Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### In General
#### Graphical Representations of Polar Equations
The image above illustrates two important polar equations:
1. **Left Diagram: \( r = a \cos \theta \), where \( a > 0 \)**
- This represents a circle centered on the x-axis.
- The radius of the circle is \( \frac{a}{2} \).
- The circle intersects the x-axis at \( a \) and \( -a \), and the y-axis at \( \frac{a}{2} \) and \( -\frac{a}{2} \).
2. **Right Diagram: \( r = a \sin \theta \), where \( a > 0 \)**
- This represents a circle centered on the y-axis.
- The radius of the circle is \( \frac{a}{2} \).
- The circle intersects the y-axis at \( a \) and \( -a \), and the x-axis at \( \frac{a}{2} \) and \( -\frac{a}{2} \).
#### Equations Relating Polar and Cartesian Coordinates
\[ x = r \cos \theta , \quad y = r \sin \theta , \quad r^2 = x^2 + y^2 , \quad \tan \theta = \frac{y}{x} \]
#### Exercises
**Sketch the graph of each polar equation.**
**Also, convert each into a rectangular coordinate equation.**
**13.** \( r = 2 \sin \theta \)
**14.** \( r = 3 \cos \theta \)
**15.** \( r = -\sin \theta \)
**16.** \( r = -\cos \theta \)
**17.** \( r \cos \theta = -4 \)
**Provided Blank Graphs:**
There are five blank coordinate graphs provided below the exercises for sketching the graphs of the given polar equations. Each graph has the standard x and y axes.
---
These explanations and exercises are designed to help you understand and practice converting between polar and Cartesian coordinates, and to graph polar equations appropriately.](https://content.bartleby.com/qna-images/question/894ba220-0d5e-4e50-906d-a5e96c05bfb6/5cf2d5c4-182d-471f-96fd-4ef82a9de3a5/oq7ot2_processed.jpeg)
Transcribed Image Text:### In General
#### Graphical Representations of Polar Equations
The image above illustrates two important polar equations:
1. **Left Diagram: \( r = a \cos \theta \), where \( a > 0 \)**
- This represents a circle centered on the x-axis.
- The radius of the circle is \( \frac{a}{2} \).
- The circle intersects the x-axis at \( a \) and \( -a \), and the y-axis at \( \frac{a}{2} \) and \( -\frac{a}{2} \).
2. **Right Diagram: \( r = a \sin \theta \), where \( a > 0 \)**
- This represents a circle centered on the y-axis.
- The radius of the circle is \( \frac{a}{2} \).
- The circle intersects the y-axis at \( a \) and \( -a \), and the x-axis at \( \frac{a}{2} \) and \( -\frac{a}{2} \).
#### Equations Relating Polar and Cartesian Coordinates
\[ x = r \cos \theta , \quad y = r \sin \theta , \quad r^2 = x^2 + y^2 , \quad \tan \theta = \frac{y}{x} \]
#### Exercises
**Sketch the graph of each polar equation.**
**Also, convert each into a rectangular coordinate equation.**
**13.** \( r = 2 \sin \theta \)
**14.** \( r = 3 \cos \theta \)
**15.** \( r = -\sin \theta \)
**16.** \( r = -\cos \theta \)
**17.** \( r \cos \theta = -4 \)
**Provided Blank Graphs:**
There are five blank coordinate graphs provided below the exercises for sketching the graphs of the given polar equations. Each graph has the standard x and y axes.
---
These explanations and exercises are designed to help you understand and practice converting between polar and Cartesian coordinates, and to graph polar equations appropriately.
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