Homework, Page 562

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 1

Homework, Page 562

1.

Let 2, 1 , 4,2 , and w 1, 3 . Find the expression.u v ��

u v

u v

2 4 , 1 2 2, 3


Homework, Page 562

5.

Let 2, 1 , 4,2 , and w 1, 3 . Find the expression.u v ��

u v

u v 2 4 1 2 8 2 6


Homework, Page 562Let 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��

BD��

1 3 , 5 1 2, 6

AC BD��

6,3 2, 6 8, 3

AC BD�� 2 2

8 3 7364 9

4 2 , 2 1 6,3


Homework, Page 562

13. 4,3 , 2,5u v

Find a the direction angles of and and

b the angle between and .

u v

u v

1 3tan

4u 0.644 1 5

tan2v

1.190

v u 1.190 0.644 0.546 OR

u v 4 2 3 5 23 2 24 3u

5 2 22 5v

1 23cos

5 29

29

1cosu v

u v

0.5467


Homework, Page 562Convert the polar coordinates to rectangular coordinates.

17. 2, 4

2, 4 2 cos ,sin4 4

2 22 ,2

2 2

2, 2


Homework, Page 562Rectangular 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. 2, 3P

222 3r 1 3

tan2

0.983

2.159 ,2 5.300 ,3 8.442 ,4 11.584 13 cos2.159,sin 2.159a P , 13 cos5.300,sin5.300P

13 cos 0.983 ,sin 0.983b P , 13 cos2.159,sin 2.159P

13 cos2.159,sin 2.159c P , 13 cos5.300,sin5.300P

13 cos8.442,sin8.442P , 13 cos11.584,sin11.584P

13


Homework, Page 562Eliminate the parameter t and identify the graph.

25. 3 5 , 4 3x t y t

3 5 , 4 3x t y t 5 3t x 3

5

xt

34 3

5

xy

5 20 3 3y x 5 20 9 3y x

29 3

5 5y x

3 The graph is a line with slope and

5-intercept 5.8.y


Homework, Page 562Eliminate the parameter t and identify the graph.

29.2 1, t tx e y e

22 1, 1t t tx e y e x e 2 1x y

The graph is a parabola opening to the right.


Homework, Page 562Refer to the complex number shown in the figure.

33. If z1 = a + bi, find a, b, and |z1|.

1, z

Z1 4

-3

Imaginary Axis

Real Axis

3a , 4b 2 23 4 25 5

13, 4, 5a b z


Homework, Page 562Write the complex number in standard form.

37. 4 4

2.5 cos sin3 3

i

4 42.5 cos sin

3 3i

5 1 5 3

2 2 2 2i

5 5 3

4 4i


Homework, Page 562Write the complex number in trigonometric form where 0 ≤ θ ≤ 2π. Then write three other possible trigonometric forms for the number.

41. 3 5i

223 5r 1 5tan

3

1.030

3 5i 34 cos 1.030 sin 1.030i

34 cos 2.111 sin 2.111i

34 cos 5.253 sin 5.253 2i

34 cos 4.172 sin 4.172i

34


Homework, Page 562Use DeMoivre’s Theorem to find the indicated power of the complex number. Write the answer in (a) trigonometric form and (b) standard form.

45. 5

3 cos sin4 4

i

5

3 cos sin4 4

i

5 5243 cos sin

4 4i

243 2 243 2

2 2i


Homework, Page 562Find and graph the nth roots of the complex number for the specified value of n.

49.

Continued on next slide.

3 3 , 4i n

2 23 3r 3 2 1 3tan

3

4

3 2 cos sin4 4

z i


Homework, Page 56249. Cont’d

41

4 43 2 cos sin4 4

z i

4 3 2 cos sin16 16

i

42

2 24 43 2 cos sin4 4

z i

4 9 93 2 cos sin

16 16i

43

4 44 43 2 cos sin4 4

z i

4 17 173 2 cos sin

16 16i

44

6 64 43 2 cos sin4 4

z i

4 25 253 2 cos sin

16 16i


Homework, Page 56249. Continued

41 18 cos sin

16 16z i

x

y

x

y

x

y

x

y

x

y

42

9 918 cos sin

16 16z i

43

17 1718 cos sin

16 16z i

44

25 2518 cos sin

16 16z i


Homework, Page 562Decide whether the graph of the polar function appears among the four.

53.

b.

3sin 4r


Homework, Page 562Decide whether the graph of the polar function appears among the four.

57.

Not shown.

2 2sinr


Homework, Page 562Convert the polar equation to rectangular form and identify the graph.

61. 2r 2r 22 2r 2 2 4x y

The graph is a circle of radius 2 centered at the origin.


Homework, Page 562Convert the rectangular equation to polar form and graph the polar equation.

65. 4y

4y sin 4r

4

sinr

4cscr


Homework, Page 562Analyze the graph of the polar curve.

69.

Domain: All real numbers

Range: –3 ≤ r ≤ 7

Continuity: Continuous

Symmetry: Symmetric about the y-axis.

Boundedness: Bounded

Maximum r-value: 7

Asymptotes: None

2 5sinr


Homework, Page 56273. a Explain why sec is a polar form of .r a x a

sec cos

cos cos .

Multiply both sides of r a by and the equation

becomes r a and r x

b Explain why csc is a polar form of .r a y a csc sin

sin sin .

Multiply both sides of r a by and the equation

becomes r a and r y

c Let . Prove that is a polar formsin cos

for the line. What is the domain of ?

by mx b r

mr

sin cosSubstituting y mx b becomes r mr b

sin cosr mr b sin cosr m b

sin cos

br

m

: : 1Domain m


Homework, Page 56273.

d Illustrate the result of part c by graphing the line

2 3 using the polar form from part c .y x 3

2 3sin 2cos

y x r


Homework, Page 56277. 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.

(b) Find the component of the force perpendicular to the ground.

3000 sin16F

3000 cos16F

lb

lb


Homework, Page 56281. 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.

40sinx bt2

15pb

2

15b

2

40sin15

x t

50 40cosy bt 2

50 40cos15

y t


Homework, Page 56285. 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?

The ball hits the ground about

1.06 secs after it is thrown,

68.431 ft from Diego.

cos ,ox v t 23.5 sin 16oy v t t 66cos12 ,x t 23.5 66sin12 16y t t


Homework, Page 56289. 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. 80 100cos 70ballx t

2100sin 70 16bally t t

60cos30Kathyx t 60 60sin 30Kathyy t

2 2distance ball Kathy ball Kathyx x y y

36012 30

12p b


Homework, Page 56289. Continued

Minimum distance between the ball

and Kathy is about 17.654 feet

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Homework, Page 562