Deck 6: Additional Topics In Trigonometery

Question 1

Plot the complex number and find its absolute value.
-5 + 10i

A) -5 + 10i| = $\sqrt { 95 }$
B) |-5 + 10i| = $2 \sqrt { 7 }$
C) |-5 + 10i| = $5 \sqrt { 5 }$
D) |-5 + 10i| = $\sqrt { 22 }$
E) |-5 + 10i| = $\sqrt { 105 }$

Answer 1

|-5 + 10i| = $5 \sqrt { 5 }$

Question 2

Find the absolute value of the complex number -3 + 5i.

A) $2 \sqrt { 2 }$
B) $4 \sqrt { 34 }$
C) 34
D) $\sqrt { 34 }$
E) $4 \sqrt { 2 }$

Answer 2

$\sqrt { 34 }$

Question 3

Write the complex number in trigonometric form.

A) $Z = 3 ( \cos \pi - i \cos \pi )$
B) $Z = 3 ( \cos \pi + i \sin \pi )$
C) $Z = - 3 ( \cos \pi + i \sin \pi )$
D) $Z = 3 \left( \cos \frac { \pi } { 2 } - i \sin \frac { \pi } { 2 } \right)$
E) $Z = 3 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$

Answer 3

$Z = 3 ( \cos \pi + i \sin \pi )$

Question 4

Plot the complex number and find its absolute value.
-5 + 9i

A) |-5 + 9i| = $2 \sqrt { 19 }$
B) |-5 + 9i| = $2 \sqrt { 7 }$
C) |-5 + 9i| = $\sqrt { 106 }$
D) |-5 + 9i| = $\sqrt { 22 }$
E) |-5 + 9i| = $\sqrt { 86 }$

Question 5

Find the trigonometric form of the complex number represented graphically.

A)3 $\sqrt { 2 }$ (cos(3 $\pi$ /4) + isin(3 $\pi$ /4))
B)-3(cos(7 $\pi$ /4) + isin(7 $\pi$ /4))
C)3 $\sqrt { 2 }$ (cos(7 $\pi$ /4) + isin(7 $\pi$ /4))
D)-3 $\sqrt { 2 }$ (cos(3 $\pi$ /4) + isin(3 $\pi$ /4))
E) 3 $\sqrt { 2 }$ (cos(5 $\pi$ /4) + isin(5 $\pi$ /4))

Question 6

Write the complex number in trigonometric form.

A) $Z = 2 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
B) $Z = 2 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
C) $Z = 2 \left( \cos \frac { 7 \pi } { 2 } + i \sin \frac { 7 \pi } { 2 } \right)$
D) $Z = 2 \left( \cos \frac { \pi } { 2 } - i \sin \frac { \pi } { 2 } \right)$
E) $Z = 2 \left( \cos \frac { 3 \pi } { 2 } - i \sin \frac { 3 \pi } { 2 } \right)$

Question 7

Plot the following complex number. -3

A)
B)
C)
D)
E) ​

Question 8

Plot the complex number and find its absolute value.
-3 + 9i

A) $| - 3 + 9 i | = \sqrt { 76 }$
B) $| - 3 + 9 i | = \sqrt { 12 }$
C) $| - 3 + 9 i | = \sqrt { 90 }$
D) $| - 3 + 9 i | = \sqrt { 2 }$
E) $| - 3 + 9 i | = \sqrt { 86 }$

Question 9

Represent the complex number graphically, and find the trigonometric form of the number.
5 + 5i

A) $z = \sqrt { 5 } \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$
B) $z = \sqrt { 50 } \left( \cos \frac { \pi } { 4 } - i \sin \frac { \pi } { 4 } \right)$
C) $z = \sqrt { 50 } \left( \cos \frac { \pi } { 4 } + \sin \frac { \pi } { 4 } \right)$
D) $z = \sqrt { 5 } \left( \cos \frac { \pi } { 4 } - i \sin \frac { \pi } { 4 } \right)$
E) $z = \sqrt { 50 } \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$

Question 10

Represent the complex number graphically, and find the trigonometric form of the number.
5 - 5i

A) $5 \sqrt { 2 } \left( \cos \frac { 7 \pi } { 4 } + i \sin \frac { 7 \pi } { 4 } \right)$
B) $5 \sqrt { 2 } \left( \cos \frac { 7 \pi } { 4 } + i \sin \frac { 7 \pi } { 4 } \right)$
C) $\sqrt { 50 } \left( \cos \frac { \pi } { 4 } + \sin \frac { \pi } { 4 } \right)$
D) $\sqrt { 50 } \left( \cos \frac { \pi } { 4 } - i \sin \frac { \pi } { 4 } \right)$
E) $\sqrt { 2 } \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$

Question 11

Find the trigonometric form of $- 14 \sqrt { 3 } + 14 i$ .

A) $14 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)$
B) $196 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$
C) $14 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$
D) $28 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$
E) $28 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)$

Question 12

Plot the complex number and find its absolute value.
-5 + 2i

A) $| - 5 + 2 i | = \sqrt { - 1 }$
B) $| - 5 + 2 i | = \sqrt { 28 }$
C) $| - 5 + 2 i | = \sqrt { 29 }$
D) $| - 5 + 2 i | = \sqrt { 2 }$
E) $| - 5 + 2 i | = \sqrt { 9 }$

Question 13

Find the absolute value of the complex number 5 + 3i.

A) $\sqrt { 34 }$
B) 34
C) $2 \sqrt { 34 }$
D) 5 + 3 i
E) $2 \sqrt { 2 }$

Question 14

Plot the complex number and find its absolute value. -2

A) $| - 2 | = - 3$
B) $| - 2 | = 2$
C) $| - 2 | = - 4$
D) $| - 2 | = 4$
E) $| - 2 | = 3$

Question 15

Write the complex number shown below in trigonometric form.

A) $6 \left( \cos \frac { 4 \pi } { 3 } + i \sin \frac { 4 \pi } { 3 } \right)$
B) $6 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$
C) $6 \left( \cos \frac { 5 \pi } { 3 } + i \sin \frac { 5 \pi } { 3 } \right)$
D) $6 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$
E) $6 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)$

Question 16

Write the complex number in trigonometric form.

A) $z = \sqrt { 2 } \left( \cos \frac { 5 \pi } { 4 } - i \sin \frac { 5 \pi } { 4 } \right)$
B) $Z = 6 ( \cos \pi + i \sin \pi )$
C) $Z = 6 \left( \cos \frac { \pi } { 2 } - i \sin \frac { \pi } { 2 } \right)$
D) $Z = 6 \sqrt { 2 } \left( \cos \frac { 5 \pi } { 4 } - i \sin \frac { 5 \pi } { 4 } \right)$
E) $Z = 6 \sqrt { 2 } \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$

Question 17

Find the trigonometric form of - 21 - 21i.

A) $\frac { 21 } { 2 } \left( \cos \frac { 5 \pi } { 2 } + i \sin \frac { 5 \pi } { 2 } \right)$
B) $21 \left( \cos \frac { 5 \pi } { 2 } + i \sin \frac { 5 \pi } { 2 } \right)$
C) $21 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$
D) $21 \sqrt { 2 } \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$
E) $\frac { 441 } { 4 } \left( \cos \frac { 5 \pi } { 8 } + i \sin \frac { 5 \pi } { 8 } \right)$

Question 18

Plot the complex number 2. ​

A)
B) ​
C)
D)
E)

Question 19

Plot the complex number and find its absolute value.
-4i

A) $| - 4 i | = - 5$
B) $| - 4 i | = 6$
C) $| - 4 i | = - 6$
D) $| - 4 i | = 5$
E) $| - 4 i | = 4$

Question 20

Find the trigonometric form of $- 5 + 5 \sqrt { 3 } i$ .

A) $5 \sqrt { 2 } \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$
B) $5 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$
C) $5 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$
D) $10 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$
E) $10 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$

Question 21

Find the standard form of the complex number $6 \left( \cos 17 ^ { \circ } 26 ^ { \prime } + i \sin 17 ^ { \circ } 26 ^ { \prime } \right)$ .Round values to the nearest hundredth.

A)5.99 + 0.78i
B) 5.72 + 1.8i
C) 4.06 + 2.76i
D) -4.06 - 2.76i
E) -5.72 - 1.8i

Question 22

Find the standard form of the complex number shown below. $3 \left( \cos \frac { 7 \pi } { 6 } + i \sin \frac { 7 \pi } { 6 } \right)$

A) $- \frac { 3 \sqrt { 3 } } { 4 } - \frac { 3 } { 4 } i$
B) $- \frac { 2 } { 3 } - \frac { 2 \sqrt { 3 } } { 3 } i$
C) $- \frac { 4 \sqrt { 3 } } { 3 } - \frac { 4 } { 3 } i$
D) $- \frac { 3 \sqrt { 3 } } { 2 } - \frac { 3 } { 2 } i$
E) $- \frac { 3 } { 2 } - \frac { 3 \sqrt { 3 } } { 2 } i$

Question 23

Find the standard form of the complex number.Then represent the complex number graphically. $6 ( \cos 0 + i \sin 0 )$

A) 6
B) 6
C) -6
D) -6
E) 6

Question 24

Represent the complex number below graphically. -3+ 3i

A)
B)​
C)​
D)
E)​

Question 25

Find the standard form of the complex number.Then represent the complex number graphically. $7 ( \cos 0 + i \sin 0 )$

A) 7
B) 7
C) -7
D) -7
E)None of the above

Question 26

Find the standard form of the complex number $8 \left( \cos 330 ^ { \circ } \right) + i \sin 330 ^ { \circ }$ .

A) 8 $\sqrt { 3 }$ - 8i
B) -4 $\sqrt { 3 }$ + 4i
C) 4 $\sqrt { 3 }$ - 4i
D) -8 $\sqrt { 3 }$ + 8i
E) 4 + 4 $\sqrt { 3 }$ i

Question 27

Find the standard form of the complex number $10 \left( \cos \frac { \pi } { 12 } + i \sin \frac { \pi } { 12 } \right)$ .Round values to the nearest hundredth.

A)9.66 - 2.59i
B) 8.66 + 5i
C) -9.66 - 2.59i
D) -8.66 - 5i
E) 9.66 + 2.59i

Question 28

Find the trigonometric form of the complex number shown below.

A) $6 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$
B) $6 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
C) $6 ( \cos 0 + i \sin 0 )$
D) $6 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
E) $6 ( \cos \pi + i \sin \pi )$

Question 29

Find the trigonometric form of the complex number shown below. 10i

A) $10 ( \cos \pi + i \sin \pi )$
B) $10 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
C) $10 ( \cos 0 + i \sin 0 )$
D) $10 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$
E) $10 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$

Question 30

Find the standard form of the complex number shown below.Round answers to two decimal places. $2 \left( \cos 230 ^ { \circ } + i \sin 230 ^ { \circ } \right)$

A) -1.29 - 1.53i
B) -2.53 - 2.88i
C) -2.04 - 2.63i
D) -1.52 - 1.92i
E) -2.71 - 3.06i

Question 31

Represent the complex number graphically, and find the trigonometric form of the number.
$6 + 6 i$

A) $6 \sqrt { 2 }$ $\left( \cos \frac { \pi } { 4 } + \sin \frac { \pi } { 4 } \right)$
B) $\sqrt { 6 }$ $\left( \cos \frac { \pi } { 4 } - i \sin \frac { \pi } { 4 } \right)$
C) $\sqrt { 6 }$ $\left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$
D) $6 \sqrt { 2 }$ $\left( \cos \frac { \pi } { 4 } - i \sin \frac { \pi } { 4 } \right)$
E) $6 \sqrt { 2 }$ $\left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$

Question 32

Represent the complex number graphically, and find the trigonometric form of the number.
4

A) $4 ( \cos 0 + i \sin 0 )$
B) $4 ( \cos 4 + i \sin 4 )$
C) $4 ( \cos 4 - i \sin 4 )$
D) $4 ( \cos 0 - i \sin 0 )$
E) $4 ( \cos 4 - i \sin 4 )$

Question 33

Represent the complex number graphically, and find the trigonometric form of the number.
12i

A) $12 ( \cos \pi + i \sin \pi )$
B) $12 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
C) $12 ( \cos \pi + i \sin \pi )$
D) $12 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
E) $12 ( \cos \pi + i \sin \pi )$

Question 34

Find the trigonometric form of the complex number shown below.
-7 - i

A) $5 \sqrt { 2 } \left( \cos \left( \arctan \left( \frac { 2 } { 21 } \right) \right) + i \sin \left( \arctan \left( \frac { 2 } { 21 } \right) \right) \right)$
B) $5 \sqrt { 2 } \left( \cos \left( \arctan \left( - \frac { 3 } { 7 } \right) \right) + i \sin \left( \arctan \left( - \frac { 3 } { 7 } \right) \right) \right)$
C) $5 \sqrt { 2 } \left( \cos \left( \arctan \left( - \frac { 3 } { 35 } \right) \right) + i \sin \left( \arctan \left( - \frac { 3 } { 35 } \right) \right) \right)$
D) $5 \sqrt { 2 } \left( \cos \left( \arctan \left( \frac { 1 } { 7 } \right) \right) + i \sin \left( \arctan \left( \frac { 1 } { 7 } \right) \right) \right)$
E) $5 \sqrt { 2 } \left( \cos \left( \arctan \left( \frac { 1 } { 14 } \right) \right) + i \sin \left( \arctan \left( \frac { 1 } { 14 } \right) \right) \right)$

Question 35

Represent the complex number graphically.

A) $Z = 2 ( \cos 0 - i \sin 0 )$
B) $Z = 2 ( \cos 0 + i \sin 0 )$
C) $Z = 2 ( \cos 0 - i \sin 0 )$
D) $Z = 2 ( \cos 0 + i \sin 0 )$
E) $Z = 2 ( \cos 0 - i \sin 0 )$

Question 36

Represent the following complex number graphically.
$$3 \left( \cos 160 ^ { \circ } + i \sin 160 ^ { \circ } \right)$$

A)
B)
C)
D)
E)

Question 37

Find the trigonometric form of the complex number shown below. -4 + 4 $\sqrt { 3 }$ i

A)4 $\left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$
B)2 $\left( \cos \frac { 4 \pi } { 3 } + i \sin \frac { 4 \pi } { 3 } \right)$
C)8 $\left( \cos \frac { 2 \pi } { 6 } + i \sin \frac { 2 \pi } { 6 } \right)$
D)4 $\left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$
E)8 $\left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$

Question 38

Represent the complex number graphically, and find the trigonometric form of the number.
-8i

A) $8 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
B) $8 ( \cos \pi - i \sin \pi )$
C) $8 \left( \cos \frac { 3 \pi } { 2 } - i \sin \frac { 3 \pi } { 2 } \right)$
D) $8 ( \cos \pi + i \sin \pi )$
E) $8 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$

Question 39

Represent the complex number below graphically. 2 $\sqrt { 2 }$ $\left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$

A)
B)
C)
D)
E)

Question 40

Find the trigonometric form of the complex number shown below.
-8

A) $8 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
B) $8 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$
C) $8 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
D) $8 ( \cos 0 + i \sin 0 )$
E) $8 ( \cos \pi + i \sin \pi )$

Question 41

Perform the operation shown below and leave the result in trigonometric form. $\left[ 3 \left( \cos 100 ^ { \circ } + i \sin 100 ^ { \circ } \right) \right] \left[ 6 \left( \cos 230 ^ { \circ } + i \sin 230 ^ { \circ } \right) \right]$

A) $18 \left( \cos 230 ^ { \circ } + i \sin 230 ^ { \circ } \right)$
B) $18 \left( \cos 130 ^ { \circ } + i \sin 130 ^ { \circ } \right)$
C) $18 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$
D) $9 \left( \cos 330 ^ { \circ } + \sin 330 ^ { \circ } \right)$
E) $9 \left( \cos 130 ^ { \circ } + i \sin 130 ^ { \circ } \right)$

Question 42

Perform the operation and leave the result in trigonometric form. $\frac { 5 \left( \cos 4.5 ^ { \circ } + i \sin 4.5 ^ { \circ } \right) } { 4 \left( \cos 2.4 ^ { \circ } + i \sin 2.4 ^ { \circ } \right) }$

A) $\frac { 5 } { 4 } \left( \cos 4.5 ^ { \circ } + i \sin 4.5 ^ { \circ } \right)$
B) $\frac { 5 } { 4 } \left( \cos 2.1 ^ { \circ } - i \sin 2.1 ^ { \circ } \right)$
C) $\frac { 5 } { 4 } \left( \cos 2.1 ^ { \circ } + i \sin 2.1 ^ { \circ } \right)$
D) $\frac { 5 } { 4 } \left( \cos 4.5 ^ { \circ } - i \sin 4.5 ^ { \circ } \right)$
E) $\frac { 5 } { 4 } \left( \cos 2.4 ^ { \circ } + i \sin 2.4 ^ { \circ } \right)$

Question 43

Perform the operation and leave the result in trigonometric form. $\frac { \left( \cos 110 ^ { \circ } + i \sin 110 ^ { \circ } \right) } { 2 \left( \cos 50 ^ { \circ } + i \sin 50 ^ { \circ } \right) }$

A) $\frac { 1 } { 2 } \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$
B) $\frac { 1 } { 2 } \left( \cos 60 ^ { \circ } - i \sin 60 ^ { \circ } \right)$
C) $\frac { 1 } { 2 } \left( \cos 110 ^ { \circ } + i \sin 110 ^ { \circ } \right)$
D) $\frac { 1 } { 2 } \left( \cos 110 ^ { \circ } - i \sin 110 ^ { \circ } \right)$
E) $\frac { 1 } { 2 } \left( \cos 50 ^ { \circ } + i \sin 50 ^ { \circ } \right)$

Question 44

Perform the operation shown below and leave the result in trigonometric form. $\left[ 4 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) \right] \left[ 3 \left( \cos \frac { 3 \pi } { 7 } + i \sin \frac { 3 \pi } { 7 } \right) \right]$

A) $7 \left( \cos \frac { 23 \pi } { 21 } + i \sin \frac { 23 \pi } { 21 } \right)$
B) $12 \left( \cos \frac { 23 \pi } { 21 } + i \sin \frac { 23 \pi } { 21 } \right)$
C) $12 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$
D) $12 \left( \cos \frac { 2 \pi } { 7 } + i \sin \frac { 2 \pi } { 7 } \right)$
E) $7 \left( \cos \frac { 2 \pi } { 7 } + i \sin \frac { 2 \pi } { 7 } \right)$

Question 45

Use a graphing utility to represent the complex number in standard form.Round your answer to 4 decimal places. $5 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$

A)4.3301 + 2.5i
B)2.5 + 4.3301i
C)2.5 - 4.3301i
D)4.3301 - 2.5i
E)None of the above

Question 46

Divide the complex numbers below and leave the result in trigonometric form. $\frac { 4 ( \cos ( 7 \pi / 10 ) + i \sin ( 7 \pi / 10 ) ) } { 10 ( \cos ( 3 \pi / 7 ) + i \sin ( 3 \pi / 7 ) ) }$

A) $\cos \frac { 79 } { 70 } \pi + i \sin \frac { 79 } { 70 } \pi$
B) $\cos \frac { 19 } { 70 } \pi + i \sin \frac { 19 } { 70 } \pi$
C) $\frac { 2 } { 5 } \left( \cos \frac { 79 } { 70 } \pi + i \sin \frac { 79 } { 70 } \pi \right)$
D) $\frac { 2 } { 5 } \left( \cos \frac { 2 } { 35 } \pi + i \sin \frac { 2 } { 35 } \pi \right)$
E) $\frac { 2 } { 5 } \left( \cos \frac { 19 } { 70 } \pi + i \sin \frac { 19 } { 70 } \pi \right)$

Question 47

Divide the complex numbers below and leave the result in trigonometric form. $\frac { 10 \left( \cos 220 ^ { \circ } + i \sin 220 ^ { \circ } \right) } { 7 \left( \cos 210 ^ { \circ } + i \sin 210 ^ { \circ } \right) }$

A) $70 \left( \cos 10 ^ { \circ } + i \sin 10 ^ { \circ } \right)$
B) $\frac { 10 } { 7 } \left( \cos 10 ^ { \circ } + i \sin 70 ^ { \circ } \right)$
C) $\frac { 10 } { 7 } \left( \cos 70 ^ { \circ } + i \sin 70 ^ { \circ } \right)$
D) $\frac { 10 } { 7 } \left( \cos 10 ^ { \circ } + i \sin 10 ^ { \circ } \right)$
E) $70 \left( \cos 70 ^ { \circ } + i \sin 70 ^ { \circ } \right)$

Question 48

Perform the operation and leave the result in trigonometric form. $\left( \cos 80 ^ { \circ } + i \sin 80 ^ { \circ } \right) \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$

A) $\cos 50 ^ { \circ } - i \sin 50 ^ { \circ }$
B) $\frac { 1 } { 2 } \left( \cos 50 ^ { \circ } - i \sin 50 ^ { \circ } \right)$
C) $\frac { 1 } { 2 } \left( \cos 80 ^ { \circ } + i \sin 80 ^ { \circ } \right)$
D) $\frac { 1 } { 2 } \left( \cos 50 ^ { \circ } + i \sin 50 ^ { \circ } \right)$
E) $\cos 50 ^ { \circ } + i \sin 50 ^ { \circ }$

Question 49

Multiply the complex numbers below and leave the result in trigonometric form. $\left[ 8 \left( \cos \frac { 2 \pi } { 7 } + i \sin \frac { 2 \pi } { 7 } \right) \right] \left[ 10 \left( \cos \frac { 5 \pi } { 9 } + i \sin \frac { 5 \pi } { 9 } \right) \right]$

A) $18 \left( \cos \frac { 53 } { 63 } \pi + i \sin \frac { 53 } { 63 } \pi \right)$
B) $80 \left( \cos \frac { 23 } { 63 } \pi + i \sin \frac { 23 } { 63 } \pi \right)$
C) $80 \left( \cos \frac { 53 } { 63 } \pi + i \sin \frac { 53 } { 63 } \pi \right)$
D) $18 \left( \cos \frac { 1 } { 9 } \pi + i \sin \frac { 1 } { 9 } \pi \right)$
E) $80 \left( \cos \frac { 1 } { 9 } \pi + i \sin \frac { 1 } { 9 } \pi \right)$

Question 50

Use a graphing utility to represent the complex number in standard form.Round your answer to 4 decimal places. $14 \left( \cos \frac { 2 \pi } { 5 } + i \sin \frac { 2 \pi } { 5 } \right)$

A)4.3262 + 13.3148i
B)13.3148 + 4.3262i
C)13.3148 - 4.3262i
D)4.3262 - 13.3148i
E)None of the above

Question 51

Perform the operation and leave the result in trigonometric form. $\frac { 16 \left( \cos 98 ^ { \circ } + i \sin 98 ^ { \circ } \right) } { 2 \left( \cos 128 ^ { \circ } + i \sin 128 ^ { \circ } \right) }$

A) $8 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$
B) $8 \left( \cos 98 ^ { \circ } - i \sin 98 ^ { \circ } \right)$
C) $8 \left( \cos 128 ^ { \circ } + i \sin 128 ^ { \circ } \right)$
D) $8 \left( \cos 330 ^ { \circ } - i \sin 330 ^ { \circ } \right)$
E) $8 \left( \cos 98 ^ { \circ } + i \sin 98 ^ { \circ } \right)$

Question 52

Multiply the complex numbers below and leave the result in trigonometric form. $\left[ 0.1 \left( \cos 260 ^ { \circ } + i \sin 260 ^ { \circ } \right) \right] \left[ 0.9 \left( \cos 220 ^ { \circ } + i \sin 220 ^ { \circ } \right) \right]$

A) $0.09 \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right)$
B) $0.09 \left( \cos 120 ^ { \circ } + i \sin 40 ^ { \circ } \right)$
C) $\cos 40 ^ { \circ } + i \sin 40 ^ { \circ }$
D) $0.09 \left( \cos 40 ^ { \circ } + i \sin 40 ^ { \circ } \right)$
E) $\cos 120 ^ { \circ } + i \sin 120 ^ { \circ }$

Question 53

Use a graphing utility to represent the complex number in standard form.Round your answer to 4 decimal places. $2 \left( \cos 154 ^ { \circ } + i \sin 154 ^ { \circ } \right)$

A)0.8767-1.7976i
B)-1.7976 - 0.8767i
C) -1.7976 + 0.8767i
D) -0.8767-1.7976i
E)None of the above

Question 54

Perform the operation and leave the result in trigonometric form. $\frac { \cos \pi + i \sin \pi } { \cos ( \pi / 9 ) + i \sin ( \pi / 9 ) }$

A) $\cos \frac { 9 \pi } { 8 } - i \sin \frac { 9 \pi } { 8 }$
B) $\cos \frac { 9 \pi } { 8 } - \sin \frac { 9 \pi } { 8 }$
C) $\cos \frac { 8 \pi } { 9 } + i \sin \frac { 8 \pi } { 9 }$
D) $\cos \frac { 8 \pi } { 9 } - i \sin \frac { 8 \pi } { 9 }$
E) $\cos \frac { 9 \pi } { 8 } + i \sin \frac { 9 \pi } { 8 }$

Question 55

Divide the complex numbers below and leave the result in trigonometric form. $\frac { 5 \left( \cos 100 ^ { \circ } + i \sin 100 ^ { \circ } \right) } { 9 \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right) }$

A) $\frac { 5 } { 9 } \left( \cos 220 ^ { \circ } + i \sin 220 ^ { \circ } \right)$
B) $\frac { 5 } { 9 } \left( \cos 340 ^ { \circ } + i \sin 340 ^ { \circ } \right)$
C) $45 \left( \cos 220 ^ { \circ } + i \sin 220 ^ { \circ } \right)$
D) $\frac { 5 } { 9 } \left( \cos 340 ^ { \circ } + i \sin 220 ^ { \circ } \right)$
E) $45 \left( \cos 340 ^ { \circ } + i \sin 340 ^ { \circ } \right)$

Question 56

Multiply the complex numbers.Write the answer in trigonometric form. $9 \left( \cos \frac { 7 \pi } { 6 } + i \sin \frac { 7 \pi } { 6 } \right) 4 \left( \cos \frac { 3 \pi } { 7 } + i \sin \frac { 3 \pi } { 7 } \right)$

A) $36 \left( \cos \frac { 67 \pi } { 42 } + i \sin \frac { 67 \pi } { 42 } \right)$
B) $36 \left( \cos \frac { 10 \pi } { 13 } + i \sin \frac { 10 \pi } { 13 } \right)$
C) $36 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
D) $13 \left( \cos \frac { 67 \pi } { 42 } + i \sin \frac { 67 \pi } { 42 } \right)$
E) $13 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$

Question 57

Use a graphing utility to represent the complex number in standard form.Round your answer to 4 decimal places. $9 \left( \cos 62 ^ { \circ } + i \sin 62 ^ { \circ } \right)$

A)7.9465 + 4.2252i
B)4.2252 + 7.9465i
C)4.2252 - 7.9465i
D)7.9465 - 4.2252i
E)None of the above

Question 58

Perform the operation and leave the result in trigonometric form. $\left( \cos 7 ^ { \circ } + i \sin 7 ^ { \circ } \right) \left( \cos 24 ^ { \circ } + i \sin 24 ^ { \circ } \right)$

A)cos 17° - isin 17°
B)cos 31° - isin 31°
C)cos 17° + isin 17°
D)cos 31° + icos 31°
E)cos 31° + isin 31°

Question 59

Perform the operation shown below and leave the result in trigonometric form.
$\frac { 12 ( \cos 3.1 + i \sin 3.1 ) } { 6 ( \cos 2.7 + i \sin 2.7 ) }$

A) $6 ( \cos 0.4 + i \sin 0.4 )$
B) $6 ( \cos 1.1 + i \sin 1.1 )$
C) $2 ( \cos 0.4 + i \sin 0.4 )$
D) $2 ( \cos 8.37 + i \sin 8.37 )$
E) $2 ( \cos 5.8 + i \sin 5.8 )$

Question 60

Perform the operation and leave the result in trigonometric form. $\frac { 2 \left( \cos 70 ^ { \circ } + i \sin 70 ^ { \circ } \right) } { 4 \left( \cos 40 ^ { \circ } + i \sin 40 ^ { \circ } \right) }$

A) $\frac { 1 } { 2 } \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$
B) $\frac { 1 } { 2 } \left( \cos 40 ^ { \circ } + i \sin 40 ^ { \circ } \right)$
C) $\frac { 1 } { 2 } \left( \cos 30 ^ { \circ } - i \sin 30 ^ { \circ } \right)$
D) $\frac { 1 } { 2 } \left( \cos 70 ^ { \circ } - i \sin 70 ^ { \circ } \right)$
E) $\frac { 1 } { 2 } \left( \cos 70 ^ { \circ } + i \sin 70 ^ { \circ } \right)$

Question 61

Find the trigonometric form of $- 9 + 9 i$ .

A) $9 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$
B) $\frac { 9 } { 2 } \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
C) $9 \sqrt { 2 } \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
D) $9 \sqrt { 2 } \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$
E) $\frac { 9 } { 2 } \left( \cos \frac { 3 \pi } { 8 } + i \sin \frac { 3 \pi } { 8 } \right)$

Question 62

Use the formula to find the missing quantity for the given conditions.The formula E=IZ where E represents voltage, I represents current, and Z represents impedance (a measure of opposition to a sinusoidal electric current), is used in electrical engineering.The variable is a complex number. $\begin{array} { l } I = 13 + 5 i \\Z = 4 + 3 i\end{array}$

A) $37 + 59 i$
B) $37 - 37 i$
C) $59 - 37 i$
D) $37 - 59 i$
E) $59 + 37 i$

Question 63

Find the trigonometric form of the complex number shown below. $- 10 i$

A) $10 ( \cos 0 + i \sin 0 )$
B) $10 ( \cos \pi + i \sin \pi )$
C) $10 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
D) $10 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$
E) $10 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$

Question 64

Find the trigonometric form of the complex number shown below. $6 i$

A) $6 ( \cos \pi + i \sin \pi )$
B) $6 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
C) $6 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
D) $6 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$
E) $6 ( \cos 0 + i \sin 0 )$

Question 65

Write the complex number shown below in trigonometric form.

A) $6 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$
B) $6 \left( \cos \frac { 5 \pi } { 3 } + i \sin \frac { 5 \pi } { 3 } \right)$
C) $6 \left( \cos \frac { 11 \pi } { 6 } + i \sin \frac { 11 \pi } { 6 } \right)$
D) $6 \left( \cos \frac { 4 \pi } { 3 } + i \sin \frac { 4 \pi } { 3 } \right)$
E) $6 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$

Question 66

Perform the indicated operation using the standard form. $( \sqrt { 7 } + i ) ( 1 + i )$

A) $( \sqrt { 7 } + i ) ( 1 + i ) \approx 1 + i$
B) $( \sqrt { 7 } + i ) ( 1 + i ) \approx 3.646 - 1.646 i$
C) $( \sqrt { 7 } + i ) ( 1 + i ) \approx 1.646 - 3.646 i$
D) $( \sqrt { 7 } + i ) ( 1 + i ) \approx 1.646 + 3.646 i$
E) $( \sqrt { 7 } + i ) ( 1 + i ) \approx 3.646 + 1.646 i$

Question 67

Find the product $( - 5 + 5 i ) ( 7 - 7 i )$ using trigonometric forms.Leave the result in trigonometric form.

A) $35 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$
B) $70 \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right)$
C) $35 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right)$
D) $70 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$
E) $70 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right)$

Question 68

Use the formula to find the missing quantity for the given conditions.The formula E=IZ where E represents voltage, I represents current, and Z represents impedance (a measure of opposition to a sinusoidal electric current), is used in electrical engineering.The variable is a complex number. $\begin{array} { l } I = 8 + 2 i \\Z = 4 + 3 i\end{array}$

A) $32 - 26 i$
B) $32 + 26 i$
C) $26 - 32 i$
D) $26 - 26 i$
E) $26 + 32 i$

Question 69

Perform the indicated operation in trigonometric form.Write the solution in standard form rounded to four decimal places. $\frac { 4 + i \sqrt { 3 } } { 8 - 3 i }$

A)0.4682 + 0.2027i
B)0.4777 - 0.1791i
C)0.7197 + 0.6943i
D)0.5095 + 0.0254i
E)0.3672 + 0.3542i

Question 70

Find the trigonometric form of the complex number shown below. $\sqrt { 3 } + i$

A) $\sqrt { 3 } \left( \cos \left( \frac { \pi } { 2 } \right) + i \sin \left( \frac { \pi } { 2 } \right) \right)$
B) $2 \left( \cos \left( \frac { \sqrt { 3 } \pi } { 2 } \right) + i \sin \left( \frac { \sqrt { 3 } \pi } { 2 } \right) \right)$
C) $\sqrt { 3 } \left( \cos \left( \frac { \pi } { 3 } \right) + i \sin \left( \frac { \pi } { 3 } \right) \right)$
D) $2 \left( \cos \left( \frac { \pi } { 6 } \right) + i \sin \left( \frac { \pi } { 6 } \right) \right)$
E) $2 \sqrt { 3 } \left( \cos \left( \frac { \pi } { 6 } \right) + i \sin \left( \frac { \pi } { 6 } \right) \right)$

Question 71

Sketch the graph of all complex numbers z satisfying the given condition. $| z | = 7$

A)
B)
C)
D)
E)

Question 72

Perform the indicated operation using the standard form. $5 i ( 1 - \sqrt { 2 } i )$

A) $5 i ( 1 - \sqrt { 2 } i ) \approx 7.07 + 5 i$
B) $5 i ( 1 - \sqrt { 2 } i ) \approx 5 + 7.07 i$
C) $5 i ( 1 - \sqrt { 2 } i ) \approx 1 + 5 \sqrt { 2 i }$
D) $5 i ( 1 - \sqrt { 2 } i ) \approx 5 - 7.07 i$
E) $5 i ( 1 - \sqrt { 2 } i ) \approx 7.07 - 5 i$

Question 73

Perform the indicated operation using the standard form. $( 2 + 2 i ) ( 1 - i )$

A) $( 2 + 2 i ) ( 1 - i ) = 2$
B) $( 2 + 2 i ) ( 1 - i ) = 1$
C) $( 2 + 2 i ) ( 1 - i ) = 4$
D) $( 2 + 2 i ) ( 1 - i ) = - 2$
E) $( 2 + 2 i ) ( 1 - i ) = - 4$

Question 74

Perform the operation and leave the result in trigonometric form. $\frac { 2 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) } { 3 \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right) }$

A) $\frac { 2 } { 3 } \left( \cos 270 ^ { \circ } + i \sin 270 ^ { \circ } \right)$
B) $\frac { 2 } { 3 } \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$
C) $\frac { 2 } { 3 } \left( \cos 270 ^ { \circ } - i \sin 270 ^ { \circ } \right)$
D) $\frac { 2 } { 3 } \left( \cos 30 ^ { \circ } - i \sin 30 ^ { \circ } \right)$
E) $\frac { 2 } { 3 } \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right)$

Question 75

Perform the indicated operation using trigonometric form.Leave the answer in trigonometric forms. $( 2 + 2 i ) ( - 1 - i )$

A) $- 2 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
B) $4 \left( \cos \frac { 3 \pi } { 2 } - i \sin \frac { 3 \pi } { 2 } \right)$
C) $4 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$
D) $- 2 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$
E) $2 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right)$

Question 76

Select the graph of all complex numbers z satisfying the given condition. $| z | = 4$

A)
B)
C)
D)
E)

Question 77

Find the absolute value of the complex number $8 + 2 i$ .

A) $\sqrt { 10 }$
B) $3 \sqrt { 10 }$
C) 68
D) $4 \sqrt { 10 }$
E) $2 \sqrt { 17 }$

Question 78

Represent the complex number below graphically. $\sqrt { 3 } + i$

A)
B)
C)
D)
E)

Question 79

Perform the indicated operation using the standard form. $- 6 i ( 1 + i )$

A) $- 6 i ( 1 + i ) = 6 - 6 i$
B) $- 6 i ( 1 + i ) = 12 - 6 i$
C) $- 6 i ( 1 + i ) = 6 + 12 i$
D) $- 6 i ( 1 + i ) = 6 - 12 i$
E) $- 6 i ( 1 + i ) = 6 + 6 i$

Question 80

Find the trigonometric form of the complex number shown below. $- 3 + 4 i$

A) $\sqrt { 25 }$ (cos (arctan (- $\frac { 1 } { 6 }$ )) + i sin (arctan (- $\frac { 1 } { 6 }$ )))
B) $\sqrt { 25 }$ (cos (arctan ( $\frac { 4 } { 5 }$ )) + i sin (arctan ( $\frac { 4 } { 5 }$ )))
C) $\sqrt { 25 }$ (cos (arctan (- $\frac { 4 } { 3 }$ )) + i sin (arctan (- $\frac { 4 } { 3 }$ )))
D) $\sqrt { 25 }$ (cos (arctan (- $\frac { 2 } { 3 }$ )) + i sin (arctan (- $\frac { 2 } { 3 }$ )))
E) $\sqrt { 25 }$ (cos (arctan (4)) + i sin (arctan (4)))