Sketch a graph of the polar equation. C-8 sin(e) /2 /2 10 10 -10 -5 10 -5 -5 -10- /2 /2 10 -10 10 -10 -10- -10- Express the equation in rectangular coordinates. (Use the variables x and y.)
Sketch a graph of the polar equation. C-8 sin(e) /2 /2 10 10 -10 -5 10 -5 -5 -10- /2 /2 10 -10 10 -10 -10- -10- Express the equation in rectangular coordinates. (Use the variables x and y.)
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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![### Sketch a Graph of the Polar Equation
**Polar Equation:**
\[ r = 8 \sin(\theta) \]
**Graph Descriptions:**
1. **Top Left Graph:**
- The graph displays a circle centered at (0, 4) on the polar axis with a radius of 4.
- The circle intersects the vertical axis at points (0, 0) and (0, 8).
2. **Top Right Graph:**
- This graph shows a circle centered at (0, -4) with a radius of 4.
- The circle touches the vertical axis at (0, -8) and (0, 0).
3. **Bottom Left Graph:**
- The circle is centered at (0, -4) and has a radius of 4.
- It intersects the vertical line at (0, -8) and (0, 0).
4. **Bottom Right Graph:**
- This graph features a circle centered at (0, -4) with a radius of 4.
- It crosses the line at (0, -8) and (0, 0).
Overall, you need to find which one corresponds to the given polar equation \( r = 8 \sin(\theta) \).
**Task:**
Express the equation in rectangular coordinates using the variables \( x \) and \( y \).
**Rectangular Coordinates Conversion:**
- Conversion Formulas:
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)
- \( r^2 = x^2 + y^2 \)
To express in rectangular coordinates:
\[ y = 8 \sin(\theta) \]
Since \( y = r \sin(\theta) \) and \( r = \sqrt{x^2 + y^2} \),
\[ y = 8 \frac{y}{\sqrt{x^2 + y^2}} \]
Multiply both sides by \( \sqrt{x^2 + y^2} \) to eliminate the fraction:
\[ y \sqrt{x^2 + y^2} = 8y \]
Since \( y \neq 0 \), divide both sides by \( y \):
\[ \sqrt{x^2 + y^2} = 8 \]
Square both sides:
\[ x^2 + y^2 = 64 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40127f56-323b-4291-ad5f-c789e533574c%2F15478138-7903-42bb-a0be-9bdfeb11ba4f%2Fiywj5h3_processed.png&w=3840&q=75)
Transcribed Image Text:### Sketch a Graph of the Polar Equation
**Polar Equation:**
\[ r = 8 \sin(\theta) \]
**Graph Descriptions:**
1. **Top Left Graph:**
- The graph displays a circle centered at (0, 4) on the polar axis with a radius of 4.
- The circle intersects the vertical axis at points (0, 0) and (0, 8).
2. **Top Right Graph:**
- This graph shows a circle centered at (0, -4) with a radius of 4.
- The circle touches the vertical axis at (0, -8) and (0, 0).
3. **Bottom Left Graph:**
- The circle is centered at (0, -4) and has a radius of 4.
- It intersects the vertical line at (0, -8) and (0, 0).
4. **Bottom Right Graph:**
- This graph features a circle centered at (0, -4) with a radius of 4.
- It crosses the line at (0, -8) and (0, 0).
Overall, you need to find which one corresponds to the given polar equation \( r = 8 \sin(\theta) \).
**Task:**
Express the equation in rectangular coordinates using the variables \( x \) and \( y \).
**Rectangular Coordinates Conversion:**
- Conversion Formulas:
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)
- \( r^2 = x^2 + y^2 \)
To express in rectangular coordinates:
\[ y = 8 \sin(\theta) \]
Since \( y = r \sin(\theta) \) and \( r = \sqrt{x^2 + y^2} \),
\[ y = 8 \frac{y}{\sqrt{x^2 + y^2}} \]
Multiply both sides by \( \sqrt{x^2 + y^2} \) to eliminate the fraction:
\[ y \sqrt{x^2 + y^2} = 8y \]
Since \( y \neq 0 \), divide both sides by \( y \):
\[ \sqrt{x^2 + y^2} = 8 \]
Square both sides:
\[ x^2 + y^2 = 64 \
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