Deck 7: Vectors, the Complex Plane, and Polar Coordinates

Question 1

Convert the point to exact polar coordinates. Assume 0 ≤ θ ≤ 2π.

Answer 1

Question 2

Convert the point to exact polar coordinates. Assume 0 ≤ θ ≤ 2π.

Answer 2

Question 3

Convert the point to exact polar coordinates. Assume 0 ≤ θ ≤ 2π.

Answer 3

Question 4

Convert the point to exact polar coordinates. Assume 0 ≤ θ ≤ 2π.

Question 5

Convert the point to exact rectangular coordinates.

Question 6

Convert the point to exact rectangular coordinates.

Question 7

Convert the point to exact rectangular coordinates.

Question 8

Convert the point to exact rectangular coordinates.
(2, $\pi$ /2)

Question 9

Convert the point to exact rectangular coordinates.

Question 10

Convert the point to exact rectangular coordinates.

Question 11

Convert the point to exact rectangular coordinates.

Question 12

Graph the equation.
r = cos 2 θ

Question 13

Graph the equation.

Question 14

Graph the equation.
r² = 4 cos 5θ

Question 15

Graph the equation.

Question 16

Graph the equation.

Question 17

Spirals are seen in nature - in the swirl of a pine cone; they are also used in machinery to convert motions. An Archimedes Spiral has the general equation r = a θ. A more general form for the quation of a spiral is r = aθ^1/n where n is a constant that determines how tightly the spiral is wrapped. Compare the Archimedes Sprial r = θ with the spiral

by graphing both on the same polar graph.

Question 18

The lemniscate motion occurs naturally in the flapping of birds' wings. The bird's vertical lift and wing sweep create the distinctive figure-eight pattern. The patterns vary with the different wing profiles. Compare the two possible lemniscate patterns by graphing them on the same polar graph: r² = cos 2θ and r² = cos (2θ + 1).

Question 19

Many microphones advertise that their exceptional pickup capabilities hat isolate the sound source and minimize background noise. The name of these microphones comes from the pattern formed by the range of the pickup. Graph the cardioid curve to see what the range looks like.

Question 20

Convert the point to exact polar coordinates. Assume 0 $\le$ $\theta$ $\le$ 2 $\pi$ .

A)
B)
C)
D)

Question 21

Convert the point to exact polar coordinates. Assume 0 $\le$ $\theta$ $\le$ 2 $\pi$ .

A)
B)
C)
D)

Question 22

Convert the point to exact polar coordinates. Assume 0 $\le$ $\theta$ $\le$ 2 $\pi$ .

A)
B)
C)
D)

Question 23

Convert the point to exact polar coordinates. Assume 0 $\le$$\theta$ $\le$ 2 $\pi$ .

A) (4, 3 $\pi$ /2)
B) (3 $\pi$ /2, 4)
C) (-4, 3 $\pi$ /2)
D) (16, $\pi$ )

Question 24

Convert the point to exact rectangular coordinates.

A)
B)
C)
D)

Question 25

Convert the point to exact rectangular coordinates.

A)
B)
C)
D)

Question 26

Convert the point to exact rectangular coordinates.

A)
B)
C)
D)

Question 27

Convert the point to exact rectangular coordinates.
(1, $\pi$ )

A)
B)
C)
D)

Question 28

Convert the point to exact rectangular coordinates.

A)
B)
C)
D)

Question 29

Convert the point to exact rectangular coordinates.

A)
B)
C)
D)

Question 30

Convert the point to exact rectangular coordinates.

A)
B)
C)
D)

Question 31

Graph the equation.

A)
B)
C)
D)

Question 32

Graph the equation.

A)
B)
C)
D)

Question 33

Graph the equation.
R² = cos 5 $\theta$

A)
B)
C)
D)

Question 34

Graph the equation.

A)
B)
C)
D)

Question 35

Graph the equation.

A)
B)
C)
D)

Question 36

The lemniscate motion occurs naturally in the flapping of birds' wings. The bird's vertical lift and wing sweep create the distinctive figure-eight pattern. The patterns vary with the different wing profiles. Compare the two possbile lemniscate patterns by graphing them on the same polar graph: r² = cos 2 $\theta$ and r² = cos (2 $\theta$ + 2).

A)
B)
C)
D)

Question 37

Many microphones advertise that their exceptional pickup capabilities hat isolate the sound source and minimize background noise. The name of these microphones comes from the pattern formed by the range of the pickup. Graph the cardioid curve to see what the range looks like.

A)
B)
C)
D)

Question 38

Plot the point

in a polar coordinate system.

Question 39

Plot the point

in a polar coordinate system.

Question 40

Plot the point

in a polar coordinate system.

Question 41

Convert the equation

from polar to rectangular form.

Question 42

Convert the equation

from polar to rectangular form.

Question 43

Convert the equation

from polar to rectangular form.

Question 44

Find the product, z₁z₂ in rectangular form.
z₁ = 3 [ cos 225° + i sin 225° ] and z₂ = 16 [ cos 15° + i sin 15° ]

Question 45

Find the product, z₁z₂ in rectangular form.
z₁ = 2 [ cos 84° + i sin 84° ] and z₂ = 12 [ cos 66° + i sin 66° ]

Question 46

Find the product, z₁z₂ in rectangular form.
z₁ = 3 [ cos 197° + i sin 197° ] and z₂ = 10 [ cos 28° + i sin 28° ]

Question 47

Find the product, z₁z₂ in rectangular form.
and

Question 48

Find the product, z₁z₂ in rectangular form.
and

Question 49

Find the product, z₁z₂ in rectangular form.
and

Question 50

Find the quotient,

in rectangular form.
z₁ = 24 [ cos 78° + i sin 78° ] and z₂ = 4 [ cos 18° + i sin 18° ]

Question 51

Find the quotient,

in rectangular form.
z₁ = 104 [ cos 173° + i sin 173° ] and z₂ = 4 [ cos 23° + i sin 23° ]

Question 52

Find the quotient,

in rectangular form.
z₁ = 208 [ cos 180° + i sin 180° ] and z₂ = 8 [ cos 45° + i sin 45° ]

Question 53

Find the quotient,

in rectangular form.
and

Question 54

Find the quotient,

in rectangular form.
and

Question 55

Find the quotient,

in rectangular form.
and

Question 56

Find the result of the expression using De Moivre's theorem. Write the answer in rectangular form.

Question 57

Find the result of the expression using De Moivre's theorem. Write the answer in rectangular form.

Question 58

Find the result of the expression using De Moivre's theorem. Write the answer in rectangular form.

Question 59

Find the result of the expression using De Moivre's theorem. Write the answer in rectangular form.

Question 60

Find the result of the expression using De Moivre's theorem. Write the answer in rectangular form.

Question 61

Find all nth roots of z. Write answers in polar form and plot roots in complex plane.

Question 62

Find all nth roots of z. Write answers in polar form and plot roots in complex plane.

Question 63

Find all nth roots of z. Write answers in polar form and plot roots in complex plane.

Question 64

Find all nth roots of z. Write answers in polar form.

Question 65

Find all nth roots of z. Write answers in polar form.

Question 66

Find all nth roots of z. Write answers in polar form.

Question 67

Find all nth roots of z. Write answers in polar form.

Question 68

Find all nth roots of z. Write answers in polar form.

Question 69

Find all nth roots of z. Write answers in polar form.

Question 70

Find all nth roots of z. Write answers in polar form.

Question 71

Find all complex solutions to the equation.Write answers in rectangular form.

Question 72

Find all complex solutions to the equation.Write answers in rectangular form.

Question 73

Find all complex solutions to the equation.Write answers in rectangular form.

Question 74

Find all complex solutions to the equation.Write answers in rectangular form.

Question 75

Find all complex solutions to the equation.Write answers in rectangular form.

Question 76

Find all complex solutions to the equation.Write answers in rectangular form.

Question 77

Find all complex solutions to the equation.

Question 78

When you graph the five fifth roots of

and connect the points, you form a pentagon. Find the roots and draw the pentagon.

Question 79

Find the product, z₁z₂ in rectangular form.
Z₁ = 8 [ cos 23° + i sin 23° ] and z₂ = 4 [ cos 37° + i sin 37° ]

A)
B)
C)
D)

Question 80

Find the product, z₁z₂ in rectangular form.
Z₁ = 10 [ cos 306° + i sin 306° ] and z₂ = 18 [ cos 24° + i sin 24° ]

A)
B)
C)
D)