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: 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 \$$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?

Accepted Answer

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