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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![**Converting Rectangular Coordinates to Polar Coordinates**
**Instructions:**
Convert the rectangular coordinates of each point to polar coordinates. Use degrees for θ.
1. \( (\sqrt{3}, 3) \)
2. \( (-2, 2) \)
3. \( (0, 2) \)
4. \( (-3, -3) \)
5. \( (4, 4) \)
6. \( (-2, 2\sqrt{3}) \)
**Formulas:**
\[ x^2 + y^2 = r^2 \]
\[ \tan \theta = \frac{y}{x} \]
**Example:**
**Sketch the graph of the polar equation \( r = 2 \cos \theta \)**
**Step 1 => Create table**
\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\theta & 0^\circ & 30^\circ & 45^\circ & 60^\circ & 90^\circ & 120^\circ & 135^\circ & 150^\circ & 180^\circ \\
\hline
r & & & & & & & & & \\
\hline
\end{array}
\]
**Step 2 => Plot points and draw curve**
**Explanation of Diagram:**
The diagram below the steps illustrates the graph of the polar equation \( r = 2 \cos \theta \). The horizontal x-axis and vertical y-axis guide the placement of these points. Various points are plotted such as:
- \( r = 2, \theta = 0^\circ \) at (2, 0)
- \( r = \sqrt{3}, \theta = 30^\circ \) at \((\sqrt{3}, 30^\circ)\)
- \( r = 1, \theta = 60^\circ \) at (1, 60^\circ)
- \( r = 0, \theta = 90^\circ \) at (0, 90^\circ)
The points are joined to reveal the curve described by the polar equation \( r = 2 \cos \theta \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F894ba220-0d5e-4e50-906d-a5e96c05bfb6%2F775db3c7-0358-458b-8aaa-237753c0467a%2Fh5wa13_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Converting Rectangular Coordinates to Polar Coordinates**
**Instructions:**
Convert the rectangular coordinates of each point to polar coordinates. Use degrees for θ.
1. \( (\sqrt{3}, 3) \)
2. \( (-2, 2) \)
3. \( (0, 2) \)
4. \( (-3, -3) \)
5. \( (4, 4) \)
6. \( (-2, 2\sqrt{3}) \)
**Formulas:**
\[ x^2 + y^2 = r^2 \]
\[ \tan \theta = \frac{y}{x} \]
**Example:**
**Sketch the graph of the polar equation \( r = 2 \cos \theta \)**
**Step 1 => Create table**
\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\theta & 0^\circ & 30^\circ & 45^\circ & 60^\circ & 90^\circ & 120^\circ & 135^\circ & 150^\circ & 180^\circ \\
\hline
r & & & & & & & & & \\
\hline
\end{array}
\]
**Step 2 => Plot points and draw curve**
**Explanation of Diagram:**
The diagram below the steps illustrates the graph of the polar equation \( r = 2 \cos \theta \). The horizontal x-axis and vertical y-axis guide the placement of these points. Various points are plotted such as:
- \( r = 2, \theta = 0^\circ \) at (2, 0)
- \( r = \sqrt{3}, \theta = 30^\circ \) at \((\sqrt{3}, 30^\circ)\)
- \( r = 1, \theta = 60^\circ \) at (1, 60^\circ)
- \( r = 0, \theta = 90^\circ \) at (0, 90^\circ)
The points are joined to reveal the curve described by the polar equation \( r = 2 \cos \theta \).
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