Q: What is the argument $\theta$ of the complex number $z = -1 + i$?

$\theta = \arctan\left(\frac{1}{-1}\right) + \pi = \frac{3\pi}{4}$

Q: Which of the following is the correct formula for converting a complex number $z = a + bi$ to polar form?

$z = r(\cos\theta + i\sin\theta)$, where $r = \sqrt{a^2 + b^2}$ and $\theta = \arctan\left(\frac{b}{a}\right)$

Q: Which of the following is the correct geometric interpretation of multiplying two complex numbers in polar form?

The moduli multiply and the arguments add: $r_1 r_2 (\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2))$

Q: A medical imaging device uses the complex number $z = 6e^{i\frac{\pi}{6}}$ to represent a signal. What is the real part of $z$?

$6 \cos\left(\frac{\pi}{6}\right) = 6 \times \frac{\sqrt{3}}{2} = 3\sqrt{3}$

Question 1

Given a complex number $z = 3 + 4i$, what is its modulus $|z|$?

Accepted Answer

$|z| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$

Question 2

What is the argument $	heta$ of the complex number $z = -1 + i$?

Accepted Answer

$	heta = \arctan\left(\frac{1}{-1}ight) + \pi = \frac{3\pi}{4}$

Question 3

Express the complex number $z = 2 - 2i$ in polar form.

Accepted Answer

$z = 2\sqrt{2} \left( \cos\left(-\frac{\pi}{4}\right) + i \sin\left(-\frac{\pi}{4}\right) \right)$

Question 4

If $z_1 = 1 + i$ and $z_2 = 1 - i$, what is $z_1 \cdot z_2$?

Accepted Answer

$z_1 \cdot z_2 = (1 + i)(1 - i) = 1 - i^2 = 2$

Question 5

Which of the following is the correct formula for converting a complex number $z = a + bi$ to polar form?

Accepted Answer

$z = r(\cos	heta + i\sin	heta)$, where $r = \sqrt{a^2 + b^2}$ and $	heta = \arctan\left(\frac{b}{a}ight)$

Question 6

A signal in electrical engineering is represented by $z = 5e^{i\frac{\pi}{3}}$. What are the rectangular coordinates of $z$?

Accepted Answer

$z = 5 \cos\left(\frac{\pi}{3}\right) + i 5 \sin\left(\frac{\pi}{3}\right) = 2.5 + i 4.33$

Question 7

If $z = 4(\cos 60^\circ + i \sin 60^\circ)$, what is the value of $z$ in rectangular form?

Accepted Answer

$z = 4 \left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right) = 2 + i 2\sqrt{3}$

Question 8

A drone moves from the origin to the point $(3, 4)$ in the complex plane. What is the distance it travels and the angle it makes with the positive $x$-axis?

Accepted Answer

Distance: $5$, Angle: $\arctan\left(\frac{4}{3}\right) \approx 53.13^\circ$

Question 9

Which of the following is the correct geometric interpretation of multiplying two complex numbers in polar form?

Accepted Answer

The moduli multiply and the arguments add: $r_1 r_2 (\cos(	heta_1 + 	heta_2) + i \sin(	heta_1 + 	heta_2))$

Question 10

A medical imaging device uses the complex number $z = 6e^{i\frac{\pi}{6}}$ to represent a signal. What is the real part of $z$?

Accepted Answer

$6 \cos\left(\frac{\pi}{6}ight) = 6 	imes \frac{\sqrt{3}}{2} = 3\sqrt{3}$

Question 11

Which of the following is the correct rectangular form for $z = 7(\cos 120^\circ + i \sin 120^\circ)$?

Accepted Answer

$z = 7 \left( -\frac{1}{2} + i \frac{\sqrt{3}}{2} \right) = -3.5 + i 6.06$

Question 12

A ship navigates using a vector represented by $z = 8e^{i\frac{\pi}{2}}$. What is the direction and magnitude of the ship's movement?

Accepted Answer

Magnitude: $8$, Direction: $90^\circ$ (positive $y$-axis)

Question 13

If $z = 5(\cos 45^\circ + i \sin 45^\circ)$, what is the imaginary part of $z$?

Accepted Answer

$5 \sin 45^\circ = 5 	imes \frac{\sqrt{2}}{2} = \frac{5\sqrt{2}}{2}$

Question 14

A pilot flies from $(0,0)$ to $(-5, 12)$ in the complex plane. What is the modulus and argument of the displacement vector?

Accepted Answer

Modulus: $\sqrt{(-5)^2 + 12^2} = 13$, Argument: $\arctan\left(\frac{12}{-5}\right) + \pi \approx 112.62^\circ$

Polar Form of Complex Numbers and Trigonometric Connections

Study Guide - Practice Questions

Study Guide - Flashcards