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Chapter 2 Acute Angles and Right Triangles Section 2.1 Trigonometric Functions of Acute Angles 1. 21 20 21 ; ; 29 29 20 2. 45 28 45 ; ; 53 53 28 2 21 2 21 21 5 21 5 ; ; ; ; ; 5 5 21 2 21 2 20. sin θ = cos (90-θ ); cos θ = sin (90-θ ); tan θ = cot (90-θ );cot θ = tan (90-θ ); sec θ = csc (90-θ ); csc θ = sec (90-θ ) 21. sin 60 n m n 3. ; ; p p m 4. 19. a = 21 22. cos 45 23. sec30 k y k ; ; z z y 24. tan17 25. csc51 For Exercises 5−10, refer to the Function Values of Special Angles chart on page 54 of the text. 26. cot 64.6 27. cos51.3 5. C 6. H 7. B 8. G 28. sin (70- ) 9. E 10. A 29. csc (75- ) 11. c = 13 12. c = 5 12 5 12 5 13 13 ; ; ; ; ; 13 13 5 12 5 12 30. They are always the same values. 4 3 4 3 5 5 ; ; ; ; ; 5 5 3 4 3 4 13. b = 13 13 6 13 6 13 7 7 13 ; ; ; ; ; 7 7 6 13 6 17 7 95 7 95 95 12 95 12 14. a = 95 ; ; ; ; ; 12 12 95 7 95 7 For exercises 31−40, if the functions in the equations are cofunctions, then the equations are true if the sum of the angles is 90º. 31. 40 32. 40 33. 20 34. 20 15. b = 91 91 3 91 3 91 10 10 91 ; ; ; ; ; 10 10 3 91 3 91 35. 12 36. 12 37. 35 8 57 8 57 57 11 57 11 ; ; ; ; ; 16. a = 57 11 11 57 8 57 8 38. 15 39. 18 17. b = 3 3 1 3 2 3 ; ; 3; ; 2; 2 2 3 3 40. 35 18. b = 2 2 2 ; ;1;1; 2; 2 2 2 42. true 41. true 43. false Copyright © 2013 Pearson Education, Inc. 12 Section 2.1 Trigonometric Functions of Acute Angles 44. false 13 66. 45. true 46. false 47. true 48. false 49. 3 3 50. 3 51. 1 2 67. The legs of the right triangle provide the ( coordinates of P, 2 2, 2 2 68. 52. 3 2 53. 2 3 3 54. 2 (1, 3) 69. sin x; tan x 70. cos x; csc x 71. 60 72. 0.7071067812 is a rational approximation for the exact value 55. 2 56. 2 æ 2 2 ö÷ ÷÷ ; 45°. , 73. ççç çè 2 2 ø÷ 57. 2 2 74. y 3 x 58. 1 75. y = 59. 1 60. 61. 62. 2 (an irrational value). 2 3 x. 3 76. 30° 2 2 3 2 1 2 77. 60° 78. (a) 45º (b) 2k (c) 2 79. (a) 60 (b) k 63. 3 64. 2 3 3 65. ) (c) 3k (d) 2; 3; 30º; 60º 80. a = 12; b = 12 3; d = 12 3; c = 12 6 9 9 3 3 3 ;z = ;w = 3 3 81. y = ; x = 2 2 2 82. m = 7 3 14 3 14 3 14 6 ; a= n= ; q= . 3 3 3 3 83. p = 15; r = 15 2; q = 5 6; t = 10 6 Copyright © 2013 Pearson Education, Inc. 14 Chapter 2 Acute Angles and Right Triangles 84. A = s2 3 4 85. A = s2 2 7. 2 is a good choice for r because in a 30 - 60 right triangle, the hypotenuse is twice the length of the shorter side (the side opposite to the 30 angle). By choosing 2, one avoids introducing a fraction (or decimal) when determining the length of the shorter side. Choosing any even positive integer for r would have this result; however, 2 is the most convenient value. 86. Yes Section 2.2 Trigonometric Functions of Non-Acute Angles 1. C 2. F 8. Answers may vary 3. A 9. Answers may vary 4. B 10. Answers may vary 5. D 6. B θ sin θ cos θ tan θ 3 3 1 11. 30 1 2 3 2 12. 45 2 2 13. 60 3 2 2 2 1 2 3 2 14. 120 2 2 15. 135 16. 150 sin150 = sin 30 1 = 2 3 cos 120 = - cos 60 =- 1 2 - - - 3 cot θ sec θ csc θ 3 2 3 3 2 1 2 3 3 cot 120 = - cot 60 3 3 tan135 cot135 = - tan 45 = - cot 45 = -1 = -1 =- 2 2 3 2 - 3 3 cot150 = - cot 30 =- 3 cos 210 17. 18. 210 240 - - 1 2 3 2 = - cos 30 3 =2 - 1 2 3 3 tan 240 = tan 60 = 3 3 cot 240 = cot 60 3 = 3 2 2 2 3 3 sec120 = - sec 60 = -2 2 3 3 - 2 2 sec150 = - sec30 2 2 3 =3 sec 210 = - sec30 2 3 =3 -2 Copyright © 2013 Pearson Education, Inc. -2 - 2 3 3 Section 2.2 Trigonometric Functions of Non-Acute Angles 19. - 3 1 3 2 3 ; ; - 3; ; 2; 2 2 3 3 2 2 ; ; -1; -1; 2; - 2 2 2 2 2 ; ;1;1; 2; 2 2 2 3 1 3 2 3 ; ; 3; ; 2; 2 2 3 3 3 2 38. 20. - 39. - 21. 40. 22. 3 42. - 2 3 1 3 2 3 ; - ; - 3; ; -2; 2 2 3 3 24. 2 2 ;; -1; -1; - 2; 2 2 2 43. - 1 3 3 44. - 1 3 3 2 3 ; ; 3; ; -2 25. - ; 2 2 3 3 1 3 3 2 3 ; ; ; 3; ;2 2 2 3 3 27. - 3 2 41. - 2 23. 26. 2 2 ;;1;1; - 2; - 2 2 2 45. 1 46. 1 47. 23 4 48. 9 2 7 2 28. 3 1 3 2 3 ; ; 3; ; 2; 2 2 3 3 49. 29. 3 1 3 2 3 ; ; 3; ; 2; 2 2 3 3 50. 1 51. - 29 12 1 3 3 2 3 ;; - 3; ; -2 30. - ; 2 2 3 3 52. 10 1 3 3 2 3 ; ; 3; ; -2 31. - ; 2 2 3 3 53. Since 0 ¹ 32. 33. 15 3 1 3 2 3 ; ; 3; ; 2; 2 2 3 3 54. Since 1 3 3 2 3 ;;; - 3; ;2 2 2 3 3 55. Since 3 +1 , the statement is false 2 1+ 3 ¹ 1, the given statement is false 2 1 ¹ 3, the statement is false 2 56. true 34. - 2 2 ;;1;1; - 2; - 2 2 2 35. - 3 1 3 2 3 ; ; - 3; ; 2; 2 2 3 3 36. - 2 2 ; ; -1; -1; 2; - 2 2 2 37. - 2 2 57. true 58. true 59. false 60. true 61. (-3 3,3 . ) 62. (-5 2, -5 2 . ) 63. yes, in quadrant IV Copyright © 2013 Pearson Education, Inc. 16 Chapter 2 Acute Angles and Right Triangles 64. there is no angle θ 89. 45;315 65. positive 90. 45;315 66. positive Section 2.3 Finding Trigonometric Function Values Using a Calculator 67. positive 68. negative 69. negative 1. sin; 1 70. negative. 2. approximate 71. θ and θ + n ⋅ 360 are coterminal angles, so the sine of each of these will result in the same value. 3. reciprocal; reciprocal 72. θ and θ + n ⋅ 360 are coterminal angles, so the cosine of each of these will result in the same value. 73. If n is even, θ and θ + n ⋅180 are coterminal angles, so the tangent of each of these will result in the same value. If n is odd, θ and θ + n ⋅180 have the same reference angle, but are positioned two quadrants apart. The tangent is positive for angles located in quadrants I and III, and negative for angles located in quadrants II and IV, so the tangent of each angle is the same value. 4. sine; inverse For Exercises 5−15, the calculation for decimal degrees is indicated for calculators that do not accept degree, minutes, and seconds. 5. sin 3842¢ » 0.62524266 6. cos 4124¢ » 0.75011107 7. sec1315¢ » 1.0273488 8. csc14545¢ » 1.7768146 9. cot18348¢ » 15.055723 10. tan 42130¢ » 1.8417709 11. sin (-31212 ¢) » 0.74080460 74. −0.4. 12. tan (-8006 ¢) » -5.7297416 75. sin 115° is closest to 0.9. 13. csc (-31736 ¢) » 1.4830142 76. 135; 315 77. 45; 225 14. cot (-51220 ¢) » 1.9074147 (b) 369 ft 15. tan 23.4 » 0.43273864 16. cos14.8 » 0.96682339 17. cot 77 » 0.23086819 (c) Answers will vary. 18. tan 33 » 0.64940759 78. (a) 550 ft 79. 30;150 80. 30;330 81. 60;300 82. 45; 225 83. 45;315 84. 60;300 85. 30;330 86. 60;300 87. 30;210 . 88. 60; 240 19. tan 4.72 » 0.08256640 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. sin 3.69 » 0.06435814 sin 39 » 0.629320391 tan 22 » 0.40402623 » 55.845496 » 81.168073 » 16.166641 » 57.997172 » 38.491580 » 46.173582 » 68.673241 » 30.502748 » 45.526434 Copyright © 2013 Pearson Education, Inc. Section 2.3 Finding Trigonometric Functions Values Using a Calculator 32. 33. 34. 35. » 31.199998 » 12.227282 » 77.831359 A common mistake is to have the calculator in radian mode, when it should be in degree mode (and vice verse). 17 57. true 58. false 59. 65.96 lbs 60. -100.5lbs 61. » -2.9 62. » 2.9 63. » 2547 lbs 64. » 2771 lbs 65. The 2200-lb car on a 2° uphill grade has the greater grade resistance. 66. 36. If the calculator allowed an angle θ where 0 £ θ < 360 , then we would need to find an angle within this interval that is coterminal with 2000º by subtracting a multiple of 360º: 2000 - 5 ⋅ 360 = 2000- 1800 = 200 . If the calculator was more restrictive on evaluating angles (such as 0 £ θ < 90 ), then reference angles would need to be used. 37. 38. 39. 40. 41. 42. 43. 44. 45. A = 56º. A = 0.3746065934º 1 -1 1 0 0 1 A: 68.94 mph; B: 65.78 mph 46. r = a cos θ . 47. false 48. false θ sin θ tan θ 0º 0.0000 0.0000 πθ 180 0.0000 0.5º 0.0087 0.0087 0.0087 1º 0.0175 0.0175 0.0175 1.5º 0.0262 0.0262 0.0262 2º 0.0349 0.0349 0.0349 2.5º 0.0436 0.0437 0.0436 3º 0.0523 0.0524 0.0524 3.5º 0.0610 0.0612 0.0611 4º 0.0698 0.0699 0.0698 (a) From the table, we see that if θ is small, πθ . sin θ = tan θ = 180 (b) F = W sin θ = W tan θ = W πθ 180 (c) 80 lb (d) 117.81 lb 49. true 67. (a) 703 ft 50. true (b) 1701 ft (c) R will decrease; 644 ft; 1559 ft. 68. 55 mph 69. (a) 2 ´108 m per sec. 51. false 52. false 53. false 54. false 55. true 56. true (b) 2 ´108 m per sec. 70. (a) 19 (b) 50 71. 48.7 Copyright © 2013 Pearson Education, Inc. 18 Chapter 2 Acute Angles and Right Triangles 4. Answers will vary. No; the number of points scored will be a whole number. 72. 40.8; 7.9 73. (a) 155 ft (b) 194 ft 5. Answers will vary. It would be cumbersome to write 2 as 2.00 or 2.000, for example, if the measurements had 3 or 4 significant digits (depending on the problem). In the formula, it is understood that 2 is an exact value. Since the radius measurement, 54.98 cm, has four significant digits, an appropriate answer would be 345.4 cm. 74. 78 mph Chapter 2 Quiz (Sections 2.1−2.3) 3 4 3 4 5 5 ; ; ; ; ; 5 5 4 3 4 3 1. 2. θ sin θ cos θ tan θ 30 1 2 3 2 3 3 45 2 2 60 3 2 2 2 1 2 cot θ sec θ csc θ 3 2 3 3 2 1 1 2 3 3 3 2 2 2 3 3 6. 23.0 ft indicates 3 significant digits and 23.00 ft indicates four significant digits. 7. 0.05 8. 0.5 9. B = 5340 ¢; a » 571m; b » 777 m 10. Y = 42.2; x » 66.4 cm; y » 60.2 cm 11. M = 38.8; n » 154 m; p » 198 m 3. w = 18; x = 18 3; y = 18; z = 18 2 12. B = 5820¢; c » 68.4 km; b » 58.2 km 4. 3x2 sin . 13. 5. A = 47.9108; c » 84.816 cm; a » 62.942 cm 14. A = 21.4858; b » 3330.68 m; a » 1311.04 m 2 2 ;; -1; -1; - 2; 2 2 2 1 3 3 2 3 ; ; 3; ; -2 6. - ; 2 2 3 3 15. c » 20.5 ft; A » 3740 ¢; B » 5220¢ 3 1 3 2 3 ; ; - 3; ; 2; 2 2 3 3 8. 60;120 . 17. No; You need to have at least one side to solve the triangle. 9. 45; 225 19. Answers will vary. If you know one acute angle, the other acute angle may be found by subtracting the given acute angle from 90°. If you know one of the sides, then choose two of the trigonometric ratios involving sine, cosine or tangent that involve the known side in order to find the two unknown sides. 7. - 10. 11. 12. 13. 14. 0.67301251 -1.1817633 » 69.497888 » 24.777233 false 15. true Section 2.4 Solving Right Triangles 1. 22,894.5 to 22,895.5 2. 28,999.5 to 29,000.5 3. 8958.5 to 8959.5 16. b » 14.5 m; A » 1820 ¢; B » 7140¢ 18. The other acute angle requires the least work. 20. Answers will vary. If you know the lengths of two sides, you can set up a trigonometric ratio to solve for one of the acute angles. The other acute angle may be found by subtracting the calculated acute angle from 90°. With either of the two acute angles that have been determined, you can set up a trigonometric ratio along with one of the known sides to solve for the missing side. 21. B = 62.0; a » 8.17 ft; b » 15.4 ft 22. A = 44.0; a » 20.6 m; b » 21.4 m Copyright © 2013 Pearson Education, Inc. Section 2.5 Further Applications of Right Triangles 23. A = 17.0; a = 39.1 in;c = 134 in 24. B = 27.5; b » 6.61 m;c » 14.3 m 25. B = 29.0; b » 70.7 cm;c » 80.9 cm 26. A = 38.3; b » 35.6 ft;c » 45.3 ft 27. b » 18 m; A » 36; B » 54 29. c » 85.9 yd; A » 6250¢; B » 2710¢ 30. c » 1080 m; A » 6300¢; B » 2700¢ 31. b » 42.3 cm; A » 2410 ¢; B » 6550 ¢ 32. a » 609 cm; A » 7010¢; B » 1950¢ 33. B » 3636 ¢; b » 310.8 ft;c » 230.8 ft 34. B » 7613¢; a » 306.2 m;b » 1248 m 35. B » 5051¢; b » 0.3934 m;a » 0.4832 m 36. B = 709 ¢; b » 4.787 m; a » 0.6006 m 37. The angle of elevation from X to Y (with Y above X) is the acute angle formed by ray XY and a horizontal ray with endpoint at X. 38. No 39. Answers will vary. The angle of elevation and the angle of depression are measured between the line of sight and a horizontal line. So, in the diagram, lines AD and CB are both horizontal. Hence, they are parallel. The line formed by AB is a transversal and angles DAB and ABC are alternate interior angle and thus have the same measure. 40. The angle of depression is measured between the line of sight and a horizontal line. This angle is measured between the line of sight and a vertical line. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 33.4 m 128 ft 864,900 mi 26.92 in 134.7 cm 22 583 ft. 28.0 m 469 m 13.3 ft 42,600 ft 3735¢ 146 m 42.18 63.39 (a) 29,000 ft. (b) shorter 58. 34.0 mi 28. a » 40 ft; A » 51; B » 39 41. 9.35 m 53. 54. 55. 56. 57. 19 Section 2.5 Further Applications of Right Triangles 1. It should be shown as an angle measured clockwise from due north. 2. It should be shown measured from north (or south) in the east (or west) direction. 3. A sketch is important to show the relationships among the given data and the unknowns. 4. The angle of elevation (or depression) from X to Y is measured from the horizontal line through X to the ray XY. 5. 270°; N 90° W, or S 90° W 6. 7. 8. 9. 10. 11. 12. 90°; N 90° E, or S 90° E 0°; N 0° E or N 0º W 180°; S 0° E or S 0º W 315°; N 45° W 225°; S 45° W 135°; S 45° E 45°; N 45° E 3 3 13. y= 14. y = - 3 x, x ³ 0 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. x, x £ 0 220 mi 150 km 47 nautical mi 70 nautical mi 2203 ft 5856 m 148 mi 2.01 mi 430 mi 350 mi 140 mi 130 mi 27. x = b a -c Copyright © 2013 Pearson Education, Inc. 20 Chapter 2 Acute Angles and Right Triangles 28. tan θ = - tan (180-θ ) = - tan θ ¢ . This is because the angle represented by 180- θ terminates in quadrant II if 0 < θ < 90 . If 90 < θ < 180 , then the angle represented by 180- θ terminates in quadrant I. Thus, tan θ and tan (180-θ ) are opposite in sign. The slope of the line is m = - ba , and tan θ = - tan (180-θ ) = - tan θ ¢ = - ba . Thus, m = - ba = tan θ . The point-slope form of the equation of a line is y - y1 = m ( x - x1 ) Substituting tan θ for m into y - y1 = m ( x - x1 ) , we have 44. (a)As θ increases, D increases and then decreases. (b) As v increases, D increases (c) The velocity affects the distance more. The shot-putter should concentrate on achieving as large a value of v as possible. Chapter 2 Review Exercises 60 11 60 11 61 61 ; ; ; ; ; 61 61 11 60 11 60 20 21 20 21 29 29 2. ; ; ; ; ; 29 29 21 20 21 20 1. 3. 10 y - y1 = - tan θ ( x - x1 ) . The line passes through (a, 0), so y - y1 = tan θ ( x - x1 ) y - 0 = tan θ ( x - a ) y = tan θ ( x - a) . 4. 10 5. 7 6. 30 7. true 29. y = tan 35 ( x - 25) 8. false 30. y = tan15 ( x - 5) 9. true 31. 433 ft 32. 448 m 33. 114 ft 10. false 11. The sum of the measures of angles A and B is 90, and, thus, they are complementary angles. Since sine and cosine are cofunctions, we have sin B = cos (90- B) = cos A. 34. 147 m 35. 5.18 m 36. 2.47 km 12. D, tan 140°, is the only one which cannot be determined exactly using the methods of this chapter. 3 1 3 2 3 13. - ; ; - 3; - ; 2; 2 2 3 3 bæ ö 37. (a) d = ççcot + cot ÷÷÷ 2 çè 2 2ø (b) 345.4 cm 38. 1.95 mi 39. 10.8 ft 14. 40. A » 35.987 » 3559 ¢10 ¢¢ B » 54.013 » 5400 ¢50 ¢¢ 3 1 3 2 3 ; - ; - 3; ; -2; 2 2 3 3 1 3 3 2 3 ;; - 3; ; -2 15. - ; 2 2 3 3 41. (a) 320 ft æ ö (b) R çç1- cos ÷÷÷ çè 2ø 16. 42. 84.7 m 43. (a) 23 ft 18. 30;330 2 2 ;; -1; -1; - 2; 2 2 2 17. 60; 240 19. 30;210 (b) 48 ft (c) The faster the speed, the more land needs to be cleared on the inside of the curve. 20. 45;315 21. 3 - 2 3 3 Copyright © 2013 Pearson Education, Inc. Chapter 2 Test 22. 1 23. 60. 110 km 61. 140 mi 7 2 62. k ( tan B - tan A) 63.−64. Answers will vary. 2 2 24. (a) ;;1 2 2 65. (a) 716 mi (b) 1104 mi 3 1 ; ;- 3 2 2 -1.3563417 0.95371695 1.0210339 -0.71592968 0.20834446 1.9362132 55.673870° 41.635092° 12.733938° 37.695528° 63.008286° 5.9998273° 47.1; 132.9 54.2º; 234.2º false true (b) - 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 66. (a) xQ = xP + d sin ; yQ = yP + d cos . (b) (181.34, 523.02) Chapter 2 1. 4. (a) true (b) false (c) true 5. - 2 2 ;;1;1; - 2; - 2 2 2 7. -1;0; undefined;0; undefined; -1 8. 135; 225 11. Take the reciprocal of tan to get cot 0.59600119. 48. quadrant I 49. B = 3130 ¢; a » 638; b » 391 50. c » 390.3; A » 1925¢; B » 7035¢ 51. B = 50.28; a » 32.38 m; c » 50.66 m 52. A = 4207 ¢; a » 270.0 m; c » 402.5 m 55. 56. 57. 58. 59. 73.7 ft 20.4 m 18.75 cm 50.24 m 1200 m 9. 240; 300 10. 45; 225 Answers may vary. quadrant III quadrant II quadrant II 54. r = 13; » 23 3 1 3 2 3 ; - ; 3; ; -2; 2 2 3 3 6. - 1 ¹ tan-1 25. tan 25 53. 137 ft 12 5 12 5 13 13 ; ; ; ; ; 13 13 5 12 5 12 3. 15 42. false 44. 45. 46. 47. Chapter Test 2. y = 4 3; w = 8; x = 4; z = 4 2 41. true 43. No, cot 25 = 21 12. (a) 0.97939940 (b) -1.9056082 (c) 1.9362432 13. θ » 16.166641 14. B = 3130¢; b » 458; c » 877 15. 16. 17. 18. 19. 20. 67.1 or 6710¢ 15.5 ft 8800 ft 72 nautical mi 92 km 448 m Copyright © 2013 Pearson Education, Inc.