EXAMPLE 9 Finding Roots of Unity Find the eight eighth roots of unity.
GALLERY OF FUNCTIONS
Rose Curves:ra cos n and rasin n
r4 sin 3 r3 sin 4
Limaçon Curves:rab sin and rabcos with a0 and b0
Limaçon with an inner loop:a
b 1 Cardioid: a b 1 Dimpled limaçon: 1 a b 2 Convex limaçon: a b2
Spiral of Archimedes: Lemniscate Curves:r2a2sin 2 and r2a2cos 2
r, 0 45 r24 cos 2
[–4.7, 4.7] by [–3.1, 3.1] [–30, 30] by [–20, 20]
[–4.7, 4.7] by [–3.1, 3.1] [–6,6] by [–4, 4]
Head Minus Tail Rule for Vectors 503 Resolving a Vector 507
Product and Quotient of Complex Numbers 551 nth Root of a Complex Number 556
Component Form of a Vector 503
The Magnitude or Length of a Vector 504 Vector Addition and Scalar Multiplication 505 Unit Vector in the Direction of the Vector v 506 Dot Product of Two Vectors 514
Properties of the Dot Product 514
Theorem Angle Between Two Vectors 515 Projection of the Vector uonto the Vector v 517
Work 518
Coordinate Conversion Equations 535 Symmetry Tests for Polar Graphs 541 The Complex Plane 551
Modulus or Absolute Value of a Complex
Number 551
Trigonometric Form of a Complex Number 551 De Moivre’s Theorem 554
Exercises 33 and 34 refer to the complex number z1shown in the figure.
33.If z1abi, find a,b, and z1. a 3,b 4,z1 5
34.Find the trigonometric form of z1.
In Exercises 35 – 38, write the complex number in standard form.
35.6cos 30°isin 30° 36. 3cos 150°isin 150°
37.2.5
(
cos 4 3isin 4 3
)
38. 4cos 2.5isin 2.5 In Exercises 39 – 42, write the complex number in trigonometric form where 0 2. Then write three other possible trigonometric forms for the number.39.33i 40. 1i2
41.35i 42. 22i
In Exercises 43 and 44, write the complex numbers z1•z2and z1z2in
trigonometric form.
43.z13cos 30°isin 30°and z24cos 60°isin 60°
44.z15cos 20°isin 20°and z2 2cos 45°isin 45° 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.
45.
3(
cos 4 isin 4)
5 46. 2(
cos 1 2 isin 1 2)
8 47.5(
cos 5 3 isin 5 3)
3 48. 7(
cos 2 4 isin 2 4)
6 In Exercises 49 – 52, find and graph the nth roots of the complex num- ber for the specified value of n.49.33i, n4 50. 8, n3 51.1, n5 52. 1, n6 4 Imaginary axis Real axis –3 z1
CHAPTER 6
Review Exercises
The collection of exercises marked in redcould be used as a chapter test.
In Exercises 1 – 6, let u2,1,v4, 2, and w1,3be vectors. Find the indicated expression.
1.uv 2,3 2. 2u3w 1, 7
3.uv 37 4. w2u 10
5.u•v6 6. u•w 5
In Exercises 7–10, let A2,1,B3, 1,C4, 2, and D 1,5. Find the component form and magnitude of the vector.
7.3AB 3, 6; 35 8. ABCD6,5; 61 9.ACBD 8,3; 73 10. CDAB 4,9; 97
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.A4, 0,B2, 1
12.A3, 1,B5, 1 12. (a) 1, 0 (b) 3, 0
In Exercises 13 and 14, find (a)the direction angles of uand vand (b)
the angle between uand v.
13.u4, 3,v2, 5 14. u2, 4,v6, 4 In Exercises 15 – 18, convert the polar coordinates to rectangular coor- dinates.
15.2.5, 25° (2.27,1.06) 16. 3.1, 135° (1.552,1.552) 17.2,4 (2,2) 18. 3.6, 34 (1.82, 1.82)
In Exercises 19 and 20, polar coordinates of point Pare given. Find all of its polar coordinates.
19.P1,23 20. P2, 56
In Exercises 21 – 24, rectangular coordinates of point Pare given. Find polar coordinates of Pthat satisfy these conditions:
(a)0 2 (b) (c) 0 4
21.P2,3 22. P10, 0
23.P5, 0 24. P0,2
In Exercises 25 – 30, eliminate the parameter tand identify the graph.
25.x35t,y43t
26.x4t,y 85t,3 t 5
27.x2t23,yt1 28. x3 cos t,y3 sin t
29.xe2t1,yet 30. xt3,yln t,t0 In Exercises 31 and 32, find a parametrization for the curve.
31.The line through the points 1,2and 3, 4.
In Exercises 53 – 60, decide whether the graph of the given polar equa- tion appears among the four graphs shown.
53.r3 sin 4(b) 54. r2sin not shown 55.r22 sin (a) 56. r3sin 3 not shown 57.r22 sin not shown 58. r12 cos (d) 59.r3 cos 5(c) 60. r32 tan not shown
In Exercises 61– 64, convert the polar equation to rectangular form and identify the graph.
61.r 2 62. r 2 sin
63.r 3 cos 2 sin 64. r3 sec
In Exercises 65 – 68, convert the rectangular equation to polar form. Graph the polar equation.
65.y 4 66. x5
67.x32y1210 68. 2x3y4
In Exercises 69 –72, analyze the graph of the polar curve.
69.r25 sin 70. r44 cos
71.r2 sin 3 72. r22 sin 2, 0 2
73.Graphing Lines Using Polar Equations
(a)Explain why rasec is a polar form for the line xa.
(b)Explain why rbcsc is a polar form for the line yb.
(c)Let ymxb. Prove that r
sin b
mcos
is a polar form for the line. What is the domain of r?
(d)Illustrate the result in part (c) by graphing the line y2x3 using the polar form from part (c).
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.
(d) (c)
(b) (a)
(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.
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. 411.89 lb; 2.07°
77. Braking Force A 3000 pound car is parked on a street that makes an angle of 16° with the horizontal (see figure).
(a)Find the force required to keep the car from rolling down the hill. 826.91 pounds
(b)Find the component of the force perpendicular to the street.
78.Work Find the work done by a force Fof 36 pounds acting in the direction given by the vector 3, 5in moving an object 10 feet from 0, 0to 10, 0. 185.22 foot-pounds
79.Height of an Arrow Stewart shoots an arrow straight up from the top of a building with initial velocity of 245 ftsec. The arrow leaves from a point 200 ft above level ground.
(a)Write an equation that models the height of the arrow as a function of time t. h 16t2245t200
(b)Use parametric equations to simulate the height of the arrow.
(c)Use parametric equations to graph height against time.
(d)How high is the arrow after 4 sec? 924 ft
(e)What is the maximum height of the arrow? When does it reach its maximum height?1138 ft; t7.66
(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 low- est point of the Ferris wheel (6 o’clock) is 15 ft above ground level at the point 0, 15of a rectangular coordinate system. Find parametric equations for the position of Lucinda as a function of time tin seconds if Lucinda starts t0at 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 tin seconds if his starting position t0is the point 0, 10and the wheel turns at the rate of one revolution every 15 sec.
16° 2883.79 pounds x 40 sin
2 1 5 t,y 50 40 cos2 1 5 t82.Ferris Wheel Problem Sarah rides the Ferris wheel described in Exercise 81. Find parametric equations for Sarah’s position as a function of time tin seconds if her starting position t0is the point 0, 90and the wheel turns at the rate of one revolution every 18 sec.
83.Epicycloid The graph of the parametric equations x4 cos tcos 4t, y4 sin tsin 4t
is an epicycloid. The graph is the path of a point Pon a circle of radius 1 rolling along the outside of a circle of radius 3, as suggested in the 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.
84.Throwing a Baseball Sharon releases a baseball 4 ft above the ground with an initial velocity of 66 ftsec 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? t 0.71 sec,x 46.75 ft
85.Throwing a Baseball Diego releases a baseball 3.5 ft above the ground with an initial velocity of 66 ftsec 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? t 1.06 sec,x 68.65 ft
3 –3 y x –3 3 t C P 1
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 ftsec at a 45° angle with the horizontal (see figure), will Spencer make the field goal if the kick sails “true”?It clears the crossbar.
87.Hang Time An NFL place-kicker kicks a football downfield with an initial velocity of 85 ftsec. 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. 77.59 ft (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. just over 125 ft/sec
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 ftsec 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.17.65 ft
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 ftsec and at a 50° angle with the horizontal, will the dart hit the target? no
70 ft/sec
45°
40 yd
t d v t d v t d v 0 1.021 0.325 0.7 0.621 –0.869 1.4 0.687 0.966 0.1 1.038 0.013 0.8 0.544 –0.654 1.5 0.785 1.013 0.2 1.023 –0.309 0.9 0.493 –0.359 1.6 0.880 0.826 0.3 0.977 –0.598 1.0 0.473 –0.044 1.7 0.954 0.678 0.4 0.903 –0.819 1.1 0.484 0.263 1.8 1.008 0.378 0.5 0.815 –0.996 1.2 0.526 0.573 1.9 1.030 0.049 0.6 0.715 –0.979 1.3 0.596 0.822 2.0 1.020 –0.260 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
xa sin btcd
where xrepresents 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 abcos btc, where yrepresents the pendulum’s velocity and a,b, and care 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.