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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Question
![**Convert the rectangular equation to polar form.**
\[ xy = 14 \]
**Solution:**
In rectangular coordinates:
- \( x \) is the horizontal coordinate.
- \( y \) is the vertical coordinate.
In polar coordinates:
- \( r \) is the distance from the origin.
- \( \theta \) is the angle measured counterclockwise from the positive \( x \)-axis.
Using the relationships between rectangular and polar coordinates:
\[ x = r \cos(\theta) \]
\[ y = r \sin(\theta) \]
Substitute these into the given equation:
\[ (r \cos(\theta))(r \sin(\theta)) = 14 \]
This simplifies to:
\[ r^2 \cos(\theta) \sin(\theta) = 14 \]
Hence, the polar form of the given rectangular equation is:
\[ r^2 \cos(\theta) \sin(\theta) = 14 \]
**Graph/Diagram Description:**
There is no graph or diagram associated with this problem. There is a text box below the equation where the polar form solution would be entered.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc51a27f7-e004-47b6-8023-544dc66e83a5%2F16453e50-daee-4ed9-b797-fb2adacb0165%2Fzxf4lb9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Convert the rectangular equation to polar form.**
\[ xy = 14 \]
**Solution:**
In rectangular coordinates:
- \( x \) is the horizontal coordinate.
- \( y \) is the vertical coordinate.
In polar coordinates:
- \( r \) is the distance from the origin.
- \( \theta \) is the angle measured counterclockwise from the positive \( x \)-axis.
Using the relationships between rectangular and polar coordinates:
\[ x = r \cos(\theta) \]
\[ y = r \sin(\theta) \]
Substitute these into the given equation:
\[ (r \cos(\theta))(r \sin(\theta)) = 14 \]
This simplifies to:
\[ r^2 \cos(\theta) \sin(\theta) = 14 \]
Hence, the polar form of the given rectangular equation is:
\[ r^2 \cos(\theta) \sin(\theta) = 14 \]
**Graph/Diagram Description:**
There is no graph or diagram associated with this problem. There is a text box below the equation where the polar form solution would be entered.
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