Q: If $z = r(\cos \theta + i \sin \theta)$, what is $z^n$ according to De Moivre's Theorem?

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

Q: If $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, what is $\dfrac{z_1}{z_2}$ in trigonometric form?

$\dfrac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2))$

Q: Which formula is used to raise a complex number in trigonometric form to a power?

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

Question 1

Given $z_1 = 8 \cos(150^\circ) + i8 \sin(150^\circ)$ and $z_2 = 4 \cos(105^\circ) + i4 \sin(105^\circ)$, what is $\dfrac{z_1}{z_2}$ in trigonometric form?

Accepted Answer

$2 \cos(45^\circ) + i2 \sin(45^\circ)$

Question 2

If $z = r(\cos 	heta + i \sin 	heta)$, what is $z^n$ according to De Moivre's Theorem?

Accepted Answer

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

Question 3

Which of the following is the correct application of De Moivre's Theorem to $z = 2(\cos 30^\circ + i \sin 30^\circ)$ for $z^3$?

Accepted Answer

$8(\cos 90^\circ + i \sin 90^\circ)$

Question 4

A complex number $z = 5(\cos 60^\circ + i \sin 60^\circ)$ is raised to the power of 2. What is $z^2$?

Accepted Answer

$25(\cos 120^\circ + i \sin 120^\circ)$

Question 5

If $z_1 = r_1(\cos 	heta_1 + i \sin 	heta_1)$ and $z_2 = r_2(\cos 	heta_2 + i \sin 	heta_2)$, what is $\dfrac{z_1}{z_2}$ in trigonometric form?

Accepted Answer

$\dfrac{r_1}{r_2}(\cos(	heta_1 - 	heta_2) + i \sin(	heta_1 - 	heta_2))$

Question 6

A medical imaging device uses the complex number $z = 3(\cos 45^\circ + i \sin 45^\circ)$. What is $z^4$?

Accepted Answer

$81(\cos 180^\circ + i \sin 180^\circ)$

Question 7

Which of the following is NOT a correct step in dividing two complex numbers in trigonometric form?

Accepted Answer

Add the angles

Question 8

A signal in electrical engineering is represented by $z = 4(\cos 120^\circ + i \sin 120^\circ)$. What is $z^3$?

Accepted Answer

$64(\cos 360^\circ + i \sin 360^\circ)$

Question 9

If $z = 2(\cos 75^\circ + i \sin 75^\circ)$, what is $z^5$?

Accepted Answer

$32(\cos 375^\circ + i \sin 375^\circ)$

Question 10

A chemist uses $z_1 = 6(\cos 80^\circ + i \sin 80^\circ)$ and $z_2 = 2(\cos 20^\circ + i \sin 20^\circ)$ to model a reaction. What is $\dfrac{z_1}{z_2}$?

Accepted Answer

$3(\cos 60^\circ + i \sin 60^\circ)$

Question 11

Which formula is used to raise a complex number in trigonometric form to a power?

Accepted Answer

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

Question 12

A navigation system uses $z = 5(\cos 72^\circ + i \sin 72^\circ)$. What is $z^2$?

Accepted Answer

$25(\cos 144^\circ + i \sin 144^\circ)$

Question 13

If $z_1 = 10(\cos 200^\circ + i \sin 200^\circ)$ and $z_2 = 2(\cos 50^\circ + i \sin 50^\circ)$, what is $\dfrac{z_1}{z_2}$?

Accepted Answer

$5(\cos 150^\circ + i \sin 150^\circ)$

Question 14

Which of the following is a correct step in multiplying two complex numbers in trigonometric form?

Accepted Answer

Add the angles

Question 15

A drone's flight path is modeled by $z = 2(\cos 60^\circ + i \sin 60^\circ)$. What is $z^3$?

Accepted Answer

$8(\cos 180^\circ + i \sin 180^\circ)$

Question 16

If $z = 3(\cos 90^\circ + i \sin 90^\circ)$, what is $z^2$?

Accepted Answer

$9(\cos 180^\circ + i \sin 180^\circ)$

Powers and Roots of Complex Numbers (DeMoivre’s Theorem)

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