﻿ Complex Numbers Worksheet | Problems & Solutions

# Complex Numbers Worksheet

Complex Numbers Worksheet
• Page 1
1.
Find the product of 5 (cos 30° + $i$ sin 30°) and 4 (cos 90° + $i$ sin 90°).
 a. 10 (1 + $i$ $\sqrt{3}$) b. 10 (- 1 + $i$ $\sqrt{3}$) c. 10 (1 - $i$ $\sqrt{3}$) d. - 10 (1 + $i$ $\sqrt{3}$)

#### Solution:

5 (cos 30° + i sin 30°) × 4 (cos 90° + i sin 90°).

= 20 (cos 120° + i sin 120°)
[(cosθ1+i sinθ1)×(cosθ2+i sinθ2) = cos (θ1+θ2) + i sin (θ1+θ2).]

= 20 ( cos(180° - 60°) + i sin (180° - 60°))
[Rewrite 120° as (180° - 60°).]

= 20 (- cos 60° + i sin 60°)

= 20 (-1 / 2+ i 32)
[Use cos 60° = 1 / 2, sin 60° = 32.]

= 10 (- 1 + i 3)
[Simplify.]

So, the product of 5 (cos 30° + i sin 30°) and 4 (cos 90° + i sin 90°) = 10(- 1 + i 3).

2.
What is standard form of the complex number?
(5 + $i$)3
 a. 110$i$ + 74 b. 110 + 74$i$ c. 110$i$ - 74 d. 110 - 74$i$

#### Solution:

(5 + i)3 = 125 + 75i + 15i2 + i3
[Use (a+b)3 = a3 + 3a2b + 3ab2 + b3.]

= 125 + 75i - 15 - i
[Use i2 = - 1, i3 = - i.]

= 110 + 74i
[Simplify.]

So, the standard form of the given complex number is 110 + 74i.

3.
Simplify:
 a. $\frac{1}{7}$ b. - 7$i$ c. 7$i$ d. 7

4.
Simplify:
$\sqrt{-16}$
 a. - 4$i$ b. $\frac{1}{4}$ c. 4$i$ d. 4

5.
Simplify:
$\sqrt{-13}$
 a. - $i$$\sqrt{13}$ b. $i$$\sqrt{13}$ c. $\sqrt{13}$ d. $\frac{1}{\sqrt{13}}$

6.
Simplify:
$\sqrt{-11}$
 a. $\sqrt{11}$ b. $\frac{1}{\sqrt{11}}$ c. $i$$\sqrt{11}$ d. - $i$$\sqrt{11}$

7.
Simplify:
$\sqrt{-7}$
 a. $i$$\sqrt{7}$ b. $\sqrt{7}$ c. - $i$$\sqrt{7}$ d. $\frac{1}{\sqrt{7}}$

8.
Simplify:
(4 - 7$i$) + (- 3 + 2$i$)
 a. 1 - 5$i$ b. - 1 - 5$i$ c. - 1 + 5$i$ d. 1 + 5$i$

9.
Simplify:
(- 6 + 5$i$) - (3 - 2$i$)
 a. - 6 + 7$i$ b. 7 - 9$i$ c. - 9 + 7$i$ d. - 6

(7 - 3$i$)(7 + 3$i$)