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: 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(\$$theta_1$$ + \$$theta_2) + i$$ \sin(\$$theta_1$$ + \$$theta_2$$))$$

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

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