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Exploring Periodic Data

Exploring Periodic Data. ALGEBRA 2 LESSON 13-1. (For help, go to Lesson 2-1.). Determine whether each relation is a function. 1. {(2, 4), (1, 3), (–3, –1), (4, 6)} 2. {(2, 6), (–3, 1), (–2, 2)} 3. {( x , y )| x = 3} 4. {( x , y )| y = 8}

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Exploring Periodic Data

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  1. Exploring Periodic Data ALGEBRA 2 LESSON 13-1 (For help, go to Lesson 2-1.) Determine whether each relation is a function. 1. {(2, 4), (1, 3), (–3, –1), (4, 6)} 2. {(2, 6), (–3, 1), (–2, 2)} 3. {(x, y)| x = 3} 4. {(x, y)| y = 8} 5. {(x, y)| x = y2} 6. {(x, y)| x2 + y2 = 36} 7. {(a, b)| a = b3} 8. {(w, z)| w = z – 36} 13-1

  2. Exploring Periodic Data ALGEBRA 2 LESSON 13-1 Solutions 1. {(2, 4), (1, 3), (–3, –1), (4, 6)}; yes, this is a function because each element of the domain is paired with exactly one element in the range. 2. {(2, 6), (–3, 1), (–2, 2)}; yes, this is a function because each element of the domain is paired with exactly one element in the range. 3. {(x, y)| x = 3}; no, this is not a function because it is a vertical line and fails the vertical line test. 4. {(x, y)| y = 8}; yes, this is a function because it is a horizontal line and passes the vertical line test. 13-1

  3. Exploring Periodic Data ALGEBRA 2 LESSON 13-1 Solutions (continued) 5. {(x, y)| x = y2}; no, this is not a function because an element of the domain is paired with more than one element in the range. Example: 4 = 22 and 4 = (–2)2 6. {(x, y)| x2 + y2 = 36}; no, this is not a function because it is a circle and fails the vertical line test. 7. {(a, b)| a = b3}; yes, this is a function because each element of the domain is paired with exactly one element in the range. 8. {(w, z)| w = z – 36}; yes, this is a function because each element of the domain is paired with exactly one element in the range. 13-1

  4. Begin at any point on the graph. Trace one complete pattern. The beginning and ending x-values of each cycle determine the period of the function. Exploring Periodic Data ALGEBRA 2 LESSON 13-1 Analyze this periodic function. Identify one cycle in two different ways. Then determine the period of the function. Each cycle is 7 units long. The period of the function is 7. 13-1

  5. a. Exploring Periodic Data ALGEBRA 2 LESSON 13-1 Determine whether each function is or is not periodic. If it is, find the period. The pattern of y-values in one section repeats exactly in other sections. The function is periodic. Find points at the beginning and end of one cycle. Subtract the x-values of the points: 2 – 0 = 2. The pattern of the graph repeats every 2 units, so the period is 2. 13-1

  6. Exploring Periodic Data ALGEBRA 2 LESSON 13-1 (continued) b. The pattern of y-values in one section repeats exactly in other sections. The function is periodic. Find points at the beginning and end of one cycle. Subtract the x-values of the points: 3 – 0 = 3. The pattern of the graph repeats every 3 units, so the period is 3. 13-1

  7. 1 2 a. amplitude = (maximum value – minimum value)   Use definition of amplitude. 1 2 = [2 – (–2)] Substitute. 1 2 = (4) = 2 Subtract within parentheses and simplify. Exploring Periodic Data ALGEBRA 2 LESSON 13-1 Find the amplitudes of the two functions in Additional Example 2. The amplitude of the function is 2. 13-1

  8. 1 2 b. amplitude = (maximum value – minimum value)   Use definition of amplitude. 1 2 = [6 – 0] Substitute. 1 2 = (6) = 3 Subtract within parentheses and simplify. Exploring Periodic Data ALGEBRA 2 LESSON 13-1 (continued) The amplitude of the function is 3. 13-1

  9. 1 unit on the t-axis = s 1 360 Exploring Periodic Data ALGEBRA 2 LESSON 13-1 The oscilloscope screen below shows the graph of the alternating current electricity supplied to homes in the United States. Find the period and amplitude. 13-1

  10. amplitude = [120 – (–120)] = (240) = 120 1 60 1 2 period = – 0Use the definitions. = Simplify. 1 2 1 60 1 60 1 60 The period of the electric current is s. Exploring Periodic Data ALGEBRA 2 LESSON 13-1 (continued) One cycle of the electric current occurs from 0 s to s. The maximum value of the function is 120, and the minimum is –120. The amplitude is 120 volts. 13-1

  11. Exploring Periodic Data ALGEBRA 2 LESSON 13-1 pages 699–702  Exercises 1. x = –2 to x = 3, x = 2 to x = 7; 5 2.x = 0 to x = 4, x = 5 to x = 9; 4 3.x = 0 to x = 4, x = 2 to x = 6; 4 4. not periodic 5. periodic; 12 6. not periodic 7. not periodic 8. periodic; 8 9. periodic; 7 10. 4 11. 3 12. 1 13. 2 14. 15. 16.a.y b. x 13-1

  12. Exploring Periodic Data ALGEBRA 2 LESSON 13-1 22. Check students’ work. 23. 3, –3, 4; 24. 5, 0, 8; 25. 4, –4, 8; 17. Answers may vary. Sample: Yes; average monthly temperatures for three years should be cyclical due to the variation of the seasons. 18. Answers may vary. Sample: No; population usually increases or decreases but is not cyclical. 19. Answers may vary. Sample: Yes; traffic that passes through an intersection should be at the same levels for the same times of day for two consecutive work days. 20. 60 beats per min 21.a. 1 s b. 1.5 mV 13-1

  13. Exploring Periodic Data ALGEBRA 2 LESSON 13-1 26. 1 yr 27. 2 weeks 28. 3 months 29. 1 hour 30. 1 day 31. 2, 2, 2 32.a. 67 b. 70 c. 70 d. 67 33. a. b. 5 s, 1 ft c. Answers may vary. Sample: about 1 s 34. a. 24.22 days b. 0.78 day c. 0.22 day d. Answers may vary. Sample: The calendar year is meant to predict events in the solar year. Keeping the difference between the two minimal is necessary for the calendar year to be useful. 1 3 13-1

  14. 35. C 36. G 37. B 38. D 39. [2] A period is the length of 1 cycle, so = Then xm = n, or x = . The period is seconds. [1] answer only with no explanation n seconds m cycles x seconds 1 cycle n m n m Exploring Periodic Data ALGEBRA 2 LESSON 13-1 40.[4] 64 seconds. The first two functions are at the beginning of their cycles together every 6 • 7 = 42 seconds: 42, 84, 126, . … The third function is at the beginning of its cycle every 8 seconds, starting at 20 seconds: 20, 28, 36, 44, 52, 60, 68, 76, 84, . … The three functions are all at the beginning of their cycles at 84 seconds, which is 64 seconds after the third function begins. [3] minor error in calculation [2] incomplete explanation [1] answer only, with no explanation 13-1

  15. (y + 3)2 4 (y + 2)2 16 (x – 5)2 5 (y – 1)2 4 Exploring Periodic Data ALGEBRA 2 LESSON 13-1 45. (x + 3)2 = 20(y – 2) 46. – = 1 47. – = 1 41. 42. 11, 13; an = 2n – 1; explicit or a1 = 1, an = an – 1 + 2; recursive 43. 14, 16; an = 2n + 2; explicit or a1 = 4, an = an – 1 + 2; recursive 44. 38, 51; a1 = 3, an = an – 1 + 2n – 1; recursive or an = n2 + 2; explicit 13-1

  16. Exploring Periodic Data ALGEBRA 2 LESSON 13-1 1. Determine whether the function shown is periodic. Find the period and the amplitude of the periodic functions shown. 3. yes 6; 1.5 2. Determine whether the function shown is periodic. 4. no 3; 0.5 13-1

  17. Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 (For help, go to page 54.) For each measure, draw an angle with its vertex at the origin of the coordinate plane. Use the positive x-axis as one ray of the angle. 1. 90° 2. 45° 3. 30° 4. 150° 5. 135° 6. 120° 13-2

  18. Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 Solutions 1. 2. 3. 4. 5. 6. 13-2

  19. Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 Find the measure of the angle. The angle measures 60° more than a right angle of 90°. Since 90 + 60 = 150, the measure of the angle is 150°. 13-2

  20. Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 Sketch each angle in standard position. a. 48° b. 310° c. –170° 13-2

  21. 16 20 The terminal side of the angle is of a full rotation from the initial side. 16 20 • 360° = 288° Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 The Aztec calendar stone has 20 divisions for the 20 days in each month of the Aztec year. An angle on the Aztec calendar shows the passage of 16 days. Find the measures of the two coterminal angles that coincide with the angle. To find a coterminal angle, subtract one full rotation. 288° – 360° = –72° Two coterminal angle measures for an angle on the Aztec calendar that show the passage of 16 days are 288° and –72°. 13-2

  22. From the figure, the x-coordinate of point A is – , so cos 135° = – , or about –0.71. 2 2 2 2 opposite leg = adjacent leg 2 2 =    Substitute. 0.71   Simplify. The coordinates of the point at which the terminal side of a 135° angle intersects are about (–0.71, 0.71), so cos 135° –0.71 and sin 135° 0.71. Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 Find the cosine and sine of 135°. Use a 45°-45°-90° triangle to find sin 135°. 13-2

  23. Step 1:  Sketch an angle of –150° in standard position. Sketch a unit circle. Step 2:  Sketch a right triangle. Place the hypotenuse on the terminal side of the angle. Place one leg on the x-axis. (The other leg will be parallel to the y-axis.) x-coordinate = cos (–150°) y-coordinate = sin (–150°) Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 Find the exact values of cos (–150°) and sin (–150°). 13-2

  24. 1 2 shorter leg = The shorter leg is half the hypotenuse. 1 2 longer leg = 3 = The longer leg is 3 times the shorter leg. Since the point lies in Quadrant III, both coordinates are negative. The longer leg lies along the x-axis, so cos (–150°) = – , and sin (–150°) = – . 1 2 3 2 3 2 Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 (continued) The triangle contains angles of 30°, 60°, and 90°. Step 3: Find the length of each side of the triangle. hypotenuse = 1 The hypotenuse is a radius of the unit circle. 13-2

  25. Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 pages 708–710  Exercises 1. –315° 2. –135° 3. 240° 4. 115° 5. –110° 6. –340° 7. 11. 12. 25° 13. 215° 14. 315° 15. 4° 16. 140° 17. 150° 18. 55° 19. 180° 20. 220°, –140° 8. 9. 10. 13-2

  26. 21. , – ; 0.50, –0.87 22. – , – ; –0.71, –0.71 23. , – ; 0.87, –0.50 24. – , ; –0.50, 0.87 25. , ; 0.87, 0.50 26. , – ; 0.71, –0.71 27. , – ; 0.87, –0.50 28. – , ; –0.71, 0.71 29. 1.00, 0.00 30. 0.85, 0.53 1 2 1 2 1 2 1 2 2 2 3 2 2 2 2 2 2 2 3 2 3 2 2 2 2 2 3 2 3 2 1 2 Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 31. 0.71, –0.71 32. –0.87, 0.50 33. –0.09, –1.00 34. 0.98, –0.17 35. –0.90, 0.44 36. 0.00, 1.00 37–44.Answers may vary. Samples: 37. 405°, –315° 38. 235°, –485° 39. 45°, –315° 40. 40°, –320° 41. 275°, –445° 42. 295°, –65° 43. 573°, –147° 44. 303°, –417° 45. II 46. III 47. negative x-axis 48. IV 49. positive x-axis 13-2

  27. 3 2 Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 51.a. 0.77, 0.77, 0.77 b. The cosines of the three angles are equal because the angles are coterminal. 52. The x-coordinate of the point on the ray defined by angle is equal to cos ; similarly for the y-coordinate and sin . The angles 0°, 180°, and 360° lie on the x-axis, and thus their sines are all 0 and their cosines are ±1. The angles 90° and 270° lie on the y-axis, so their cosines are 0 and their sines are ±1. 53. , 50. a. b. II c. If the terminal side of an angle is in Quadrants I or II, then the sine of the angle is positive; otherwise it is not. If the terminal side of an angle is in Quadrants I or IV, then the cosine of the angle is positive; otherwise it is not. 1 2 13-2

  28. 1 2 54. – , 55. – , – 56. , – 1 2 3 2 3 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 1 2 Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 57. , – 58. , – 59. – , 60. Answers may vary. Sample: 30°, 150°, –210°, 390° 13-2

  29. 3 2 3 2 Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 61. No; yes; if the sin and cos are both negative, is in Quadrant III. –120° is in Quadrant III. 62.a. Check students’ work. b. –20° 63. A 64. H 65. D 66. H 67.[2] The terminal side forms an angle of 30° with the negative x-axis. Using the unit circle, x = – and y = . So cos (–210°) = – . [1] answer only, with no work shown 1 2 13-2

  30. 2 2 2 2 2 2 2 2 Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 69. periodic; 3 70. not periodic 71. periodic; 6 72. 73. 68.[4] The terminal side forms an angle of 45° with the negative x-axis, so sin(–135°) = – and cos(–135°) = – . Then [sin (–135°)]2 + [cos (–135°)]2 = – + – = + = = 1. (OR a convincing argument using x2 + y2 = 1) [3] one computational error [2] incomplete explanation with correct answer [1] answer only, with no work shown 2 2 4 4 2 4 2 4 13-2

  31. Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 74. (0, 2 5 ), (0, –2 5) 75. (0, 5 5 ), (0, –5 5) 76. ( 85, 0), (– 85, 0) 77. ( 145, 0), (– 145, 0) 13-2

  32. 1 2 – – ; ; 2 2 2 2 3 2 Angles and the Unit Circle ALGEBRA 2 LESSON 13-2 Sketch each angle in standard position. Use a right triangle to find the exact values of the cosine and sine of the angle. 1. 45° 2. –120° 3. What angle is less than 360° and coterminal with 45°? –315° 13-2

  33. Radian Measure ALGEBRA 2 LESSON 13-3 (For help, go to page 870.) Find the circumference of a circle with the given radius or diameter. Round your answer to the nearest tenth. 1. radius 4 in. 2. diameter 70 m 3. radius 8 mi 4. diameter 3.4 ft 5. radius 5 mm 6. diameter 6.3 cm 13-3

  34. Radian Measure ALGEBRA 2 LESSON 13-3 1.C = 2 r = 2 (4 in.) 25.1 in. 2.C = d = (70 m) 219.9 m 3.C = 2 r = 2 (8 mi) 50.3 mi 4.C = d = (3.4 ft) 10.7 ft 5.C = 2 r = 2 (5 mm) 31.4 mm 6.C = d = (6.3 cm) 19.8 cm Solutions 13-3

  35. Write a proportion. 45° 180° = Write the cross-products. r radians radians 45 • 45 • = 180 • r 180 r = Divide each side by 45. 4 = 0.785 Simplify. Radian Measure ALGEBRA 2 LESSON 13-3 a. Find the radian measure of angle of 45°. An angle of 45° measures about 0.785 radians. 13-3

  36. 13 13 b. Find the degree measure of . 6 6 13 13 = Write a proportion. d° 180 6 6 radians • 180 = • dWrite the cross-product. d = Divide each side by . 30 13 • 180 6 • 1 An angle of radians measures 390°. Radian Measure ALGEBRA 2 LESSON 13-3 (continued) = 390° Simplify. 13-3

  37. 90 . 180° – radians • = – radians • Multiply by radians 1 3 3 3 3 2 2 2 2 An angle of – radians measures –270°. 180° 180° radians radians Radian Measure ALGEBRA 2 LESSON 13-3 a. Find the degree measure of an angle of – radians. = –270° 13-3

  38. 3 3 = radians Simplify. 3 54° • radians = 54° • radians Multiply by radians. 180° An angle of 54° measures radians. 10 180° 180° 10 10 Radian Measure ALGEBRA 2 LESSON 13-3 (continued) b. Find the radian measure of an angle of 54°. 13-3

  39. radians radians radians radians radians • = 60°    Convert to degrees. Draw the angle. Complete a 30°-60°-90° triangle. 3 3 3 3 3 1 2 180° The shorter leg is the length of the hypotenuse, and the longer leg is 3 times the length of the shorter leg. radians 1 2 Thus, cos = 3 2 and sin = . Radian Measure ALGEBRA 2 LESSON 13-3 Find the exact values of cos and sin . The hypotenuse has length 1. 13-3

  40. 7 6 = 6 • Substitute 6 for r and for . 7 6 = 7 Simplify. 22.0 Use a calculator. Radian Measure ALGEBRA 2 LESSON 13-3 Use this circle to find length s to the nearest tenth. s = rUse the formula. The arc has length 22.0 in. 13-3

  41. 1 4 Since one complete rotation (orbit) takes 4 h, the satellite completes of a rotation in 1 h. Radian Measure ALGEBRA 2 LESSON 13-3 Another satellite completes one orbit around Earth every 4 h. The satellite orbits 2900 km above Earth’s surface. How far does the satellite travel in 1 h? Step 1: Find the radius of the satellite’s orbit. r = 6400 + 2900 Add the radius of Earth and the distance from Earth’s surface to the satellite. = 9300 13-3

  42. Step 2:  Find the measure of the central angle the satellite travels through in 1 h. = • 2 Multiply the fraction of the rotation by the number of radians in one complete rotation. = • Simplify. 1 4 1 2 2 Step 3:  Find s for = . s = rUse the formula. = 9300 • Substitute 9300 for r and for . 14608 Simplify. 2 2 Radian Measure ALGEBRA 2 LESSON 13-3 (continued) The satellite travels about 14,608 km in 1 h. 13-3

  43. 5 3 5 6 8 9 2 3 9 Radian Measure ALGEBRA 2 LESSON 13-3 pages 715–719  Exercises 1. – , –5.24 2. , 2.62 3. – , –1.57 4. – , –1.05 5. , 2.79 6. , 0.35 7. 540° 8. 198° 9. –120° 10. –172° 11. 90° 12. 270° 13. 13-3

  44. 3 2 3 2 3 2 3 2 12 Radian Measure ALGEBRA 2 LESSON 13-3 14. , 15. , 16. 0, 1 17. – , 18. – , 19. 0, –1 20. 3.1 cm 21. 10.5 m 22. 51.8 ft 23. 25.1 in. 24. 4.7 m 31. III 32. II 33. positive y-axis 34. II 35. negative x-axis 36. III 37. 0.71, –0.71 1 2 25. 43.2 cm 26. 107 in. 27. 32 ft 28.a. 11,048 km b. 33,144 km c. 27,620 km d. 276,198 km e. 18.1 h 29. 42.2 in. 30.a. 15°, radians b. 1036.7 mi c. 413.6 mi 1 2 1 2 1 2 13-3

  45. Radian Measure ALGEBRA 2 LESSON 13-3 38. –0.50, –0.87 39. 0.00, 1.00 40. 1.00, 0.00 41. –0.87, –0.50 13-3

  46. 3 10 4 5 4 5 3 10 6 5 9 5 9 5 7 6 6 5 17 6 5 5 Radian Measure ALGEBRA 2 LESSON 13-3 44. Check students’ work. 45. 11 radians 46. The student forgot to include parentheses. 47. 798 ft; 55°, 665° 48. 23.6 in.; – , 49. If two angles measured in radians are coterminal, the difference of their measures will be evenly divisible by 2 . 42. 0.81, –0.59 43. a–b. c. All five triangles are congruent by SSS. All have a hypotenuse of 1 unit, a long leg of about 0.81 unit, and a short leg of 0.59 unit. cos = 0.81, sin = 0.59; sin = 0.81, cos = 0.59; cos = –0.81, sin = 0.59; cos = –0.81, sin = –0.59; cos = 0.81, sin = –0.59 13-3

  47. 3 2 4 3 11 3 35 6 2 2 s 2 r s 2 r x2 2! x8 8! x6 6! x4 4! Radian Measure ALGEBRA 2 LESSON 13-3 57.a. 0.5017962; 0.4999646; the first four terms b. 1 – + – + – . . . c. 0.951; 18° 58. C 59. G 60. D 61. G 62.[2] For a central angle of 1 radian, the length of the intercepted arc is the length of the radius. [1] incomplete explanation 50. 6.3 cm 51. 4008.7 mi 52. – radians 53. – radians 54. radians 55. radians 56. = • 2 r = • 2 r r = s s = r 13-3

  48. Radian Measure ALGEBRA 2 LESSON 13-3 63. 64. 65. 66. 67. 68. 9.1, 5.41 69. 12.9, 3.53 70. 30, 8.09 71.x2 + y2 = 64 72. x2 + (y + 5)2 = 16 73. (x – 3)2 + (y – 7)2 = 42.25 74. (x + 8)2 + (y – 4)2 = 9 13-3

  49. radians 5 4 2 Radian Measure ALGEBRA 2 LESSON 13-3 1. Rewrite each angle measure using the other unit, either degrees or radians. a.  225° b.  radians 2. A wrench turns through an angle of 1.5 radians. If the wrench is 14 in. long, what is the distance that the end of the wrench moves? 3. A jogger runs 100 m around a circular track with a radius of 40 m. Through what angle does the jogger move? Express your answer in both radians and degrees. 90° 21 in. 2.5 radians; about 143.2° 13-3

  50. The Sine Function ALGEBRA 2 LESSON 13-4 (For help, go to Lesson 13-1.) Use the graph. Find the value(s) of each of the following. 1. the period 2. the domain 3. the amplitude 4. the range 13-4

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