Q: If $$z = 2 \left( \cos \theta + i \sin \theta \right)$$, what is $$z^3 in $$exponential form?

$$8 \left( \cos 3\theta + i \sin 3\theta \right)$$

Q: Which of the following is the correct statement of De Moivre's Theorem?

$$[r (\cos \theta + i \sin \theta)]^n = r^n (\cos n\theta + i \sin n\theta)$$

Question 1

Given the complex number $$z = 4 \left( \cos \left( \frac{\pi}{3} \right) + i \sin \left( \frac{\pi}{3} \right) \right)$$, what is $$z^2 in $$polar form?

Accepted Answer

$$16 \left( \cos \left( \frac{2\pi}{3} \right) + i \sin \left( \frac{2\pi}{3} \right) \right)$$

Question 2

If $$z = 2 \left( \cos 	heta + i \sin 	heta ight)$$, what is $$z^3 in $$exponential form?

Accepted Answer

$$8 \left( \cos 3	heta + i \sin 3	heta ight)$$

Question 3

Which theorem allows you to compute powers of complex numbers in polar form efficiently?

Accepted Answer

De Moivre's Theorem

Question 4

Given $$z = 3 \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)$$, find $$z^4 in $$rectangular form.

Accepted Answer

$$81 \left( \cos \pi + i \sin \pi \right) = -81$$

Question 5

If $$z = r \left( \cos 	heta + i \sin 	heta ight)$$, what is the modulus of $$z^n$$?

Accepted Answer

$$r^n$$

Question 6

A complex number $$z = 2 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right). $$What is $$z^3 in $$rectangular form?

Accepted Answer

$$8 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right) = 8i$$

Question 7

Which of the following is the correct statement of De Moivre's Theorem?

Accepted Answer

$$[r (\cos 	heta + i \sin 	heta)]^n = r^n (\cos n	heta + i \sin n	heta)$$

Question 8

If $$z = 1 + i$$, what is $$z^4$$?

Accepted Answer

$$-4$$

Question 9

A phasor in electrical engineering is represented as $$z = 5 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right). $$What is its value in rectangular form?

Accepted Answer

$$5i$$

Question 10

If $$z = 2 \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)$$, what is $$z^2 in $$rectangular form?

Accepted Answer

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

Question 11

Which of the following is the rectangular form of $$z = 3 \left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right)$$?

Accepted Answer

$$-1.5 + 2.598i$$

Question 12

If $$z = r (\cos 	heta + i \sin 	heta)$$, what is the argument of $$z^n$$?

Accepted Answer

$$n	heta$$

Question 13

A signal rotates in the complex plane with $$z = 2 (\cos \omega t + i \sin \omega t). $$What is $$z^5$$?

Accepted Answer

$$32 (\cos 5\omega t + i \sin 5\omega t)$$

Question 14

If $$z = 1 - i$$, what is $$z^2$$?

Accepted Answer

$$0 - 2$$

Question 15

Given $$z = 2 (\cos \frac{\pi}{3} + i \sin \frac{\pi}{3})$$, what is $$z^6$$?

Accepted Answer

$$64$$

Powers of Complex Numbers (De Moivre’s Theorem)

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