Multiple Choice If z1=r1 e^i θ1 and z2=r2 e^i θ2 are complex numbers, then (z1)/(z2), z2 ≠0, equ

Name: Problem 12, Chapter 10: Algebra and Trigonometry
Uploaded: 2021-08-06T15:16:34-08:00
Duration: 1 min 5 s
Channel: Yujie Wang
Description: Multiple Choice If z1=r1 e^i θ1 and z2=r2 e^i θ2 are complex numbers, then (z1)/(z2), z2 ≠0, equals: (a) (r1)/(r2) e^i(θ1-θ2) (b) (r1)/(r2) e^i(θ1 ·θ2) (c) (r1)/(r2) e^i(θ1+θ2) (d) (r1)/(r2) e^i(θ1 / θ2)

step 1 Let's look at this problem number 12. So we have Z1, Z2. So we want to find the value of Z1 over

Question

Multiple Choice If $z_{1}=r_{1} e^{i \theta_{1}}$ and $z_{2}=r_{2} e^{i \theta_{2}}$ are complex numbers, then $\frac{z_{1}}{z_{2}}, z_{2} \neq 0,$ equals: (a) $\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1}-\theta_{2}\right)}$ (b) $\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1} \cdot \theta_{2}\right)}$ (c) $\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1}+\theta_{2}\right)}$ (d) $\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1} / \theta_{2}\right)}$

   Multiple Choice If $z_{1}=r_{1} e^{i \theta_{1}}$ and $z_{2}=r_{2} e^{i \theta_{2}}$ are complex numbers, then $\frac{z_{1}}{z_{2}}, z_{2} \neq 0,$ equals:
(a) $\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1}-\theta_{2}\right)}$
(b) $\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1} \cdot \theta_{2}\right)}$
(c) $\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1}+\theta_{2}\right)}$
(d) $\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1} / \theta_{2}\right)}$

Algebra and Trigonometry

Michael Sullivan 11th Edition

Chapter 10, Problem 12

Instant Answer

Solved by Expert Yujie Wang

08/06/2021

Step 1

Step 1: We are given two complex numbers $z_{1}=r_{1} e^{i \theta_{1}}$ and $z_{2}=r_{2} e^{i \theta_{2}}$ and we want to find the value of $\frac{z_{1}}{z_{2}}$. Show more…

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Multiple Choice If $z_{1}=r_{1} e^{i \theta_{1}}$ and $z_{2}=r_{2} e^{i \theta_{2}}$ are complex numbers, then $\frac{z_{1}}{z_{2}}, z_{2} \neq 0,$ equals: (a) $\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1}-\theta_{2}\right)}$ (b) $\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1} \cdot \theta_{2}\right)}$ (c) $\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1}+\theta_{2}\right)}$ (d) $\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1} / \theta_{2}\right)}$

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Key Concepts

Complex Numbers in Polar Form

This concept involves expressing a complex number in the form r e^(i?), where r is the modulus (or absolute value) of the complex number, and ? is the argument (or angle). Representing complex numbers in this polar form simplifies operations like multiplication and division, as it allows one to work separately with the magnitudes and angles.

Division of Complex Numbers in Polar Form

When dividing two complex numbers expressed in polar form, the moduli are divided and the arguments are subtracted. Specifically, if z1 = r1e^(i?1) and z2 = r2e^(i?2), then the quotient z1/z2 is given by (r1/r2) e^(i(?1 - ?2)). This rule follows from the properties of exponents and simplifies the process of division.

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