$$0(\cos \theta + i \sin \theta)$$, \text{for any } \theta$

Question 1

Express the complex number $$z = 1 + i in $$polar form.

Accepted Answer

$$\sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})$$

Question 2

Express the complex number $$z = -1 + i in $$polar form.

Accepted Answer

$$\sqrt{2}(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4})$$

Question 3

Express the complex number $$z = -1 - i in $$polar form.

Accepted Answer

$$\sqrt{2}(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4})$$

Question 4

Express the complex number $$z = 1 - i in $$polar form.

Accepted Answer

$$\sqrt{2}(\cos \frac{7\pi}{4} + i \sin \frac{7\pi}{4})$$

Question 5

A signal in an electrical circuit is represented by the complex number $$z = 3 + 4i. $$What is its modulus and argument?

Accepted Answer

Modulus: $$5$$, Argument: $$\arctan(\frac{4}{3})$$

Question 6

Which of the following is the correct polar form for the complex number $$z = 0 - 2i$$?

Accepted Answer

$$2(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2})$$

Question 7

If $$z = 2(\cos 	heta + i \sin 	heta)$$ and $$	heta = \frac{\pi}{6}$$, what is the rectangular form of $$z$$?

Accepted Answer

$$\sqrt{3} + i$$

Question 8

A medical imaging device uses the complex number $$z = 5e$$^{i\frac{\pi}$${3}} to $$represent a signal. What is the rectangular form?

Accepted Answer

$$2.5 + 4.33i$$

Question 9

Which of the following is the correct polar form for the complex number $$z = -3$$?

Accepted Answer

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

Question 10

A chemical reaction is modeled by the complex number $$z = 2e$$^{i\pi}$$. What is the rectangular form?

Accepted Answer

$$-2$$

Question 11

If $$z = 4(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})$$, what is the rectangular form of $$z$$?

Accepted Answer

$$4i$$

Question 12

A rotating object in physics is described by $$z = 6e$$^{i\frac{\pi}$${2}}. $$What is the rectangular form?

Accepted Answer

$$6i$$

Question 13

Which of the following is the correct polar form for the complex number $$z = 2 + 2i$$?

Accepted Answer

$$2\sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})$$

Question 14

A signal in a communications system is represented by $$z = 4e$$^{i\frac{\pi}$${6}}. $$What is the rectangular form?

Accepted Answer

$$3.46 + 2i$$

Question 15

Express the complex number $$z = 0 + 5i in $$polar form.

Accepted Answer

$$5(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})$$

Question 16

A rotating vector in engineering is described by $$z = 7e$$^{i\pi}$$. What is the rectangular form?

Accepted Answer

$$-7$$

Question 17

Which of the following is the correct polar form for the complex number $$z = 0$$?

Accepted Answer

$$0(\cos 	heta + i \sin 	heta)$$, 	ext{for any } 	heta$

Polar Form of Complex Numbers: Practice Problems

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