Deck 40: Trigonometric Form of a Complex Number

Question 1

Plot the complex number 2. ​

A)
B) ​
C)
D)
E)

Answer 1

​

Question 2

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

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

Answer 2

​ $\sqrt { 34 }$

Question 3

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)​ ​ $$| - 3 + 9 i | = \sqrt { 86 }$$

Answer 3

​ ​ $$| - 3 + 9 i | = \sqrt { 90 }$$

Question 4

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| = $\sqrt { 105 }$

Question 5

Find the trigonometric form of the complex number represented graphically.

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

Question 6

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 7

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

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

Question 8

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 9

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 10

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 11

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 12

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 13

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 14

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 15

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)​ $Z = 3 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$

Question 16

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 17

Plot the following complex number. -3

A)
B)
C)
D)
E) ​

Question 18

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 19

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 20

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 21

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 22

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 23

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 24

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

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 $10 \left( \cos \frac { \pi } { 12 } + i \sin \frac { \pi } { 12 } \right)$ .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 27

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

A)​
B)
C)​
D)​
E)​

Question 28

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 29

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 30

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 31

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

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 32

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 33

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

A)
B)​
C)​
D)
E)​

Question 34

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 35

Represent the complex number graphically. ​
2
​

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

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 37

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 38

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 39

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 40

​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 41

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 42

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 43

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 44

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 45

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 46

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 47

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 48

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 49

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 50

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 51

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 52

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 53

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 54

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 55

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 56

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 57

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 58

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 59

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 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 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 62

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

A)​
B)​
C)​
D)​
E)​

Question 63

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 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

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 66

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 67

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 68

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 69

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 70

​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)))

Question 71

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

A)
B)
C)
D)
E)

Question 72

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 73

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 74

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 75

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 76

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 77

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 78

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 79

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

A)​
B)​
C)​
D)​
E)​

Question 80

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)​ $( \sqrt { 7 } + i ) ( 1 + i ) \approx 3.646 + 1.646 i$