Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Trigonometric Form Complex Number Operation
**Problem Statement:**
Perform the operation and leave the result in trigonometric form. (Let \( 0 \leq \theta < 360^\circ \)).
\[
\frac{15(\cos 116^\circ + i \sin 116^\circ)}{3(\cos 146^\circ + i \sin 146^\circ)}
\]
In this problem, you are given a division of two complex numbers represented in trigonometric form. The general formula for a complex number in trigonometric form is:
\[ z = r(\cos \theta + i \sin \theta) \]
### Solution Steps:
1. **Identify the Magnitudes and Arguments**:
- For the numerator: \( 15(\cos 116^\circ + i \sin 116^\circ) \)
- Magnitude \( r_1 = 15 \)
- Argument \( \theta_1 = 116^\circ \)
- For the denominator: \( 3(\cos 146^\circ + i \sin 146^\circ) \)
- Magnitude \( r_2 = 3 \)
- Argument \( \theta_2 = 146^\circ \)
2. **Divide the Magnitudes**:
\[ \frac{r_1}{r_2} = \frac{15}{3} = 5 \]
3. **Subtract the Arguments**:
\[ \theta = \theta_1 - \theta_2 = 116^\circ - 146^\circ = -30^\circ \]
Since the angle must be between \( 0^\circ \) and \( 360^\circ \), we convert \(-30^\circ\) to a positive angle:
\[ -30^\circ + 360^\circ = 330^\circ \]
4. **Construct the Result**:
\[ \frac{15(\cos 116^\circ + i \sin 116^\circ)}{3(\cos 146^\circ + i \sin 146^\circ)} = 5 \left( \cos 330^\circ + i \sin 330^\circ \right) \]
### Final Answer:
\[
\boxed{5 \left( \cos 330^\circ + i \sin 330^\circ \right)}
\]
This operation simplifies the given](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fab864d-7419-4b8c-bc50-25971d273c3c%2Fd876c9ec-1423-42ae-b737-3b0e422e1fd8%2Fzccmwrk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Trigonometric Form Complex Number Operation
**Problem Statement:**
Perform the operation and leave the result in trigonometric form. (Let \( 0 \leq \theta < 360^\circ \)).
\[
\frac{15(\cos 116^\circ + i \sin 116^\circ)}{3(\cos 146^\circ + i \sin 146^\circ)}
\]
In this problem, you are given a division of two complex numbers represented in trigonometric form. The general formula for a complex number in trigonometric form is:
\[ z = r(\cos \theta + i \sin \theta) \]
### Solution Steps:
1. **Identify the Magnitudes and Arguments**:
- For the numerator: \( 15(\cos 116^\circ + i \sin 116^\circ) \)
- Magnitude \( r_1 = 15 \)
- Argument \( \theta_1 = 116^\circ \)
- For the denominator: \( 3(\cos 146^\circ + i \sin 146^\circ) \)
- Magnitude \( r_2 = 3 \)
- Argument \( \theta_2 = 146^\circ \)
2. **Divide the Magnitudes**:
\[ \frac{r_1}{r_2} = \frac{15}{3} = 5 \]
3. **Subtract the Arguments**:
\[ \theta = \theta_1 - \theta_2 = 116^\circ - 146^\circ = -30^\circ \]
Since the angle must be between \( 0^\circ \) and \( 360^\circ \), we convert \(-30^\circ\) to a positive angle:
\[ -30^\circ + 360^\circ = 330^\circ \]
4. **Construct the Result**:
\[ \frac{15(\cos 116^\circ + i \sin 116^\circ)}{3(\cos 146^\circ + i \sin 146^\circ)} = 5 \left( \cos 330^\circ + i \sin 330^\circ \right) \]
### Final Answer:
\[
\boxed{5 \left( \cos 330^\circ + i \sin 330^\circ \right)}
\]
This operation simplifies the given
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