 1 4 1 2 1 2 4 1 , x3 Domain: all real numbers x Horizontal asymptote: y 0 Range: y > 0 x 4 2 1 0 1 2 1 y-intercept: 0, 2 f x 1 1 1 2 1 3 1 4 1 5 Horizontal asymptote: y 0 y 6 4 2 −8 −6 x −2 −4 −6 −8 2 4 6 3 2 Vertical asymptote: x 3 8 4 )0, 12 ) 1 −2 −1 x −1 1 2 3 4 566 Chapter 6 Trigonometry 118. y 3x1 2 119. y 3x2 x 3 2 y 19 9 7 3 1 0 3 5 4 x 1 y 11 y 2 9 3 0 2 4 1 1 3 1 9 7 6 5 4 3 Domain: all real numbers x Domain: all real numbers x y Range: y > 2 7 y-intercept: 0, 5 5 Horizontal asymptote: y2 3 2 Range: y > 0 6 −5 −4 −3 −2 −1 y-intercept: 0, 1 (0, 5) Horizontal asymptote: y 0 2 1 − 5 − 4 −3 − 2 − 1 x 1 2 3 120. y 3x12 y 7 x 3 2 y 1 3 3 3 0.578 1 0 1 3 1.73 1 6 2 5 4 33 5.20 3 3 2 1 Domain: all real numbers x (0, 3 ) x −5 −4 −3 −2 −1 1 2 3 Range: y > 0 y-intercept: 0, 3 0, 1.73 Horizontal asymptote: y 0 121. y lnx 1 y 3 x 1.1 1.5 2 3 4 y 2.30 0.69 0 0.69 1.10 2 1 (2, 0) x Domain: x 1 > 0 ⇒ x > 1 −1 1 2 3 4 5 6 −2 Range: all real numbers −3 x-intercept: 2, 0 Vertical asymptote: x 1 122. y ln x 4 y 12 x 6 3 1 12 1 2 1 3 6 y 7.17 4.39 0 2.77 2.77 0 4.39 7.17 9 6 (−1, 0) − 12 − 9 − 6 −3 Domain: all real numbers x Range: all real numbers y x-intercepts: 1, 0, 1, 0 Vertical asymptote: x 0 (1, 0) x 3 6 9 12 (0, 1) x 1 2 3 Section 6.4 Graphs of Sine and Cosine Functions 567 124. y log103x 123. y log10 x 2 x 1.5 1 0 1 2 x 3 2 1 13 16 y 0.301 0 0.301 0.477 0.602 y 0.95 0.78 0.48 0 0.30 Domain: x 2 > 0 ⇒ x > 2 Domain: 3x > 0 ⇒ x < 0 y Range: all real numbers Range: all real numbers y 3 x-intercept: 1, 0 2 y-intercept: 0, 0.301 (0, 0.301) (−1, 0) −3 −1 y x-intercept: 13, 0 2 Vertical asymptote: x 0 1 x 1 2 3 −3 −1 −2 x −1 1 −1 −2 −3 Section 6.4 (− 13, 0( −2 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 ■ Period: ■ Shift: Solve bx c 0 and bx c 2. ■ Key increments: 2 b 1 (period) 4 Vocabulary Check 1. cycle 2. amplitude 2 b 5. vertical shift 4. phase shift 3. 2. y 2 cos 3x 1. y 3 sin 2x Period: 2 2 Amplitude: 3 3 Period: 3. y 2 2 b 3 5 x cos 2 2 Period: Amplitude: a 2 2 4 1 2 Amplitude: 4. y 3 sin Period: x 3 2 2 1 6 b 3 Amplitude: a 3 3 5. y 1 x sin 2 3 Period: 6. y 2 6 3 Amplitude: 1 1 2 2 5 5 2 2 3 x cos 2 2 Period: 2 2 4 b 2 Amplitude: a 3 2 568 Chapter 6 Trigonometry 8. y cos 7. y 2 sin x Period: 2 2 1 Period: Amplitude: 2 2 10. y 2 2 b 8 4 13. y 1 3 14. y 12. y 1 1 2 2 Amplitude: a The graph of g is a horizontal shift to the right units of the graph of f a phase shift. 2 3 18. f x sin 3x, gx sin3x 17. f x cos 2x gx cos 2x g is a reflection of f about the y-axis. The graph of g is a reflection in the x-axis of the graph of f. 19. f x cos x 20. f x sin x, gx sin 3x The period of g is one-third the period of f. gx cos 2x 21. f x sin 2x f x 3 sin 2x The period of f is twice that of g. The graph of g is a vertical shift three units upward of the graph of f. 22. f x cos 4x, gx 2 cos 4x g is a vertical shift of f two units downward. 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 5 2 gx sinx 2 2 20 b 10 Amplitude: a 16. f x cos x, gx cosx g is a horizontal shift of f units to the left. 2 2 8 b 14 15. f x sin x 1 1 4 4 x 5 cos 2 4 Period: 2 x cos 3 10 Period: 2 10 5 Amplitude: 3 3 2 3 23 Amplitude: 2 1 2 Amplitude: Period: 2x 1 cos 2 3 Period: 1 sin 2x 4 Period: 2 2 3 b 23 11. y Amplitude: a 9. y 3 sin 10x Amplitude: a 1 1 1 sin 8x 3 Period: 2x 3 Minimum , 2 2 Intercept Maximum , 0 3 ,0 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. 3π 2 x Section 6.4 Graphs of Sine and Cosine Functions 28. f x sin x Period: y 2 2 2 b 1 2 g f Amplitude: 1 6π x Symmetry: origin Key points: Intercept 0, 0 Since gx sin Maximum Intercept Minimum Intercept 2 , 1 , 0 32, 1 2, 0 −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 π Intercept Minimum Intercept Maximum 2 , 0 , 1 32, 0 2, 1 2π x f −1 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. 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 ,0 4 , 2 2 3 ,0 4 , 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. −2 x 569 570 Chapter 6 Trigonometry 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 x 3π 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. 33. f x 2 cos x Period: y 2 2 2 b 1 3 f Amplitude: 2 π 2π x Symmetry: y-axis g Key points: Maximum 0, 2 Intercept Minimum Intercept Maximum 2 , 0 , 2 32, 0 2, 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 Key points: Minimum 0, 1 2π x f Intercept Maximum Intercept Minimum 2 , 0 , 1 32, 0 2, 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. Section 6.4 35. y 3 sin x Period: 2 4 Amplitude: 3 2 0, 0, , 3 , , 0, 2 −π 2 0, 0, 2 3 1 3 π 2 −3 Key points: π 2π −π π 2π x −1 1 2 , 4, , 0, −2 2 −3 0, 4, − 43 x 2 1 y 4 Key points: x −1 1 1 0, , , 0 , , , 3 2 3 39. y cos − 2π Amplitude: 4 1 1 1 4 Period: 2 1 3 1 , 0 , 2, 2 3 2 38. y 4 cos x y 4 3 y Key points: 3 Period: 2 x 3π 2 571 2 , 4, 2, 0 1 cos x 3 Amplitude: π 2 −4 3 , 3 , 2, 0 2 37. y Amplitude: 1 3 − π 2 1 sin x 4 Period: 2 3 Key points: 36. y y Graphs of Sine and Cosine Functions − 2π y 2 2 Period: Amplitude: 1 Amplitude: 1 2π −1 3, 0, 4, 1 −2 −4 3 , 0 , 2, 4 2 Period: 4 − 2π 4π x 2π −2 40. y sin 4x y 0, 1, , 0, 2, 1, π 2 , 0, , 4, 2 Key points: −π 1 x x π 4 Key points: 0, 0, 8 , 1, 4 , 0, 3 −2 8 , 1, 2 , 0 41. y cos 2x Period: 2 1 2 42. y sin y 2 Period: 1 Amplitude: 1 x 1 Key points: 0, 1, 14, 0, 12, 1, 34, 0 2 x 4 y 2 2 8 4 Amplitude: 1 1 −6 −2 Key points: −2 0, 0, 2, 1, 4, 0, 6, 1, 8, 0 −2 x 2 6 572 Chapter 6 43. y sin Period: Trigonometry 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 −π −2 9 x 4 3 5 x π −3 7 9 4 , 0, 4 , 1, 4 , 0, 4 , 1, 4 , 0 Key points: 47. y 3 cosx 46. y sinx Period: 2 Period: 2 Amplitude: 1 Amplitude: 3 Shift: Set x 0 and x Key points: , 0, x 2 Shift: Set x 0 x x 3 32, 1, 2, 0, 52, 1, 3, 0 x Key points: , 3, , 0 , 0, 3, , 0 , , 3 2 2 y y 6 2 4 2 −π 2 −1 −2 3π 2 x x 2 and −π π −4 −6 x Section 6.4 48. y 4 cos x 4 Graphs of Sine and Cosine Functions y 6 Period: 2 Amplitude: 4 2 0 4 Shift: Set x and x x 4 2 4 7 x 4 −π −2 π 49. y 2 sin −4 −6 2x 3 50. y 3 5 cos Period: 3 Period: Amplitude: 1 Key points: 0, 2, 3 3 9 , 1 , , 2 , , 3 , 3, 2 4 2 4 Amplitude: 5 Key points: 0, 2, 6, 3, 12, 8, 18, 3, 24, 2 y 16 12 8 4 5 4 x Period: 1 −1 2 4 3 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, 2 , 3, , 5, 32, 3, 2, 1 Key points: y 1 1 1 1 ,2 , , 1.9 , ,2 , , 2.1 0, 2.1, 120 60 40 30 1 −π π y 2.2 −4 −5 −6 −7 1.8 − 0.1 12 −8 − 12 − 16 − 20 − 24 1 51. y 2 t − 12 2 –1 t 12 2 24 12 y –2 x 2π 4 , 4, 4 , 0, 34, 4, 54, 0, 74, 4 Key points: –3 573 x 0 0.1 0.2 2π x 574 Chapter 6 Trigonometry 53. y 3 cosx 3 y Period: 2 4 2 Amplitude: 3 Shift: Set x 0 π x 2 and x x Key points: , 0, , 3 , 0, 6, , 3 , , 0 2 2 54. y 4 cos x x 2π −8 4 4 y 10 Period: 2 6 Amplitude: 4 4 Shift: Set x 0 4 x Key points: 55. y and x 2 4 4 x 2 − 2π − π 7 4 y 4 Period: 4 3 2 2 Amplitude: 3 1 x 2 2 −3 −4 5 2 7 9 2 2 , 3 , 2 , 0, 2 , 3 , 2 , 0, 2 , 3 56. y 3 cos6x y 2 6 3 3 2 Amplitude: 3 Shift: Set 6x 0 x Key points: x 4π −2 9 x 2 3 π −1 x x 0 and 2 2 4 2 4 Period: x 3π 4 , 8, 4 , 4, 34, 0, 54, 4, 74, 8 Key points: 2π −4 2 x 2 1 cos ; a , b ,c 3 2 4 3 2 4 Shift: π and π 6x 2 6 x x 6 6 , 3, 12 , 0, 0, 3, 12 , 0, 6 , 3 57. y 2 sin4x 58. y 4 sin 4 −6 6 −4 3x 3 2 8 −12 12 −8 Section 6.4 59. y cos 2x 1 2 Graphs of Sine and Cosine Functions 60. y 3 cos 3 x 2 2 2 2 −6 −3 6 3 −6 −1 61. y 0.1 sin x 10 62. y 0.12 − 20 1 sin 120 t 100 0.02 − 0.03 20 Amplitude: 64. f x a cos x d 1 3 1 2 ⇒ a 2 2 Amplitude: Vertical shift one unit upward of gx 2 cos x ⇒ d 1. Thus, f x 2 cos x 1. 1 3 2 2 1 2 cos 0 d d 1 2 1 a 2, d 1 65. f x a cos x d Amplitude: 66. f x a cos x d 1 8 0 4 2 Amplitude: 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. 2 4 1 2 Reflected in the x-axis: a 1 4 1 cos 0 d d 3 a 1, d 3 68. y a sinbx c 67. y a sinbx c Amplitude: a 3 Since the graph is reflected in the x-axis, we have a 3. 2 Period: ⇒ b2 b Amplitude: 2 ⇒ a 2 Period: 4 2 1 4 ⇒ b b 2 Phase shift: c 0 Phase shift: c 0 Thus, y 3 sin 2x. 1 a 2, b , c 0 2 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 4 c 2 4 ⇒ b b 2 4 0.03 − 0.02 − 0.12 63. f x a cos x d 575 Phase shift: c 1 ⇒ c b 2 a 2, b ,c 2 2 576 Chapter 6 Trigonometry 72. y1 cos x 2 71. y1 sin x −2 1 y2 2 y2 1 2 In the interval 2, 2, 2 y1 y2 when x , −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 (c) Amplitude: 0.85; Period: 6 3 Key points: 0, 0, , 0.85 , 3, 0, 2 74. v 1.75 sin 2 − 0.25 4 8 10 − 1.00 9 , 0.85 , 6, 0 2 t 2 (a) Period 2 4 seconds 2 (b) 1 cycle 4 seconds 60 seconds 15 cycles per minute 1 minute (c) v 3 2 1 t 1 3 5 7 −2 −3 76. P 100 20 cos 75. y 0.001 sin 880t (a) Period: (b) f 1 2 seconds 880 440 1 440 cycles per second p 1 1 77. (a) a high low 83.5 29.6 26.95 2 2 (a) Period: (b) (b) 5 t 3 6 2 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 1 1 d high low 83.5 29.6 56.55 2 2 Ct 56.55 26.95 cos —CONTINUED— 12 0 6t 3.67 The model is a good fit. Section 6.4 77. —CONTINUED— (c) Graphs of Sine and Cosine Functions 577 2 12 months. 6 This is as we expected since one full period is one year. (e) The period for both models is 100 0 (f) Chicago has the greater variability in temperature throughout the year. The amplitude, a, determines this variability since it is 12high temp low temp. 12 0 The model is a good fit. (d) Tallahassee average maximum: 77.90 Chicago average maximum: 56.55 The constant term, d, gives the average maximum temperature. 78. (a) and (c) (b) Vertical shift: y Amplitude: Percent of moon’s face illuminated 1.0 1 1 ⇒ a 2 2 0.8 0.6 Period: 0.4 0.2 20 30 88768 7.4 average length of 5 interval in data 2 47.4 29.6 b x 10 40 Day of the year b Reasonably good fit (d) Period is 29.6 days. 2 0.21 29.6 Horizontal shift: 0.213 7.4 C 0 (e) March 12 ⇒ x 71. y 0.44 44% C 0.92 The Naval observatory says that 50% of the moon’s face will be illuminated on March 12, 2007. 79. C 30.3 21.6 sin 2 t 365 10.9 2 365 Period 2 365 (a) 1 1 ⇒ d 2 2 y 80. (a) Period 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 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. (b) 1 1 sin0.21x 0.92 2 2 The radius of the wheel is 50 feet. (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 cos 2x has an amplitude that is half that 2 of y cos x. For y a cos bx, the amplitude is a . 82. False. y 578 Chapter 6 Trigonometry 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 x 3π 2 π 2 . 2 −2 86. f x sin x, gx cos x x 2 0 1 0 1 0 sin x cos x 87. (a) 2 2 y 2 Conjecture: sin x cos x f=g 1 3 2 2 0 1 0 0 1 0 − 3π 2 3π 2 π 2 2 x −2 x3 x5 x7 3! 5! 7! x2 x4 x6 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 (c) sin 1 0.5000 6 3! 5! sin 0.5 (by calculator) 6 (e) cos 1 1 1 1 0.5417 2! 4! cos 1 0.5403 (by calculator) (b) sin 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 cos 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. Section 6.5 89. log10 x 2 log10x 212 91. ln 1 log10x 2 2 t3 ln t 3 lnt 1 3 ln t lnt 1 t1 Graphs of Other Trigonometric Functions 90. log2x2x 3 log2 x2 log2x 3 2 log2 x log2x 3 z 92. ln 2 z 1 z 1 ln 2 ln z lnz2 1 1 2 z 1 2 93. 1 1 log10 x log10 y log10xy 2 2 log2 x2(xy) log10 xy ln 1 1 ln z lnz2 1 2 2 94. 2 log2 x log2xy log2 x2 log2xy log2 x3y 95. ln 3x 4 ln y ln 3x ln y4 96. 3xy 1 1 ln 2x 2 ln x 3 ln x ln 2x ln x2 ln x3 2 2 4 1 2x ln 2 ln x3 2 x 2xx ln x 2x lnx x ln 3 2 3 2 lnx22x 97. Answers will vary. Section 6.5 ■ ■ 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 4. 5. x n 6. , 1 1, 7. 2 579 580 Chapter 6 Trigonometry 1. y sec 2x 2. y tan 2 2 Period: Period: Matches graph (e). x 2 3. y 2 b 1 2 1 cot x 2 Period: 1 Matches graph (a). Asymptotes: x , x Matches graph (c). 1 x sec 2 2 2 2 Period: 4 b 2 4. y csc x 5. y Period: 2 Matches graph (d). 6. y 2 sec Period: x 2 2 2 4 b 2 Asymptotes: x 1, x 1 Asymptotes: x 1, x 1 Matches graph (f). Reflected in x-axis Matches graph (b). 7. y 1 tan x 3 Period: 3 Two consecutive asymptotes: 1 x 4 0 4 y 1 3 0 1 3 −π π 9. y tan 3x 3x 12 y 1 0 12 0 1 1 −π x 0 4 y 1 4 0 1 4 Period: π 3 x x π −3 y 10. y 3 tan x 1 ⇒ x 2 6 2 4 2 ⇒ x 2 6 3 x ,x 2 2 3 −π 3 y Two consecutive asymptotes: 4 3 x x y Two consecutive asymptotes: 3x 1 tan x 4 Period: 2 x and x 2 2 Period: 8. y y 1 Two consecutive asymptotes: x 2 −4 1 1 x ,x 2 2 x y 3 1 4 −8 0 1 4 0 3 Section 6.5 1 11. y sec x 2 y 12. y 3 Period: 2 1 −π x π 3 y 1 0 1 2 x 1 y 1 2 4 Two consecutive asymptotes: 1 2 x −1 x 0, x 1 1 2 1 6 1 2 5 6 y 2 1 2 2 2 −2 −1 x 1 −1 2 3 y 1 17. y csc Period: 1 3 y 8 6 4 2 −π 4 4 24 8 5 24 y 6 3 6 2 4 2 6 4 Two consecutive asymptotes: 0 1 3 x 0 1 y 2 x 2 18. y csc y 2 4 12 Two consecutive asymptotes: 6 Period: 4 2 π x 12 2 −π 4 0 12 0 2 π 4 x 3 π 2 x y 2 6 13 Two consecutive asymptotes: 6 4 2 π 2π x 0, x 3 x 0, x 2 x π 4 −2 y x ,x 8 8 1 1 x ,x 2 2 x 1 2 x Period: −3 1 4 16. y 2 sec 4x 2 y Two consecutive asymptotes: 3 Two consecutive asymptotes: −4 15. y sec x 1 0 2 4 2 x 0, x −3 x Period: Period: 3 −2 3 x 2π 14. y 3 csc 4x y 2 2 Period: 3 Two consecutive asymptotes: 3 13. y csc x y x ,x 2 2 x ,x 2 2 x 1 sec x 4 581 Period: 2 2 Two consecutive asymptotes: Graphs of Other Trigonometric Functions x 3 5 3 x 2 3 2 5 2 y 2 1 2 y 2 1 2 x 582 Chapter 6 19. y cot Period: Trigonometry x 2 y 20. y 3 cot 3 2 12 1 − 2π x 2π x y 2 1 3 2 0 1 1 sec 2x 2 y 1 23. y tan Period: 6 2 x −2 2 3 0 6 1 2 1 π x 4 2 x −4 4 x 1 0 1 y 1 0 1 25. y csc x Period: 2 4 Two consecutive asymptotes: 2 1 − 3π 2 6 2 5 6 y 2 1 2 0 3 y 3 2 Two consecutive asymptotes: x y 1 2 −π 4 0 4 0 π 2 x x π 1 2 y Period: 4 Two consecutive asymptotes: 2 3 1 x π x 4 0 4 y 1 0 1 Period: 3 x 3 26. y csc2x y x 0, x y x ,x 2 2 x ⇒ x 2 4 2 x ⇒ x2 4 2 3 2 24. y tanx 6 Two consecutive asymptotes: 1 x ,x 2 2 y 4 4 1 2 Period: −π x 4 x 1 22. y tan x 2 y 2 Period: 2 4 x 0, x 2 x ⇒ x 2 2 x 6 Two consecutive asymptotes: x 0 ⇒ x0 2 21. y y 2 Period: 2 2 Two consecutive asymptotes: x 2 y 2 2 Two consecutive asymptotes: x 0, x 2 −1 −2 x 12 4 5 12 y 2 1 2 π 2 π 3π 2 2π x Section 6.5 27. y 2 secx 4 Two consecutive asymptotes: 2 y 29. y Period: 3 1 x ,x 2 2 x 28. y sec x 1 y Period: 2 3 −π 4 2 1 csc x 4 4 π −1 2π 3π x 2 2π y 1 2 31. y tan 7 12 1 4 1 2 x x 3 1 1 3 0 1 3 y 1 0 1 2 x y 2 4 0 4 0 2 − 2 2 −4 4 36. y 2 3 −3 1 cot x 4 2 − 3 2 −2 3 2 1 4 tan x 3 2 cos 4x 4 3 4 35. y tan x x −2 −3 1 cos x 3π 2 33. y 2 sec 4x − 3 4 −5 34. y sec x ⇒ y 4 − 3π 2 3 5 5 3 y x ,x 2 2 32. y tan 2x −5 2 Two consecutive asymptotes: x 3 x ,x 4 4 4 1 Period: Two consecutive asymptotes: 12 2 1 3 x 30. y 2 cot x y Period: 2 x 2 2 1 1 x ,x 2 2 4 583 y Two consecutive asymptotes: 3 0 Graphs of Other Trigonometric Functions 2 3 −3 − 3 2 3 2 −3 37. y csc4x y 1 sin4x 38. y 2 sec(2x ) ⇒ 3 − 2 2 −3 y 2 cos2x 4 − −4 584 Chapter 6 39. y 0.1 tan Trigonometry 4x 4 40. y 0.6 −6 x 1 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 x 3 2 5 4 , , , 3 3 3 3 y 2 3 y 2 3 2 1 1 π 2π x π 2π π 2 x 3π 2 x −3 x 46. sec x 2 45. sec x 2 44. cot x 1 7 3 5 , , , 4 4 4 4 x± 2 4 , ± 3 3 x 5 5 , , , 3 3 3 3 y y y 3 2 − 3π 2 1 − π2 π 2 x 3π 2 −2π −π 1 π 2π x −2π −3 47. csc x 2 x 48. csc x 7 5 3 , , , 4 4 4 4 x 23 3 2 4 5 , , , 3 3 3 3 y y 3 3 2 2 1 − 3π 2 −π 2 −1 π 2 1 x 3π 2 − 3π 2 −π 2 π 2 x 3π 2 −π π 2π x Section 6.5 f x sec x 49. 1 cos x 50. f x tan x y 4 f x secx Graphs of Other Trigonometric Functions y tanx tan x 3 3 2 1 cosx −π π 2π Thus, the function is odd and the graph of y tan x is symmetric about the origin. x − 3π 2 − π2 π 2 1 cos x −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 x π (b) f > g on the interval, (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). 5 < x < 6 6 (c) As x → , f x 2 sin x → 0 and 1 gx csc x → ± since g x is the reciprocal 2 of f x. 54. y1 sin x sec x, y2 tan x 53. y1 sin x csc x and y2 1 2 −3 4 −2 3 −2 sin x csc x sin x sin x 1, sin x 0 1 sin x sec x sin x cos x 1 and y2 cot x sin x tan x cot x 2 −4 The expressions are equivalent except when sin x 0 and y1 is undefined. 55. y1 585 cos x sin x The expressions are equivalent. sin x 1 tan x cos x cos x The expressions are equivalent. 4 −2 2 −4 x 3π 2 586 Chapter 6 Trigonometry 56. y1 sec 2 x 1, y2 tan2 x 1 57. f x x cos x 3 As x → 0, f x → 0 and f x > 0. tan2 x sec x tan2 x sec x 1 2 2 − 3 2 3 2 The expressions are equivalent. Matches graph (d). −1 58. f x x sin x 59. gx x sin x Matches graph (a) as x → 0, f x → 0. As x → 0, gx → 0 and gx is odd. Matches graph (b). 61. f x sin x cos x 60. gx x cos x Matches graph (c) as x → 0, gx → 0. , gx 0 2 f x gx, The graph is the line y 0. y 3 2 1 −3 −2 −1 x 1 −1 2 3 −2 −3 62. f x sin x cos x 2 1 63. f x sin2 x, gx 1 cos 2x 2 f x gx gx 2 sin x y y 4 3 2 2 −π x π −π −4 x π –1 It appears that f x gx. That is, sin x cos x 64. f x cos2 2 sin x. 2 x 2 65. gx ex 2 sin x 2 y ex 2 ≤ gx ≤ ex 2 2 3 1 gx 1 cos x 2 −6 x −3 3 −1 2 The damping factor is 2 y ex 2. 2 It appears that f x gx. That is, x 1 1 cos x. cos2 2 2 1 6 As x → , gx →0. −8 8 −1 Section 6.5 Graphs of Other Trigonometric Functions 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 −1 −3 As x → , 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 sin x x 2 6 8 0 −2 −6 6 0 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 2 − d −1 As x → 0, f x oscillates between 1 and 1. As x → 0, f (x) → 0. 75. tan x −2 −1 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 2 −2 −6 π 4 π 2 3π 4 π x 60 Distance Ground distance 10 6 40 20 −π 2 − 10 − 14 Angle of elevation 1 x −π 4 0 π 4 Angle of camera π 2 x 587 588 Chapter 6 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 6.5 Graphs of Other Trigonometric Functions 83. As x → 82. True. y sec x 1 cos x As x → 589 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 −6 590 Chapter 6 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 x2 e3.2 1 14 e68 1.702 1029 2 x ± e3.2 1 ± 4.851 96. lnx 4 5 97. log8 x log8x 1 1 3 1 ln(x 4) 5 2 log8xx 1 1 3 xx 1 8 ln(x 4) 10 x4 e10 x e10 13 x x2 2 4 22,022.466 x2 x20 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. 98. log6 x log6(x2 1) log6(64x) log6x(x2 1 log6(64x) x(x2 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 6.6 Section 6.6 Inverse Trigonometric Functions 591 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 2 592 Chapter 6 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 3 2 < y < ⇒ y0 2 2 2 ⇒ sin y 1.5 f 3 3 ⇒ tan y 33 for < y < ⇒ y 2 2 6 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 34. arctan 0.75 0.85 1 95 1.50 7 25. arcsin 0.31 sin1 0.31 0.32 27. arccos0.41 cos10.41 1.99 26. arccos 0.26 1.31 3 2 g Graph y3 x. −1 28. arcsin0.125 0.13 2 − 2 Graph y2 tan1 x. g 23. arctan3 tan13 1.25 for ≤ y ≤ ⇒ y 2 2 4 Graph y1 tan x. −1.5 4 sin 2 16. y arccos 1 ⇒ cos y 1 for 0 ≤ y ≤ ⇒ y 0 gx arcsin x 31. arcsin 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 2 10. y arctan3 ⇒ tan y 3 for < y < ⇒ y 2 2 3 11. y arccos 2 8. y arcsin 30. arctan 2.8 1.23 29. arctan 0.92 tan1 0.92 0.74 3 1.91 32. arccos 1 33. arctan 2 tan 7 1 3.5 1.29 35. This is the graph of y arctan x. The coordinates are 3, 3 , 3 , 6 , and 1, 4 . 3 Section 6.6 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 593 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 594 Chapter 6 Trigonometry 5 52. Let u arccos cos u 5 5 sin arccos 5 ,0 < u < 5 5 5 . 13 53. Let y arcsin , . 2 sin u sin y 2 5 25 y 5 , 0 < y < 13 2 and cos y 5 12 . 13 13 5 y x 12 5 2 u 1 54. Let u arctan 5 . 5 , 12 55. Let y arctan 5 tan u , < u < 0. 12 2 5 csc arctan 12 13 csc u 5 12 u 13 and sec y 34 5 5 x y −3 34 . 3 . 3 57. Let y arccos 3 sin u , < u < 0. 4 2 3 tan arcsin 4 3 tan y , < y < 0 5 2 y −5 4 , 56. Let u arcsin 3 37 3 tan u 7 7 y 2 2 < y < cos y , 3 2 and sin y 5 3 5 3 . y −2 7 u 4 −3 x Section 6.6 5 58. Let u arctan , 8 595 59. Let y arctan x. 5 tan u , 0 < u < . 8 2 cot arctan Inverse Trigonometric Functions 89 tan y x 5 u x 1 and cot y . x 8 5 8 cot u 8 5 x2 + 1 x 1 y 1 61. Let y arcsin2x. 60. Let u arctan x, x tan u x . 1 sin y 2x sinarctan x sin u 1 2x and cos y 1 x x2 2x 1 4x2. y 1 1 − 4x 2 x2 + 1 x u 1 62. Let u arctan 3x, tan u 3x 63. Let y arccos x. 3x . 1 cos y x 9x 2 + 1 secarctan 3x sec u 9x2 1 3x x 1 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 cos y 1 x x 3 and tan y 2x x2 3 . 3 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 cot arctan 1 1 cot u x x tan y x . x2 + 2 x x 2 and csc y u x 2 x2 2 x . y 2 596 Chapter 6 68. Let u arcsin sin u Trigonometry xh , r xh . r cos arcsin r xh cos u r r 2 x h r x−h 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 tan y x −2 arcsin y These are equal because: x Let u arccos . 2 2 x tan u 2 f x tan arccos 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 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 6.6 74. If arccos x2 u, 2 then cos u arccos Inverse Trigonometric Functions 597 75. y 2 arccos x Domain: 1 ≤ x ≤ 1 x2 . 2 Range: 0 ≤ y ≤ 2 x2 arctan 2 4x x 2 This is the graph of f x arccos x with a factor of 2. x2 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 2 598 Chapter 6 82. f x arccos Trigonometry x 4 83. f x 2 arccos2x y 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 graphs are the same and 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 6.6 92. (a) tan s 750 arctan 93. arctan (a) s 750 Inverse Trigonometric Functions 599 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 arctan 95. 11 0.5743 32.9 17 1 (b) r 40 20 2 tan 41 ft (a) tan h h r 20 h 20 tan 20 20 ft θ 20 41 tan arctan 11 12.94 feet 17 (b) tan 26 41 26.0 20 h 50 tanh 50 tan 26 24.39 feet 96. (a) tan 6 x arctan 97. (a) tan 6 x tan arctan (b) x 7 miles 6 arctan 0.71 40.6 7 (b) x 5: arctan 5 14.0 20 12 31.0 20 6 1.41 80.5 1 99. False. 98. False. 5 is not in the range of arcsinx. 6 arcsin x 20 x 12: arctan x 1 mile arctan x 20 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 arcsin 1 . arccos 1 Since arctan 1 2 arcsin 1 undefined. , but 4 arccos 1 0 4 600 Chapter 6 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 ≤ . The 2 2 domain of y arcsec x is , 1 1, and the y Range: 0 < x < π 2 2 , . 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 6.6 Inverse Trigonometric Functions 601 107. 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. 10(14)2 10 10 0.051 142 196 602 Chapter 6 Trigonometry 111. 1.150 117.391 113. sin 112. 162 3 opp 4 hyp 114. tan 2 adj2 32 42 adj2 9 16 adj2 7 4 3 θ adj 7 cos 4 7 37 7 3 4 47 7 7 csc 4 3 cos 5 adj 6 hyp opp2 25 36 opp 11 2 opp 11 tan cot sec csc sin 2 5 cot 1 2 6 6 11 1 8 θ 5 3 22 cos 1 3 sin 22 3 θ 1 tan 2 2 5 5 511 11 11 6 5 6 θ opp 32 12 11 11 2 116. sec 3 opp2 52 62 sin 1 5 1 csc 5 2 sec 115. cos sec 5 7 cot hyp 12 22 5 7 3 tan 1 2.718 108 162 611 11 sec 3 csc cot 3 22 1 22 32 4 2 4 Section 6.6 117. Let x the number of people presently in the group. Each person’s share is now If two more join the group, each person’s share would then be 250,000. x2 Inverse Trigonometric Functions 250,000 . x 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 d r rate time distance ⇒ t 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. 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 22 632,000 632 742 458,504.31 0.035 4 0.035 12 0.035 365 (b) A 15,000 1 (Time to go upstream) (Time to go downstream) 4 2 er 4 410 119. (a) A 15,000 1 (c) A 15,000 1 $21,253.63 1210 36510 $21,275.17 $21,285.66 (d) A 15,000e0.03510 $21,286.01 603 604 Chapter 6 Section 6.7 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 sin B b ⇒ b c sin B c b = 10 b b 24 ⇒ a 8.26 a tan B tan 71 b b 24 sin B ⇒ 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 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 C C c = 15 15 sin 54 12.14 3. Given: B 71, b 24 a 71° 54° a 90 54 36 c 20° C tan B A 90 B B a B 25 45.58 35 a 25 a ⇒ B arccos arccos 44.42 c c 35 Section 6.7 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 605 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 A C b = 1.32 9. Given: A 1215, c 430.5 B 90 1215 7745 sin 12 15 a 430.5 a 430.5 sin 12 15 91.34 cos 12 15 10. Given: B 6512, a 14.2 A 90 B 90 6512 2448 cos B a 14.2 a ⇒ c 33.85 c cos B cos 6512 tan B b ⇒ b a tan B 14.2 tan 6512 30.73 a b 430.5 B b 430.5 cos 12 15 420.70 a = 14.2 B 65°12′ c C c = 430.5 A b a 12°15′ b C 11. tan A h 1 ⇒ h b tan 12b 2 1 h 4 tan 52 2.56 inches 2 θ θ 1 h 10 tan 18 1.62 meters 2 θ θ 1 b 2 1 b 2 1 b 2 b b 13. tan h 1 ⇒ h b tan 12b 2 h h 1 b 2 12. tan 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 606 Chapter 6 15. tan 25 Trigonometry 50 x 16. tan 20 50 50 x tan 25 25° x 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 6.7 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 artan 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 θ 28. 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 607 608 Chapter 6 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 6.7 38. cot 55 d ⇒ d 7 kilometers 10 39. tan 57 D cot 28 ⇒ D 18.8 kilometers 10 tan 16 Distance between towns: tan 16 D d 18.8 7 11.8 kilometers cot 16 28° 55° 10 km T1 55° d 28° Applications and Models a ⇒ x a cot 57 x a 16° x 55 6 550 60 x a a cot 57 55 6 a cot 57 55 6 a 55 ⇒ a 3.23 miles 6 17,054 ft h x h 2.5° 17 h x tan 2.5 x 9° x − 17 41. L1: 3x 2y 5 ⇒ y 3 5 3 ⇒ m1 x 2 2 2 L2: 3x y 1 ⇒ y x 1 ⇒ m2 1 Not drawn to scale tan h tan 9 x 17 x a 57° D 40. tan 2.5 P2 P1 H a cot 16 a cot 57 T2 609 1 32 1 132 52 12 5 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 a 2a arctan 1 m2 m1 5 2 arctan 1 1 m2m1 1 52 arctan 44. tan 43. The diagonal of the base has a length of a2 a2 2a. Now, we have 1 2 a 1 2 θ 35.3. 2a 97 52.1 a2 2 a 45. sin 36 d ⇒ d 14.69 25 Length of side: 2d 29.4 inches arctan 2 54.7º a 2 θ d a 36° 25 610 Chapter 6 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. 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 54. Displacement at t 0 is 2 ⇒ d a cos t Amplitude: a 2 Period: 4t 3 t 3 cos 3 . t 3 2 10 ⇒ 5 d 2 cos t 5 9 Section 6.7 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 16 611 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: 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 2264 528 1 1 64 64 1 792 792 612 Chapter 6 Trigonometry 60. At t 0, buoy is at its high point ⇒ d a cos t. 61. y Distance from high to low 2 a 3.5 1 cos 16t, t > 0 4 (a) 7 a 4 y 1 Returns to high point every 10 seconds: 2 10 ⇒ Period: 5 d π 8 3π 8 π 2 t −1 t 7 cos 4 5 (b) Period: (c) 62. (a) π 4 (b) 2 16 8 1 ⇒ t cos 16t 0 when 16t 4 2 32 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 sin 2 16 16 cos 4 sin 641 cos sin (d) 100 0 90 0 8 8 16 cos 60º 8 sin 60º 83.1 8 8 16 cos 70º 8 sin 70º 80.7 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. 613 S Average sales (in millions of dollars) Section 6.7 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 ft 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 70. Linear equation m 12 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 23 1 2 y 1 4 3 1 3 4 4 3 2 1 −2 −1 −1 1 3 14, 32 and 21, 13. x 1 2 3 4 5 2 1 4 3 −3 2 4 1 y x 3 3 4 4 1 y x 3 3 −2 −1 x 2 −1 −2 −3 3 614 Chapter 6 Trigonometry Review Exercises for Chapter 6 1. 40 2. 269 3. (a) 70 4. (a) 280 y y 280° 70° x x (b) The angle lies in Quadrant I. (b) The angle lies in Quadrant IV. (c) Coterminal angles: 70 360 430 70 360 290 (c) Coterminal angles: 280° 360° 640° 280° 360° 80° 6. (a) 405 5. (a) 110 y y −405° x x − 110° (b) The angle lies in Quadrant III. (b) The angle lies in Quadrant IV. (c) Coterminal angles: 110 360 250 110 360 470 (c) Coterminal angles: 405° 720° 315° 405° 360° 45° 7. (a) 11 4 8. (a) 2 9 y y 11π 4 2π 9 x x (b) The angle lies in Quadrant II. (b) The angle lies in Quadrant I. 3 11 (c) Coterminal angles: 2 4 4 (c) Coterminal angles: 3 5 2 4 4 20 2 2 9 9 2 16 2 9 9 Review Exercises for Chapter 6 9. (a) 4 3 10. (a) 23 3 y y − 23π 3 x x − 4π 3 (b) The angle lies in Quadrant I. (b) The angle lies in Quadrant II. (c) Coterminal angles: 11. 480 480 2 4 2 3 3 4 10 2 3 3 rad 8 rad 8.378 rad 180 3 13. 16.5 16.5 rad 0.288 rad 180 15. 3345 33.75 33.75 16. 9825 98.416 18. 19677 197.283 20. 7 7 5 5 22. rad 1.718 rad 180 rad 3.443 rad 180 180 24. 8.3 8.3 180 rad 330 180 475.555 rad 180 rad 326.586 12. 120 120 s 11.52 m 60 180 113 rad 5 23 6 3 3 rad 2.094 rad 180 14. 127.5 127.5 rad 2.225 rad 180 17. 8415 84.25 84.25 19. 5 rad 5 rad 7 7 21. 180 27. 138 180 rad 108 23. 3.5 rad 3.5 rad 25. 1.75 rad rad 1.470 rad 180 rad 128.571 3 3 rad rad 5 5 180 200.535 rad 1.75 180 rad 100.268 1 rad 138 23 radians 180 30 s r 20 28. s r 11 23 8 3 3 3 rad rad 0.589 rad 180 16 rad 252 11 11 6 6 26. 5.7 5.7 (c) Coterminal angles: 2330 48.17 inches 615 616 Chapter 6 Trigonometry 29. (a) Angular speed 30. Linear speed 3313 2 radians 1 minute 66 23 Linear speed rad 272 inches5 second 1 2 1 5 r 65 2 2 2 6 A 55.31 mm2 34. opp 6, adj 6, hyp 62 666 23 inches 1 minute 400 inches per minute radians per minute 212.1 inchessecond 32. A (b) 31. 120 120 2 radians 180 3 1 2 1 339.29 square inches A r 2 182 2 2 3 33. opp 4, adj 5, hyp 42 52 41 sin opp 441 4 hyp 41 41 csc hyp 41 opp 4 cos adj 5 541 hyp 41 41 sec 41 hyp adj 5 tan opp 4 adj 5 cot adj 5 opp 4 35. adj 4, hyp 8, opp 82 42 48 43 sin 2 opp 6 hyp 62 2 sin opp 43 3 hyp 8 2 cos 2 adj 6 hyp 62 2 cos adj 4 1 hyp 8 2 tan opp 1 adj tan opp 43 3 adj 4 csc hyp 62 2 opp 6 csc 8 hyp 23 opp 43 3 sec hyp 62 2 adj 6 sec hyp 8 2 adj 4 cot adj 1 opp cot 3 adj 4 opp 43 3 36. opp 5 adj hyp 9 92 52 214 sin opp 5 hyp 9 cos adj 214 hyp 9 tan 5 opp 514 adj 28 214 csc hyp 9 opp 5 sec hyp 9 914 adj 28 2 14 cot adj 214 opp 5 Review Exercises for Chapter 6 37. sin 1 3 38. tan 4 (a) cot 1 3 sin (a) csc (b) 1 tan2 sec2 (b) sin2 cos2 1 13 2 1 1 tan 4 sec2 1 16 17 cos2 1 sec 17 1 9 cos2 1 (c) cos 17 1 1 sec 17 17 cos2 8 9 cos 89 csc2 cos 22 3 csc (d) cot2 1 csc2 14 2 1 17 16 17 4 1 3 32 cos 22 4 1 2 1 sin 3 (d) tan cos 4 22 22 3 (c) sec 39. csc 4 40. csc 5 (a) sin 1 1 csc 4 (a) sin (b) sin2 cos2 1 1 4 2 cos2 (b) cot2 1 csc2 1 cot2 csc2 1 52 1 24 cos2 1 cot 26 1 16 cos2 15 16 cos 1516 cos 1 1 csc 5 (c) tan 6 1 1 cot 26 12 (d) sec90 csc 5 15 4 1 4 415 cos 15 15 1 15 sin 4 1 (d) tan cos 15 15 15 4 (c) sec 41. tan 33 0.6494 44. sec 79.3 1 5.3860 cos 79.3 42. csc 11 1 5.2408 sin 11 43. sin 34.2 0.5621 45. cot 1514 cot 15.2333 1 3.6722 tan 15.2333 617 618 Chapter 6 Trigonometry 46. cos 7811 58 cos 78.1994 47. cos 18 0.9848 48. tan 5 0.5774 6 0.2045 49. sin 110 x 3.5 m 3.5 k 1°10' x 3.5 sin 1101000 tan 52 50. x x Not drawn to scale 71.3 meters 25 x 25 19.5 feet tan 52 25 52° x 51. x 12, y 16, r 144 256 400 20 52. x 3, y 4, r 32 (4)2 5 sin y 4 r 5 csc 5 r y 4 sin 4 y r 5 csc r 5 y 4 cos x 3 r 5 sec r 5 x 3 cos x 3 r 5 sec r 5 x 3 tan y 4 x 3 cot x 3 y 4 tan y 4 x 3 cot x 3 y 4 5 2 53. x , y 3 2 r 54. x 2 3 2 5 2 2 241 6 5 2 y 15241 15 sin 241 r 241 241 6 2 x 4241 3 4 cos 241 r 241 241 6 5 y 2 15 tan x 2 4 3 241 csc r y 6 5 2 2241 241 30 15 241 sec r x 6 2 3 2 x 3 4 cot y 5 15 2 241 4 r 2 10 ,y 3 3 103 32 2 2 226 3 2 26 y 3 sin r 26 226 3 10 526 x 3 cos r 26 226 3 2 y 1 3 tan x 5 10 3 r csc y 226 3 26 2 3 226 26 3 10 5 3 10 3 x 5 cot y 2 3 r sec x Review Exercises for Chapter 6 1 9 55. x 0.5 , y 4.5 2 2 r 1 2 2 9 2 2 56. x 0.3, y 0.4 r (0.3)2 (0.4)2 0.5 82 2 92 y 982 9 r 82 822 82 82 x 12 1 cos r 82 822 82 y 92 9 tan x 12 sin csc 822 82 r y 92 9 sec 822 r 82 x 12 cot x 12 1 y 92 9 57. x, 4x , x > 0 sin y 0.4 4 r 0.5 5 csc r 0.5 5 y 0.4 4 cos x 0.3 3 r 0.5 5 sec 0.5 5 r x 0.3 3 tan y 0.4 4 x 0.3 3 cot x 0.3 3 y 0.4 4 58. x 2x, y 3x, x > 0 x x, y 4x r (2x)2 (3x ) 2 13x r x2 4x2 17 x sin y 3x 313 r 13x 13 cos x 2x 213 r 13x 13 tan y 3x 3 x 2x 2 csc 13x 13 r y 3x 3 17x r 17 sec x x sec 13x 13 r x 2x 2 x x 1 y 4x 4 cot x 2x 2 y 3x 3 sin y 4x 17x r 417 17 17 x x cos 17x r 17 y 4x 4 tan x x 17x 17 r csc y 4x 4 cot 619 6 59. sec , tan < 0 ⇒ is in Quadrant IV. 5 3 60. csc , cos < 0 ⇒ is in Quadrant II. 2 r 6, x 5, y 36 25 11 r 3, y 2, x 32 22 5 sin 11 y r 6 csc r 611 y 11 sin y 2 r 3 cos x 5 r 6 sec r 6 x 5 cos 5 x r 3 tan 11 y x 5 cot x 511 y 11 y 25 2 x 5 5 3 csc 2 tan r 3 35 x 5 5 x 5 cot y 2 sec 620 Chapter 6 Trigonometry 5 61. tan , cos < 0 ⇒ is in Quadrant III. 4 3 62. sin , cos < 0 ⇒ is in Quadrant II. 8 x 4, y 5 y 3, r 8, x 55 r (4)2 (5)2 41 y 5 541 r 41 41 4 x 441 cos r 41 41 5 tan 4 sin sin y 3 r 8 cos 55 x r 8 tan 3 y 355 55 x 55 csc 41 41 r y 5 5 csc 8 r y 3 sec 41 41 r x 4 4 sec r 8 855 x 55 55 cot x 4 4 y 5 5 cot 55 x y 3 63. tan y 12 ⇒ r 13 x 5 64. cos sin > 0 ⇒ is in Quadrant II ⇒ y 12, x 5 x 2 ⇒ y2 21 r 5 sin > 0 ⇒ is in Quadrant II ⇒ y 21 sin y 12 r 13 csc r 13 y 12 sin y 21 r 5 cos x 5 r 13 sec r 13 x 5 tan 21 y x 2 tan y 12 x 5 cot 5 x y 12 csc 5 r 521 y 21 21 sec r 5 5 x 2 2 cot x 2 221 y 21 21 65. 264 66. 635 720 85 y 264 180 84 y 85 264° 635° x x θ′ θ′ 67. 6 5 6 4 2 5 5 y 68. 17 18 3 3 3 6 θ′ 3 y 17π 3 x 4 5 5 − 6π 5 x 3 θ′ Review Exercises for Chapter 6 3 3 2 69. sin 70. sin 2 4 2 71. sin 621 5 1 sin 6 6 2 cos 1 3 2 cos 2 4 2 cos 3 5 cos 6 6 2 tan 3 3 tan 2 2 1 4 2 2 tan 3 5 tan 6 6 3 3 5 5 sin sin 2 3 3 3 2 72. sin 3 7 sin 3 3 2 7 1 cos 3 3 2 7 tan 3 3 3 73. sin cos 5 5 1 cos cos 2 3 3 3 2 cos tan 3 5 1 3 3 2 2 tan 74. 5 3 is coterminal with . 4 4 5 3 3 2 sin sin sin 4 4 4 4 2 2 5 3 3 cos cos cos 4 4 4 4 2 2 2 5 1 4 2 2 sin cos tan 75. sin 495 sin 45 2 76. sin 120 sin180 120 sin 60 2 cos 495 cos 45 2 tan 495 tan 45 1 tan 120 3 2 1 2 21 3 1 2 cos150 cos 30 tan150 tan 30 2 cos 120 cos180 120 cos 60 2 77. sin150 sin 30 3 3 2 3 3 78. 420 is coterminal with 300. sin420 sin 300 sin360 300 sin 60 cos420 cos 300 cos360 300 cos 60 tan420 79. sin 4 0.0698 3 2 3 2 1 2 1 3 2 80. tan 231 1.2349 81. sec 2.8 1 1.0613 cos 2.8 622 Chapter 6 Trigonometry 82. cos 5.5 0.7087 85. t sin 2 1 3 , x, y , 3 2 2 83. sin 17 0.4067 15 86. cos 3 2 y 3 2 sin 2 1 cos x 3 2 tan 87. t sin 25 4.3813 7 2 7 4 2 y 74 22 x 7 tan 1 4 ( 2 y 32 3 3 x 12 3 7 1 , x, y , 6 2 2 88. cos 7 1 y 6 2 sin 3 7 cos x 6 2 tan 84. tan 34 22 34 ( − 2, 2 2 2 2, 2 − 2 2 ( y ( 2 2 x 3 1 tan 4 3 12 1 7 y 6 x 32 3 3 89. y sin x 90. y cos x y Amplitude: 1 Period: 2 y 2 Amplitude: 1 1 Period: 2 x π 2 − 3π 2 −π Amplitude: 3 Period: 2 1 2 π x 2π −1 −2 −2 91. y 3 cos 2x 2 92. y 2 sin x y Amplitude: 2 3 2 Period: 1 x 1 2 2 2 y 3 1 x 1 3 −1 −2 −3 93. f x 5 sin 2x 5 Amplitude: 5 Period: 2 2 5 4x y 94. f (x) 8 cos 6 2 −2 8 Amplitude: 8 4 5 y 6π x Period: 2 1 4 8 4π −4 −6 −6 −8 8π x Review Exercises for Chapter 6 95. y 2 sin x 96. y 4 cos x y Amplitude: 1 4 Amplitude: 1 Period: 2 3 Period: 2 y −3 −2 −1 −π π −1 2π −1 −5 −6 98. g(t) 3 cos(t ) y Period: 2 3 −3 97. gt 52 sint Amplitude: 2 x −2 5 2 x 1 −2 2 Shift the graph of y sin x two units upward. 623 y 4 Amplitude: 3 4 3 Period: 2 3 2 1 −1 1 π t t π −2 −3 −3 −4 −4 99. y a sin bx 100. (a) st 18.09 1.41 sin 2 1 ⇒ b 528 (a) a 2, b 264 6t 4.60 22 y 2 sin528x (b) f 1 1 264 264 cycles per second 0 12 14 2 26 12 6 12 months 1 year, so this is expected. (b) Period: (c) Amplitude: 1.41 The amplitude represents the maximum change in time from the average time d 18.09 of sunset. 101. f x tan x 102. f (t) tan t 4 103. f x cot x Graph of tant shifted to the right by 4. y 4 3 y 4 3 y 2 2 1 3 π x 1 2 −π π 1 −π 2 π 2 t −3 −4 x 624 Chapter 6 Trigonometry Period: 2 106. h(t) sec t Graph y cos x first. Graph of sec t shifted to the right by . 4 y y 4 105. f x sec x 104. g(t) 2 cot 2t 3 y 2 1 −π π −π t x π −1 1 −2 −3 t π −4 107. f x csc x 108. f (t) 3 csc 2t Graph y sin x first. 4 109. f x x cos x Graph y1 x cosx y y2 x y 4 y3 x 2 3 2 π 1 − 3π 2 As x → , f x → . t 6 x π 2 −9 −3 9 −4 −6 110. g(x) ex cos x 21 arcsin 12 6 111. arcsin 300 Graph y ± ex first. −1 7 −150 112. arcsin(1) 2 115. sin 1 0.44 0.46 radian 118. arccos 22 4 121. arccos 0.324 1.24 radians 124. tan1(8.2) 1.45 radians 113. arcsin 0.4 0.41 radian 114. arcsin(0.213) 0.21 116. sin1(0.89) 1.10 117. arccos 119. cos 11 120. cos1 122. arccos(0.888) 2.66 radians 123. tan11.5 0.98 radian 3 2 6 23 6 Review Exercises for Chapter 6 125. f x 2 arcsin x 2 sin1x 127. f x arctan 126. y 3 arccos x 625 2x tan 2x 1 3 2 −1.5 1.5 −4 −1.5 4 1.5 0 − − 2 129. cosarctan 34 45. Use a right triangle. 128. f x arcsin 2x Let arctan 34. Then 2 5 tan 34 and cos 45. −1.5 3 θ 1.5 4 − 2 130. tanarccos 35 43 12 13 131. secarctan 5 5 Use a right triangle. Let Use a right triangle. Let u arccos 35. Then arctan 12 5 . Then 3 5 cos u and 5 13 12 13 tan 12 5 and sec 5 . 4 θ tanarccos 35 tan u 43. 5 u 3 5 132. arcsin 12 13 12 5 Use a right triangle. Let u arcsin 12 13 . Then u sin u 12 13 and 5 cotarcsin 12 13 cot u 12 . 133. Let y arccos Then cos y 2x . 4 x 2 x 2 x 2 arcsinx 1 ⇒ 4 − x2 ≤ ≤ 2 2 sin x 1 cos 12 x 12 y . −12 134. secarcsinx 1 x and 2 tan y tan arccos 13 x 1 x2 x sec 1 x2 x θ 12 − (x − 1)2 x −1 626 Chapter 6 135. tan Trigonometry 70 30 136. 137. r arctan 70 66.8 30 h 1808 ft 21° 25.2° r 25 feet tan 21 1808 4246.33 ft sin 25.2 h 25 h 25 tan 21 9.6 feet 138. sin 48 d1 ⇒ d1 483 650 cos 25 d2 ⇒ d2 734 810 d cos 48 3 ⇒ d3 435 650 sin 25 d4 ⇒ d4 342 810 N d1 d2 1217 48° W d3 d3 d4 93 A B 25° 48° 65° 810 650 D θ d1 d2 E d4 C S tan 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. 139. cos 45 140. High point at time t 0 ⇒ d a cos bt a ⇒ 60 a a 42.43 nautical miles north sin 45 b ⇒ 60 1 1.5 0.75 inches 2 Period 3 seconds N b b 42.43 nautical miles east d 0.75 cos a 45° 60 W 2 2 ⇒ b b 3 23t E S 141. False. The sine or cosine functions are often useful for modeling simple harmonic motion. 142. True. The inverse sine, y arcsin x, cannot be defined as a function over any interval that is greater than the interval defined as ≤ y ≤ . 2 2 143. False. For each there corresponds exactly one value of y. 144. False. arctan(1) 4 3 is not in the range of the arctan function. 4 Review Exercises for Chapter 6 145. y 3 sin x 146. y 3 sin x 147. y 2 sin x Amplitude: 3 Amplitude: 2 Amplitude: 3 Period: 2 Period: 2 Period: 2 a < 0 ⇒ graph is reflected in the x-axis Matches graph d. 627 Matches graph b. Matches graph (a). x 2 148. y 2 sin 149. f sec is undefined at the zeros of g cos 1 since sec . cos Amplitude: 2 Period: 4 Matches graph (c). 150. (a) 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 2 151. 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, . 1 152. y Aekt cos bt 5 et10 cos 6t 1 1 (a) If A is changed from 5 to 3, the amplitude of each oscillation is increased. 1 1 (b) If k is changed from 10 to 3, the oscillations are damped more quickly. (c) If b is changed from 6 to 9, the frequency of the oscillations increases. 153. (a) tan x 12 (b) 800 x 12 tan Area Area of triangle Area of sector 0 1 1 bh r 2 2 2 0 As ⇒ 1 1 1212 tan 122 2 2 72 tan 72 72tan x θ 12 2 , ⇒ . 2 The area increases without bound as approaches . 2 628 Chapter 6 Trigonometry 1 1 (b) A 2 r 2 210 2 50, > 0 1 1 154. (a) A 2 r 2 2 r 20.8 0.4r 2, r > 0 s r 10, > 0 s r 0.8r, r > 0 30 4 A A s 0 s 3 0 6 0 0 The area function increases more rapidly than the arc length function because it is a function of the square of the radius, while the arc length function is a function of the radius. 155. Answers will vary. Problem Solving for Chapter 6 1. (a) 8:57 6:45 2 hours 12 minutes 132 minutes 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 3000 d 3000 4767 feet sin 39 (b) tan 39 x 24 3 radians 360 270 32 2 2. Gear 1: 3000 x 3000 3705 feet tan 39 (c) tan 63 w 3705 3000 3000 tan 63 w 3705 w 3000 tan 63 3705 2183 feet 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. Problem Solving for Chapter 6 5. (a) hx cos2 x (b) hx sin2 x 3 3 −2 −2 2 2 −1 −1 h is even. h is even. 6. Given: f is an even function and g is an odd function. hx f x2 (a) (b) hx gx2 hx gx2 hx f x2 f x2 since f is even gx2 since g is odd hx gx2 hx Thus, h is an even function. Thus, h is an even function. 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. 1 1 a max min 101 1 50 2 2 1 1 d max min 101 1 51 2 2 b 8 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 3 2 Here we wanted the sine value to be 1 when t 0. 8. P 100 20 cos (a) 83t (b) Period 130 2 6 3 sec 8 8 4 3 This is the time between heartbeats. 0 5 70 (c) Amplitude: 20 The blood pressure ranges between 100 20 80 and 100 20 120. (e) Period 64 60 15 sec 64 16 64 32 60 ⇒ b 2 15 2b 60 (d) Pulse rate 60 secmin 80 beatsmin 3 4 secbeat 629 630 Chapter 6 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 20 years 5 leap years 11 remaining July days 31 t 365 20 (b) Number of days since birth until September 1, 2006: August days 1 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) (b) The graphs intersect when x 5.35 32 0.65. 6 g − 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. f −6 (b) The period of f x is 2. The period of gx is . (c) hx A cos x 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 6 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. 2 631 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. 632 Chapter 6 Chapter 6 Trigonometry Practice Test 1. Express 350° in radian measure. 2. Express 59 in degree measure. 3. Convert 1351412 to decimal form. 4. Convert 22.569 to DMS 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 magnitude of 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. 21. A 40, c 12 22. B 6.84, a 21.3 23. a 5, b 9 B c A b a C 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