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Homework, Page 562 Let u 2, 1 , v 4, 2 , and w 1, 3 . Find the expression. 1. u v u v 2 4 , 1 2 2, 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 1 Homework, Page 562 Let u 2, 1 , v 4, 2 , and w 1, 3 . Find the expression. 5. uv u v 2 4 1 2 8 2 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 2 Homework, Page 562 Let A = (2, –1), B = (3, 1), C = (–4, 2), and D = (1, –5). Find the component form and magnitude of the vector. 9. AC BD AC 4 2 , 2 1 6,3 BD 1 3 , 5 1 2, 6 AC BD 6,3 2, 6 8, 3 AC BD 8 3 2 2 64 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 73 Slide 6- 3 Homework, Page 562 Find a the direction angles of u and v and b the angle between u and v. 13. u 4,3 , v 2,5 3 1 5 u tan 0.644 v tan 1.190 4 2 v u 1.190 0.644 0.546 OR 1 2 2 29 v 2 5 5 u v 4 2 3 5 23 u 4 3 2 cos 1 2 23 cos 0.5467 5 29 uv uv 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 4 Homework, Page 562 Convert the polar coordinates to rectangular coordinates. 2, 17. 4 2, 4 2 cos 4 ,sin 4 2 2 2 , 2 2 2 2, 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 5 Homework, Page 562 Rectangular coordinates of point P are given. Find all polar coordinates of P that satisfy: (a) 0 ≤θ ≤2π (b) –π ≤ θ ≤ π (c) 0 ≤ θ ≤ 4π 21. P 2, 3 3 r 2 3 13 tan 2 0.983 2.159 , 2 5.300 ,3 8.442 , 4 11.584 a P 13 cos 2.159,sin 2.159 , P 13 cos5.300,sin 5.300 2 2 1 b P 13 cos 0.983 ,sin 0.983 , P 13 cos 2.159,sin 2.159 c P 13 cos 2.159,sin 2.159 , P 13 cos5.300,sin 5.300 P 13 cos8.442,sin8.442 , P 13 cos11.584,sin11.584 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 6 Homework, Page 562 Eliminate the parameter t and identify the graph. 25. x 3 5t , y 4 3t 3 x x 3 5t , y 4 3t 5t 3 x t 5 3 x y 4 3 5 y 20 3 3 x 5 y 20 9 3x 5 29 3 3 y x The graph is a line with slope and 5 5 5 y -intercept 5.8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 7 Homework, Page 562 Eliminate the parameter t and identify the graph. x e 2t 1, y et 29. x e 1, y e x e 2t t t 2 2 x y 1 1 The graph is a parabola opening to the right. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 8 Homework, Page 562 Imaginary Axis Z1 4 Refer to the complex number shown in the figure. 33. If z1 = a + bi, find a, b, and |z1|. 2 2 25 5 3 4 a 3 , b 4 , z1 Real Axis -3 a 3, b 4, z1 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 9 Homework, Page 562 Write the complex number in standard form. 37. 4 4 2.5 cos i sin 3 3 4 4 5 1 5 2.5 cos i sin 3 3 2 2 2 3 i 2 5 5 3 i 4 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 10 Homework, Page 562 Write the complex number in trigonometric form where 0 ≤ θ ≤ 2π. Then write three other possible trigonometric forms for the number. 41. 3 5i r 3 5 2 2 5 34 tan 1.030 3 1 3 5i 34 cos 1.030 i sin 1.030 34 cos 2.111 i sin 2.111 34 cos 5.253 i sin 5.253 2 34 cos 4.172 i sin 4.172 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 11 Homework, Page 562 Use DeMoivre’s Theorem to find the indicated power of the complex number. Write the answer in (a) trigonometric form and (b) standard form. 45. 3 cos i sin 4 4 5 5 5 5 i sin 3 cos 4 i sin 4 243 cos 4 4 243 2 243 2 i 2 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 12 Homework, Page 562 Find and graph the nth roots of the complex number for the specified value of n. 49. 3 3i, n 4 3 r 3 3 3 2 tan 3 4 z 3 2 cos i sin 4 4 2 2 1 Continued on next slide. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 13 Homework, Page 562 49. Cont’d z1 4 3 z2 4 3 z3 4 3 z4 4 3 4 3 2 cos i sin 2 cos 4 i sin 4 16 16 4 4 2 2 4 9 9 3 2 cos i sin 2 cos 4 i sin 4 16 16 4 4 4 4 4 17 17 3 2 cos 16 i sin 16 2 cos 4 i sin 4 4 4 6 6 4 25 25 3 2 cos 16 i sin 16 2 cos 4 i sin 4 4 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 14 Homework, Page 562 49. Continued z1 4 18 cos i sin 16 16 9 9 z2 4 18 cos i sin 16 16 17 17 4 z3 18 cos i sin 16 16 25 25 4 z4 18 cos i sin 16 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley y x Slide 6- 15 Homework, Page 562 Decide whether the graph of the polar function appears among the four. 53. r 3sin 4 b. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 16 Homework, Page 562 Decide whether the graph of the polar function appears among the four. 57. r 2 2sin Not shown. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 17 Homework, Page 562 Convert the polar equation to rectangular form and identify the graph. r 2 2 r 2 r 2 2 x 2 y 2 4 The graph is a circle of radius 2 centered at the origin. 61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 18 Homework, Page 562 Convert the rectangular equation to polar form and graph the polar equation. y 4 65. y 4 r sin 4 4 r r 4csc sin Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 19 Homework, Page 562 Analyze the graph of the polar curve. 69. r 2 5sin Domain: All real numbers Range: –3 ≤ r ≤ 7 Continuity: Continuous Symmetry: Symmetric about the y-axis. Boundedness: Bounded Maximum r-value: 7 Asymptotes: None Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 20 Homework, Page 562 73. a Explain why r a sec is a polar form of x a. Multiply both sides of r a sec by cos and the equation becomes r cos a and r cos x. b Explain why r a csc is a polar form of y a. Multiply both sides of r a csc by sin and the equation becomes r sin a and r sin y. b is a polar form c Let y mx b. Prove that r sin m cos for the line. What is the domain of r ? Substituting y mx b becomes r sin mr cos b r sin mr cos b r sin m cos b b r sin m cos Domain : : m 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 21 Homework, Page 562 73. d Illustrate the result of part c by graphing the line y 2 x 3 using the polar form from part c . 3 y 2x 3 r sin 2cos Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 22 Homework, Page 562 77. A 3,000-lb car is parked on a street that makes an angle of 16º With the horizontal. (a) Find the force required to keep the car from rolling down the hill. F 3000 sin16 lb (b) Find the component of the force perpendicular to the ground. F 3000 cos16 lb Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 23 Homework, Page 562 81. The lowest point on a Ferris wheel of radius 40-ft is 10-ft above the ground, and the center is on the y-axis. Find the parametric equation for Henry’s position as a function of time t in seconds, if his starting position (t = 0) is the point (0, 10) and the wheel turns at a rate of one revolution every 15 sec. 2 2 2 15 b x 40sin bt p x 40sin t b 15 15 2 y 50 40cos bt y 50 40cos t 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 24 Homework, Page 562 85. Diego releases a baseball 3.5-ft above the ground with an initial velocity of 66-fps at an angle of 12º 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? x vo cos t , y 3.5 vo sin t 16t 2 x 66cos12 t , y 3.5 66sin12 t 16t 2 The ball hits the ground about 1.06 secs after it is thrown, 68.431 ft from Diego. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 25 Homework, Page 562 89. 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 fps and an angle of 70º with the horizontal. He releases the ball at the same height as the bottom of the Ferris wheel. Find the minimum distance between the ball and Kathy. xball 80 100cos 70 t 360 yball 100sin 70 t 16t 2 p 12 b 30 12 xKathy 60cos30t yKathy 60 60sin 30t distance x ball xKathy yball yKathy 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Slide 6- 26 Homework, Page 562 89. Continued Minimum distance between the ball and Kathy is about 17.654 feet Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 27