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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Question
Which equation is the polar equivalent to the equation y=−3√x?
![### Conversion of Cartesian Equation to Polar Equation in Mathematics
#### Problem Statement
Which equation is the polar equivalent to the equation \( y = -\sqrt{3}x \)?
#### Answer Choices
1. \( \theta = \frac{\pi}{6} \)
2. \( \theta = \frac{\pi}{3} \)
3. \( \theta = \frac{2\pi}{3} \)
4. \( \theta = \frac{5\pi}{6} \)
#### Explanation
To determine which of the given polar equations is equivalent to the Cartesian equation \( y = -\sqrt{3}x \), we need to understand the relationship between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\).
The Cartesian equation \( y = -\sqrt{3}x \) represents a line passing through the origin with a specific slope:
\[
\text{slope} = -\sqrt{3}
\]
In polar coordinates, a line with slope \( m \) can be written as:
\[
\tan(\theta) = m
\]
To find the corresponding angle \( \theta \):
\[
\tan(\theta) = -\sqrt{3}
\]
The angles for which the tangent value is \(-\sqrt{3}\) are:
\[
\theta = \frac{2\pi}{3}, \frac{5\pi}{6}
\]
#### Reasoning
By comparing these angles to the given choices, we can see that the correct answer must be one of these options.
Thus, the correct equation among the given choices is:
\[
\theta = \frac{2\pi}{3} \quad \text{(Option 3)}
\]
Therefore, the polar equivalent to the given Cartesian equation \( y = -\sqrt{3}x \) is \( \theta = \frac{2\pi}{3} \).
Selecting the correct response ensures that learners can correctly convert between Cartesian and polar coordinate systems, improving their understanding of coordinate geometry.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec391479-b25a-480e-a409-64858e4cacbb%2Fbc9d5773-1afd-4c07-a718-af67c7786039%2Fjjlw3dr_processed.png&w=3840&q=75)
Transcribed Image Text:### Conversion of Cartesian Equation to Polar Equation in Mathematics
#### Problem Statement
Which equation is the polar equivalent to the equation \( y = -\sqrt{3}x \)?
#### Answer Choices
1. \( \theta = \frac{\pi}{6} \)
2. \( \theta = \frac{\pi}{3} \)
3. \( \theta = \frac{2\pi}{3} \)
4. \( \theta = \frac{5\pi}{6} \)
#### Explanation
To determine which of the given polar equations is equivalent to the Cartesian equation \( y = -\sqrt{3}x \), we need to understand the relationship between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\).
The Cartesian equation \( y = -\sqrt{3}x \) represents a line passing through the origin with a specific slope:
\[
\text{slope} = -\sqrt{3}
\]
In polar coordinates, a line with slope \( m \) can be written as:
\[
\tan(\theta) = m
\]
To find the corresponding angle \( \theta \):
\[
\tan(\theta) = -\sqrt{3}
\]
The angles for which the tangent value is \(-\sqrt{3}\) are:
\[
\theta = \frac{2\pi}{3}, \frac{5\pi}{6}
\]
#### Reasoning
By comparing these angles to the given choices, we can see that the correct answer must be one of these options.
Thus, the correct equation among the given choices is:
\[
\theta = \frac{2\pi}{3} \quad \text{(Option 3)}
\]
Therefore, the polar equivalent to the given Cartesian equation \( y = -\sqrt{3}x \) is \( \theta = \frac{2\pi}{3} \).
Selecting the correct response ensures that learners can correctly convert between Cartesian and polar coordinate systems, improving their understanding of coordinate geometry.
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