Unit 6 Pre-calc Review questions

CHAPTER 6 74. Visualizing Roots of Unity Set your graphing calculator in parametric mode with 0 … T … 8, Tstep = 1, Xmin = – 2.4, Xmax = 2.4, Ymin = – 1.6, and Ymax = 1.6. (a) Let x = cos 112p/82t2 and y = sin 112p/82t2. Use TRACE to visualize the eight eighth roots of unity. We say that 2p/8 generates the eighth roots of unity. (Try both dot mode and connected mode.) (b) Replace 2p/8 in part (a) by the arguments of other eighth roots of unity. Do any others generate the eighth roots of unity? (c) Repeat parts (a) and (b) for the fifth, sixth, and seventh roots of unity, using appropriate functions for x and y. (d) What would you conjecture about an nth root of unity that generates all the nth roots of unity in the sense of part (a)? 75. Parametric Graphing Write parametric equations that represent 112 + i2n for n = t. Draw and label an accurate spiral representing 1 12 + i2n for n = 0, 1, 2, 3, 4. 76. Parametric Graphing Write parametric equations that represent 1- 1 + i2n for n = t. Draw and label an accurate spiral representing 1-1 + i2n for n = 0, 1, 2, 3, 4. 77. Explain why the triangles formed by 0, 1, and z1, and by 0, z2, and z1z2 shown in the figure are similar triangles. 78. Compass and Straightedge Construction Using only a compass and straightedge, construct the location of z1z2 given the location of 0, 1, z1, and z2. z1z2 y z2 z1 0 1 Component Form of a Vector 456 The Magnitude or Length of a Vector 458 Vector Addition and Scalar Multiplication 458 Unit Vector in the Direction of the Vector v 459 Dot Product of Two Vectors 467 Properties of the Dot Product 467 Theorem Angle Between Two Vectors 468 Projection of the Vector u onto the Vector v 470 79. x 3 – 1 = 0 80. x 4 – 1 = 0 81. x 3 + 1 = 0 82. x 4 + 1 = 0 83. x 5 + 1 = 0 84. x 5 – 1 = 0 Work 471 Coordinate Conversion Equations 488 Symmetry Tests for Polar Graphs 494 The Complex Plane 503 Modulus or Absolute Value of a Complex Number 504 Trigonometric Form of a Complex Number 504 De Moivre’s Theorem 507 Procedures Head Minus Tail Rule for Vectors 457 Resolving a Vector 460 Product and Quotient of Complex Numbers nth Root of a Complex Number 508 Gallery of Functions Rose Curves: r = a cos nu and r = a sin nu [–6,6] by [–4, 4] [–4.7, 4.7] by [–3.1, 3.1] r = 4 sin 3u r = 3 sin 4u Limaçon Curves: r = a ⫾ b sin u and r = a ⫾ b cos u with a 7 0 and b 7 0 Limaçon with an inner loop: a 6 1 b x In Exercises 79–84, find all solutions of the equation (real and complex). CHAPTER 6 Key Ideas Properties, Theorems, and Formulas 513 Key Ideas Cardioid: a = 1 b 505 514 CHAPTER 6 Applications of Trigonometry Dimpled limaçon: 1 6 a 6 2 b Convex limaçon: a Ú 2 b Lemniscate Curves: r 2 = a 2 sin 2 u Spiral of Archimedes: [–30, 30] by [–20, 20] [–4.7, 4.7] by [–3.1, 3.1] r = u, 0 … u … 45 r 2 = 4 cos 2u and r 2 = a 2 cos 2 u CHAPTER 6 Review Exercises Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 1–6, let u = 82, – 19, v = 84, 29, and w = 81, -39 be vectors. Find the indicated expression. 1. u – v 2. 2u – 3w 3. ƒ u + v ƒ 4. ƒ w – 2u ƒ 5. u # v 6. u # w In Exercises 7–10, let A = 12, – 12, B = 13, 12, C = 1 -4, 22, and D = 11, – 52. Find the component form and magnitude of the vector. ! ! ! 7. 3AB 8. AB + CD ! ! ! ! 9. AC + BD 10. CD + AB ! In Exercises 11 and 12, find (a) a unit vector in the direction of AB and (b) a vector of magnitude 3 in the opposite direction. 11. A = 14, 02, B = 12, 12 14. u = 8-2, 49, v = 86, 49 Use an algebraic method in Exercises 15–18 to convert the polar coordinates to rectangular coordinates. Approximate exact values with a calculator when appropriate. 16. 1 – 3.1, 135°2 In Exercises 19 and 20, polar coordinates of point P are given. Find all of its polar coordinates. 19. P = 1- 1, -2p/32 20. P = 1- 2, 5p/62 Use an algebraic method in Exercises 21–24 to find the polar coordinates of the given rectangular coordinates of point P that satisfy the stated conditions. Approximate exact values with a calculator when appropriate. (a) 0 … u … 2p 21. P = 12, -32 (b) -p … u … p 23. P = 15, 02 (c) 0 … u … 4p 22. P = 1-10, 02 24. P = 10, – 22 In Exercises 25–30, eliminate the parameter t and identify the graph. Support your answer with a grapher. 26. x = 4 + t, y = – 8 – 5t, – 3 … t … 5 In Exercises 13 and 14, find (a) the direction angles of u and v and (b) the angle between u and v. 15. 1 – 2.5, 25°2 18. 13.6, 3p/42 25. x = 3 – 5t, y = 4 + 3t 12. A = 13, 12, B = 15, 12 13. u = 84, 39, v = 82, 59 17. 12, – p/42 27. x = 2t 2 + 3, y = t – 1 28. x = 3 cos t, y = 3 sin t 29. x = e2t – 1, y = et 30. x = t 3, y = ln t, t 7 0 In Exercises 31 and 32, find a parametrization for the curve. 31. The line through the points 1 – 1, – 22 and 13, 42 32. The line segment with endpoints 1 – 2, 32 and 15, 12 CHAPTER 6 Review Exercises 515 Exercises 33 and 34 refer to the complex number z1 shown in the figure. Imaginary axis z1 4 (a) (b) (c) (d) Real axis –3 33. If z 1 = a + bi, find a, b, and ƒ z 1 ƒ . 53. r = 3 sin 4 u 54. r = 2 + sin u 34. Find the trigonometric form of z 1. 55. r = 2 + 2 sin u 56. r = 3 ƒ sin 3u ƒ 57. r = 2 – 2 sin u 58. r = 1 – 2 cos u 59. r = 3 cos 5u 60. r = 3 – 2 tan u In Exercises 35–38, write the complex number in standard form. 35. 61cos 30° + i sin 30°2 37. 2.5 a cos 36. 31cos 150° + i sin 150°2 4p 4p + i sin b 38. 41cos 2.5 + i sin 2.52 3 3 Use an algebraic method in Exercises 39–42 to find the trigonometric form with 0 … u 6 2p for the given complex number. Then write three other possible trigonometric forms for the number. Approximate exact values with a calculator when appropriate. 39. 3 – 3i 40. – 1 + i 12 41. 3 – 5i 42. – 2 – 2i In Exercises 43 and 44, write the complex numbers z 1 # z 2 and z 1/z 2 in trigonometric form. In Exercises 61–64, convert the polar equation to rectangular form and identify the graph. 61. r = – 2 62. r = – 2 sin u 63. r = – 3 cos u – 2 sin u 64. r = 3 sec u In Exercises 65–68, convert the rectangular equation to polar form. Graph the polar equation. 65. y = – 4 66. x = 5 67. 1x – 322 + 1y + 122 = 10 68. 2x – 3y = 4 In Exercises 69–72, analyze the graph of the polar curve. 43. z 1 = 31cos 30° + i sin 30°) and z 2 = 41cos 60° + i sin 60°2 69. r = 2 – 5 sin u 70. r = 4 – 4 cos u 44. z 1 = 51cos 20° + i sin 20°2 and z 2 = – 21cos 45° + i sin 45°2 71. r = 2 sin 3u 72. r 2 = 2 sin 2u, 0 … u … 2p In Exercises 45–48, use De Moivre’s Theorem to find the indicated power of the complex number. Write your answer in (a) trigonometric form and (b) standard form. p p 5 45. c 3 a cos + i sin b d 4 4 p 8 p + i sin b d 12 12 47. c 5 a cos 5p 5p 3 + i sin bd 3 3 74. Flight Engineering An airplane is flying on a bearing of 80° at 540 mph. A wind is blowing with the bearing 100° at 55 mph. In Exercises 49–52, find and graph the nth roots of the complex number for the specified value of n. 51. 1, n = 5 (b) Explain why r = b csc u is a polar form for the line y = b. (d) Illustrate the result in part (c) by graphing the line y = 2x + 3 using the polar form from part (c). p 6 p + i sin b d 48. c 7 a cos 24 24 n = 4 (a) Explain why r = a sec u is a polar form for the line x = a. (c) Let y = mx + b. Prove that b r = sin u – m cos u is a polar form for the line. What is the domain of r? 46. c 2 a cos 49. 3 + 3i, 73. Graphing Lines Using Polar Equations 50. 8, 52. – 1, n = 3 n = 6 In Exercises 53–60, decide whether the graph of the given polar equation appears among the four graphs shown. (a) Find the component form of the velocity of the airplane. (b) Find the actual speed and direction of the airplane. 75. Flight Engineering An airplane is flying on a bearing of 285° at 480 mph. A wind is blowing with the bearing 265° at 30 mph. (a) Find the component form of the velocity of the airplane. (b) Find the actual speed and direction of the airplane. CHAPTER 6 Applications of Trigonometry 76. Combining Forces A force of 120 lb acts on an object at an angle of 20°. A second force of 300 lb acts on the object at an angle of – 5°. Find the direction and magnitude of the resultant force. 77. Braking Force A 3000-pound car is parked on a street that makes an angle of 16° with the horizontal (see figure). (a) Graph simultaneously this epicycloid and the circle of radius 3. (b) Suppose the large circle has a radius of 4. Experiment! How do you think the equations in part (a) should be changed to obtain defining equations? What do you think the epicycloid would look like in this case? Check your guesses. (a) Find the force required to keep the car from rolling down the hill. y (b) Find the component of the force perpendicular to the street. 3 C 1 16° 78. Work Find the work done by a force F of 36 pounds acting in the direction given by the vector 83, 59 in moving an object 10 feet from (0, 0) to (10, 0). t –3 3 79. Height of an Arrow Stewart shoots an arrow straight up from the top of a building with initial velocity of 245 ft/sec. The arrow leaves from a point 200 ft above level ground. (d) How high is the arrow after 4 sec? (e) What is the maximum height of the arrow? When does it reach its maximum height? (f ) How long will it be before the arrow hits the ground? 80. Ferris Wheel Problem Lucinda is on a Ferris wheel of radius 35 ft that turns at the rate of one revolution every 20 sec. The lowest point of the Ferris wheel (6 o’clock) is 15 ft above ground level at the point (0, 15) of a rectangular coordinate system. Find parametric equations for the position of Lucinda as a function of time t in seconds if Lucinda starts 1t = 02 at the point (35, 50). 81. Ferris Wheel Problem The lowest point of a Ferris wheel (6 o’clock) of radius 40 ft is 10 ft above the ground, and the center is on the y-axis. Find parametric equations for Henry’s position as a function of time t in seconds if his starting position 1t = 02 is the point (0, 10) and the wheel turns at the rate of one revolution every 15 sec. 84. Throwing a Baseball Sharon releases a baseball 4 ft above the ground with an initial velocity of 66 ft/sec at an angle of 5° with the horizontal. How many seconds after the ball is thrown will it hit the ground? How far from Sharon will the ball be when it hits the ground? 85. Throwing a Baseball Diego releases a baseball 3.5 ft above the ground with an initial velocity of 66 ft/sec at an angle of 12° with the horizontal. How many seconds after the ball is thrown will it hit the ground? How far from Diego will the ball be when it hits the ground? 86. Field Goal Kicking Spencer practices kicking field goals 40 yd from a goal post with a crossbar 10 ft high. If he kicks the ball with an initial velocity of 70 ft/sec at a 45° angle with the horizontal (see figure), will Spencer make the field goal if the kick sails “true”? c (c) Use parametric equations to graph height against time. –3 se (b) Use parametric equations to simulate the height of the arrow. x ft/ (a) Write an equation that models the height of the arrow as a function of time t. P 70 516 45° 40 yd 82. Ferris Wheel Problem Sarah rides the Ferris wheel described in Exercise 81. Find parametric equations for Sarah’s position as a function of time t in seconds if her starting position 1t = 02 is the point (0, 90) and the wheel turns at the rate of one revolution every 18 sec. 83. Epicycloid The graph of the parametric equations x = 4 cos t – cos 4t, y = 4 sin t – sin 4t is an epicycloid. The graph is the path of a point P on a circle of radius 1 rolling along the outside of a circle of radius 3, as suggested in the figure. 87. Hang Time An NFL place-kicker kicks a football downfield with an initial velocity of 85 ft/sec. The ball leaves his foot at the 15 yard line at an angle of 56° with the horizontal. Determine the following: (a) The ball’s maximum height above the field. (b) The “hang time” (the total time the football is in the air). 88. Baseball Hitting Brian hits a baseball straight toward a 15-ft-high fence that is 400 ft from home plate. The ball is hit when it is 2.5 ft above the ground and leaves the bat at an angle of 30° with the horizontal. Find the initial velocity needed for the ball to clear the fence. CHAPTER 6 89. Throwing a Ball at a Ferris Wheel A 60-ft-radius Ferris wheel turns counterclockwise one revolution every 12 sec. Sam stands at a point 80 ft to the left of the bottom (6 o’clock) of the wheel. At the instant Kathy is at 3 o’clock, Sam throws a ball with an initial velocity of 100 ft/sec and an angle with the horizontal of 70°. He releases the ball from the same height as the bottom of the Ferris wheel. Find the minimum distance between the ball and Kathy. Review Exercises 517 90. Yard Darts Gretta and Lois are launching yard darts 20 ft from the front edge of a circular target of radius 18 in. If Gretta releases the dart 5 ft above the ground with an initial velocity of 20 ft/sec and at a 50° angle with the horizontal, will the dart hit the target? CHAPTER 6 Project Parametrizing Ellipses As you discovered in the Chapter 4 Data Project, it is possible to model the displacement of a swinging pendulum using a sinusoidal equation of the form x = a sin 1b1t – c22 + d where x represents the pendulum’s distance from a fixed point and t represents total elapsed time. In fact, a pendulum’s velocity behaves sinusoidally as well: y = ab cos 1b1t – c22, where y represents the pendulum’s velocity and a, b, and c are constants common to both the displacement and velocity equations. Use a motion detection device to collect distance, velocity, and time data for a pendulum, then determine how a resulting plot of velocity versus displacement (called a phase-space plot) can be modeled using parametric equations. Collecting the Data Construct a simple pendulum by fastening about 1 meter of string to the end of a ball. Collect time, distance, and velocity readings for between 2 and 4 seconds (enough time to capture at least one complete swing of the pendulum). Start the pendulum swinging in front of the detector, then activate the system. The data table below shows a sample set of data collected as a pendulum swung back and forth in front of a CBR where t is total elapsed time in seconds, d = distance from the CBR in meters, v = velocity in meters/second. t d v t d v t d v 0 0.1 0.2 0.3 0.4 0.5 0.6 1.021 1.038 1.023 0.977 0.903 0.815 0.715 0.325 0.013 – 0.309 – 0.598 – 0.819 – 0.996 – 0.979 0.7 0.8 0.9 1.0 1.1 1.2 1.3 0.621 0.544 0.493 0.473 0.484 0.526 0.596 – 0.869 – 0.654 – 0.359 – 0.044 0.263 0.573 0.822 1.4 1.5 1.6 1.7 1.8 1.9 2.0 0.687 0.785 0.880 0.954 1.008 1.030 1.020 0.966 1.013 0.826 0.678 0.378 0.049 –0.260 Explorations 1. Create a scatter plot for the data you collected or the data above. 2. With your calculator/computer in function mode, find values for a, b, c, and d so that the equation y = a sin 1b1x – c22 + d (where y is distance and x is time) fits the distance versus time data plot. 3. Make a scatter plot of velocity versus time. Using the same a, b, and c values you found in (2), verify that the equation y = ab cos 1b1x – c22 (where y is velocity and x is time) fits the velocity versus time data plot. 4. What do you think a plot of velocity versus distance (with velocity on the vertical axis and distance on the horizontal axis) would look like? Make a rough sketch of your prediction, then create a scatter plot of velocity versus distance. How well did your predicted graph match the actual data plot? 5. With your calculator/computer in parametric mode, graph the parametric curve x = a sin 1b1t – c22 + d, y = ab cos 1b1t – c22, 0 … t … 2, where x represents distance, y represents velocity, and t is the time parameter. How well does this curve match the scatter plot of velocity versus time? CHAPTER 6 89. Throwing a Ball at a Ferris Wheel A 60-ft-radius Ferris wheel turns counterclockwise one revolution every 12 sec. Sam stands at a point 80 ft to the left of the bottom (6 o’clock) of the wheel. At the instant Kathy is at 3 o’clock, Sam throws a ball with an initial velocity of 100 ft/sec and an angle with the horizontal of 70°. He releases the ball from the same height as the bottom of the Ferris wheel. Find the minimum distance between the ball and Kathy. Review Exercises 517 90. Yard Darts Gretta and Lois are launching yard darts 20 ft from the front edge of a circular target of radius 18 in. If Gretta releases the dart 5 ft above the ground with an initial velocity of 20 ft/sec and at a 50° angle with the horizontal, will the dart hit the target? CHAPTER 6 Project Parametrizing Ellipses As you discovered in the Chapter 4 Data Project, it is possible to model the displacement of a swinging pendulum using a sinusoidal equation of the form x = a sin 1b1t – c22 + d where x represents the pendulum’s distance from a fixed point and t represents total elapsed time. In fact, a pendulum’s velocity behaves sinusoidally as well: y = ab cos 1b1t – c22, where y represents the pendulum’s velocity and a, b, and c are constants common to both the displacement and velocity equations. Use a motion detection device to collect distance, velocity, and time data for a pendulum, then determine how a resulting plot of velocity versus displacement (called a phase-space plot) can be modeled using parametric equations. Collecting the Data Construct a simple pendulum by fastening about 1 meter of string to the end of a ball. Collect time, distance, and velocity readings for between 2 and 4 seconds (enough time to capture at least one complete swing of the pendulum). Start the pendulum swinging in front of the detector, then activate the system. The data table below shows a sample set of data collected as a pendulum swung back and forth in front of a CBR where t is total elapsed time in seconds, d = distance from the CBR in meters, v = velocity in meters/second. t d v t d v t d v 0 0.1 0.2 0.3 0.4 0.5 0.6 1.021 1.038 1.023 0.977 0.903 0.815 0.715 0.325 0.013 – 0.309 – 0.598 – 0.819 – 0.996 – 0.979 0.7 0.8 0.9 1.0 1.1 1.2 1.3 0.621 0.544 0.493 0.473 0.484 0.526 0.596 – 0.869 – 0.654 – 0.359 – 0.044 0.263 0.573 0.822 1.4 1.5 1.6 1.7 1.8 1.9 2.0 0.687 0.785 0.880 0.954 1.008 1.030 1.020 0.966 1.013 0.826 0.678 0.378 0.049 –0.260 Explorations 1. Create a scatter plot for the data you collected or the data above. 2. With your calculator/computer in function mode, find values for a, b, c, and d so that the equation y = a sin 1b1x – c22 + d (where y is distance and x is time) fits the distance versus time data plot. 3. Make a scatter plot of velocity versus time. Using the same a, b, and c values you found in (2), verify that the equation y = ab cos 1b1x – c22 (where y is velocity and x is time) fits the velocity versus time data plot. 4. What do you think a plot of velocity versus distance (with velocity on the vertical axis and distance on the horizontal axis) would look like? Make a rough sketch of your prediction, then create a scatter plot of velocity versus distance. How well did your predicted graph match the actual data plot? 5. With your calculator/computer in parametric mode, graph the parametric curve x = a sin 1b1t – c22 + d, y = ab cos 1b1t – c22, 0 … t … 2, where x represents distance, y represents velocity, and t is the time parameter. How well does this curve match the scatter plot of velocity versus time?

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