Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter8: Polar Coordinates And Parametric Equations
Section8.CT: Chapter Test
Problem 1CT
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Can you provide the answer with step by step instructions.
![### Conversion of Polar Coordinates to Rectangular Coordinates
In this example, we start with the polar equation given by:
\[ r = 2 \sin \theta \]
#### Objective:
Express the given polar equation in rectangular coordinates.
#### Solution:
To convert the given polar equation to rectangular coordinates, we use the following relationships between polar and rectangular coordinates:
1. \( x = r \cos \theta \)
2. \( y = r \sin \theta \)
3. \( r^2 = x^2 + y^2 \)
Starting with the given polar equation:
\[ r = 2 \sin \theta \]
We know that \( \sin \theta = \frac{y}{r} \).
Substitute this into the equation:
\[ r = 2 \left( \frac{y}{r} \right) \]
Multiply both sides by \( r \) to clear the fraction:
\[ r^2 = 2y \]
Use the relationship \( r^2 = x^2 + y^2 \):
\[ x^2 + y^2 = 2y \]
This is the equation in rectangular coordinates.
#### Summary:
The polar equation \( r = 2 \sin \theta \) is equivalent to the rectangular equation \( x^2 + y^2 = 2y \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b3adb48-57c8-4732-a4e1-40416a51be41%2F9ac83cdc-2dc7-480c-9f2b-2f9af2b3e48a%2Fzqewbvf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Conversion of Polar Coordinates to Rectangular Coordinates
In this example, we start with the polar equation given by:
\[ r = 2 \sin \theta \]
#### Objective:
Express the given polar equation in rectangular coordinates.
#### Solution:
To convert the given polar equation to rectangular coordinates, we use the following relationships between polar and rectangular coordinates:
1. \( x = r \cos \theta \)
2. \( y = r \sin \theta \)
3. \( r^2 = x^2 + y^2 \)
Starting with the given polar equation:
\[ r = 2 \sin \theta \]
We know that \( \sin \theta = \frac{y}{r} \).
Substitute this into the equation:
\[ r = 2 \left( \frac{y}{r} \right) \]
Multiply both sides by \( r \) to clear the fraction:
\[ r^2 = 2y \]
Use the relationship \( r^2 = x^2 + y^2 \):
\[ x^2 + y^2 = 2y \]
This is the equation in rectangular coordinates.
#### Summary:
The polar equation \( r = 2 \sin \theta \) is equivalent to the rectangular equation \( x^2 + y^2 = 2y \).
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