Write the quotient in rectangular 3 cis 275° 27 cis 50° 3 cis 275° 27 cis 50° (Simplify your answer, including any radicals. Use integers or fractions for any numbers expression. Type your answer in the form a + bi.)
Write the quotient in rectangular 3 cis 275° 27 cis 50° 3 cis 275° 27 cis 50° (Simplify your answer, including any radicals. Use integers or fractions for any numbers expression. Type your answer in the form a + bi.)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter1: Equations And Inequalities
Section1.3: Complex Numbers
Problem 110E
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![**Complex Numbers in Polar Form**
_Quotient of Complex Numbers_
**Problem Statement:**
Write the quotient in rectangular form.
\[
\frac{3 \text{ cis } 275^\circ}{27 \text{ cis } 50^\circ}
\]
---
Below the problem statement, there is an equals sign followed by a blank box indicating where the simplified answer should be entered. The text instructs to:
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type your answer in the form \(a + bi\).)
---
**Notes:**
1. **cis** notation: In mathematics, the notation \(r \text{ cis } \theta\) represents a complex number in polar form, which can be expressed as \(r (\cos(\theta) + i \sin(\theta))\).
2. **Conversion to Rectangular Form:** To convert the polar form \(r (\cos(\theta) + i \sin(\theta))\) to rectangular form \(a + bi\):
- Identify the value of \(r\), the modulus (magnitude).
- Determine \(\theta\), the argument (angle).
- Use the trigonometric identities to find the real part \(a = r \cos(\theta)\) and the imaginary part \(b = r \sin(\theta)\).
3. **Division of Complex Numbers in Polar Form:**
\[
\frac{r_1 \text{ cis } \theta_1}{r_2 \text{ cis } \theta_2} = \left(\frac{r_1}{r_2}\right) \text{ cis } (\theta_1 - \theta_2)
\]
**Example Calculation:**
Given the expression:
\[
\frac{3 \text{ cis } 275^\circ}{27 \text{ cis } 50^\circ}
\]
Step-by-step process:
1. Divide the magnitudes:
\[
r = \frac{3}{27} = \frac{1}{9}
\]
2. Subtract the angles:
\[
\theta = 275^\circ - 50^\circ = 225^\circ
\]
3. Express the quotient in polar form:
\[
\frac{1}{9} \text{ cis } 225^\circ](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3def50f-0bee-45b6-a067-ec9125eb6917%2F6d7b8d4b-bd45-49c6-89b7-3ff1f7c2f135%2Fj03x6y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Complex Numbers in Polar Form**
_Quotient of Complex Numbers_
**Problem Statement:**
Write the quotient in rectangular form.
\[
\frac{3 \text{ cis } 275^\circ}{27 \text{ cis } 50^\circ}
\]
---
Below the problem statement, there is an equals sign followed by a blank box indicating where the simplified answer should be entered. The text instructs to:
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type your answer in the form \(a + bi\).)
---
**Notes:**
1. **cis** notation: In mathematics, the notation \(r \text{ cis } \theta\) represents a complex number in polar form, which can be expressed as \(r (\cos(\theta) + i \sin(\theta))\).
2. **Conversion to Rectangular Form:** To convert the polar form \(r (\cos(\theta) + i \sin(\theta))\) to rectangular form \(a + bi\):
- Identify the value of \(r\), the modulus (magnitude).
- Determine \(\theta\), the argument (angle).
- Use the trigonometric identities to find the real part \(a = r \cos(\theta)\) and the imaginary part \(b = r \sin(\theta)\).
3. **Division of Complex Numbers in Polar Form:**
\[
\frac{r_1 \text{ cis } \theta_1}{r_2 \text{ cis } \theta_2} = \left(\frac{r_1}{r_2}\right) \text{ cis } (\theta_1 - \theta_2)
\]
**Example Calculation:**
Given the expression:
\[
\frac{3 \text{ cis } 275^\circ}{27 \text{ cis } 50^\circ}
\]
Step-by-step process:
1. Divide the magnitudes:
\[
r = \frac{3}{27} = \frac{1}{9}
\]
2. Subtract the angles:
\[
\theta = 275^\circ - 50^\circ = 225^\circ
\]
3. Express the quotient in polar form:
\[
\frac{1}{9} \text{ cis } 225^\circ
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