The polar coordinates of a point are given. Plot the point. (s, ) T/2 л/2 6. 6. 4 4 2 -6 -4 -2 2 -6 -4 -2 4 6. -2 -2 -4 -4 7/2 T/2 6. 6. 4 4 2 2 -6 -4 -2 2 4 6 - 6 -4 2 4 -2 -2 -4 -4 -6 Find the corresponding rectangular coordinates for the point. (x, y) = 2. 2, 2.
The polar coordinates of a point are given. Plot the point. (s, ) T/2 л/2 6. 6. 4 4 2 -6 -4 -2 2 -6 -4 -2 4 6. -2 -2 -4 -4 7/2 T/2 6. 6. 4 4 2 2 -6 -4 -2 2 4 6 - 6 -4 2 4 -2 -2 -4 -4 -6 Find the corresponding rectangular coordinates for the point. (x, y) = 2. 2, 2.
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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![### Plotting Polar Coordinates and Finding Rectangular Coordinates
#### Problem Statement:
The polar coordinates of a point are given. Plot the point.
\[
\left( 6, \frac{5\pi}{4} \right)
\]
Four separate coordinate grids are displayed, each with different points marked on them. You are required to determine which graph correctly represents the given polar coordinates.
#### Detailed Explanation of Polar Coordinates:
Polar coordinates are represented in the form \((r, \theta)\) where:
- \(r\) is the radius - the distance from the origin (0,0).
- \(\theta\) is the angle, measured in radians, from the positive x-axis.
For the coordinates \(\left( 6, \frac{5\pi}{4} \right)\):
- The radius \(r\) is 6.
- The angle \(\theta\) is \(\frac{5\pi}{4}\).
Based on the angle, \(\frac{5\pi}{4}\) is in the third quadrant because it is equivalent to \(225^\circ\), which is more than \(180^\circ\) and less than \(270^\circ\).
#### Graph Descriptions:
1. **Top left graph**:
- Displays a point in the first quadrant.
2. **Top right graph**:
- Displays a point in the second quadrant.
3. **Bottom left graph**:
- Displays a point in the fourth quadrant at coordinates (6, -6).
4. **Bottom right graph**:
- Displays a point in the third quadrant approximately at (-4.2, -4.2).
The correct graph representing the given polar coordinates is the **bottom right** one.
#### Finding Rectangular Coordinates:
To convert polar coordinates \(\left( 6, \frac{5\pi}{4} \right)\) to rectangular coordinates (x, y), we use the formulae:
\[ x = r \cos(\theta) \]
\[ y = r \sin(\theta) \]
For the given coordinates:
\[ x = 6 \cos\left(\frac{5\pi}{4}\right) \]
\[ y = 6 \sin\left(\frac{5\pi}{4}\right) \]
Knowing that:
\[
\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F512ef196-f607-4f2c-ad14-c7c8491bf1a1%2F31c91e31-8d72-4de4-b563-1d7db3381d1b%2Fkj2g52_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Plotting Polar Coordinates and Finding Rectangular Coordinates
#### Problem Statement:
The polar coordinates of a point are given. Plot the point.
\[
\left( 6, \frac{5\pi}{4} \right)
\]
Four separate coordinate grids are displayed, each with different points marked on them. You are required to determine which graph correctly represents the given polar coordinates.
#### Detailed Explanation of Polar Coordinates:
Polar coordinates are represented in the form \((r, \theta)\) where:
- \(r\) is the radius - the distance from the origin (0,0).
- \(\theta\) is the angle, measured in radians, from the positive x-axis.
For the coordinates \(\left( 6, \frac{5\pi}{4} \right)\):
- The radius \(r\) is 6.
- The angle \(\theta\) is \(\frac{5\pi}{4}\).
Based on the angle, \(\frac{5\pi}{4}\) is in the third quadrant because it is equivalent to \(225^\circ\), which is more than \(180^\circ\) and less than \(270^\circ\).
#### Graph Descriptions:
1. **Top left graph**:
- Displays a point in the first quadrant.
2. **Top right graph**:
- Displays a point in the second quadrant.
3. **Bottom left graph**:
- Displays a point in the fourth quadrant at coordinates (6, -6).
4. **Bottom right graph**:
- Displays a point in the third quadrant approximately at (-4.2, -4.2).
The correct graph representing the given polar coordinates is the **bottom right** one.
#### Finding Rectangular Coordinates:
To convert polar coordinates \(\left( 6, \frac{5\pi}{4} \right)\) to rectangular coordinates (x, y), we use the formulae:
\[ x = r \cos(\theta) \]
\[ y = r \sin(\theta) \]
For the given coordinates:
\[ x = 6 \cos\left(\frac{5\pi}{4}\right) \]
\[ y = 6 \sin\left(\frac{5\pi}{4}\right) \]
Knowing that:
\[
\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt
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