Question
Question
If lower r not equal to 0, which of the following polar coordinate pairs represents the same point as the point with polar coordinates left parenthesis lower r comma theta right parenthesis? Select all that apply.
(1 point)
Responses
left parenthesis negative lower r comma theta minus 2 pi right parenthesis
Image with alt text: left parenthesis negative lower r comma theta minus 2 pi right parenthesis
left parenthesis negative lower r comma theta plus 3 pi right parenthesis
Image with alt text: left parenthesis negative lower r comma theta plus 3 pi right parenthesis
left parenthesis lower r comma theta plus 2 pi right parenthesis
Image with alt text: left parenthesis lower r comma theta plus 2 pi right parenthesis
left parenthesis lower r comma theta minus pi right parenthesis
If lower r not equal to 0, which of the following polar coordinate pairs represents the same point as the point with polar coordinates left parenthesis lower r comma theta right parenthesis? Select all that apply.
(1 point)
Responses
left parenthesis negative lower r comma theta minus 2 pi right parenthesis
Image with alt text: left parenthesis negative lower r comma theta minus 2 pi right parenthesis
left parenthesis negative lower r comma theta plus 3 pi right parenthesis
Image with alt text: left parenthesis negative lower r comma theta plus 3 pi right parenthesis
left parenthesis lower r comma theta plus 2 pi right parenthesis
Image with alt text: left parenthesis lower r comma theta plus 2 pi right parenthesis
left parenthesis lower r comma theta minus pi right parenthesis
Answers
GPT-4o mini
In polar coordinates, the point \((r, \theta)\) represents a location in the polar plane, where \(r\) is the radial distance from the origin and \(\theta\) is the angle measured from the positive x-axis.
For a given point \((r, \theta)\), the following transformations yield the same point:
1. \((r, \theta + 2k\pi)\) for any integer \(k\) (this accounts for the periodicity of the angle).
2. \((-r, \theta + \pi)\) (this reflects the point across the origin).
Given this, let's analyze each option:
1. **\((-r, \theta - 2\pi)\)**:
\[
(-r, \theta - 2\pi) \equiv (-r, \theta)
\]
This represents the point where \(r\) is negated and the angle is adjusted by the periodicity of \(2\pi\), which does not yield the same point (it reflects the point across the origin). Thus, this option **does not** represent the same point.
2. **\((-r, \theta + 3\pi)\)**:
\[
(-r, \theta + 3\pi) \equiv (-r, \theta + \pi)
\]
This also represents the point where \(r\) is negated, which reflects it. Thus, this option **does not** represent the same point.
3. **\((r, \theta + 2\pi)\)**:
This maintains the same radial distance but adds \(2\pi\) to the angle, thus representing the same point due to periodicity. This option **does** represent the same point.
4. **\((r, \theta - \pi)\)**:
This reflects the point since it keeps \(r\) and shifts the angle by \(-\pi\). This effectively negates the radial distance resulting in the same direction but at a distance of \(-r\). Thus, this option **does** represent the same point.
The correct options that represent the same point as \((r, \theta)\) are:
- **\((r, \theta + 2\pi)\)**
- **\((r, \theta - \pi)\)**
For a given point \((r, \theta)\), the following transformations yield the same point:
1. \((r, \theta + 2k\pi)\) for any integer \(k\) (this accounts for the periodicity of the angle).
2. \((-r, \theta + \pi)\) (this reflects the point across the origin).
Given this, let's analyze each option:
1. **\((-r, \theta - 2\pi)\)**:
\[
(-r, \theta - 2\pi) \equiv (-r, \theta)
\]
This represents the point where \(r\) is negated and the angle is adjusted by the periodicity of \(2\pi\), which does not yield the same point (it reflects the point across the origin). Thus, this option **does not** represent the same point.
2. **\((-r, \theta + 3\pi)\)**:
\[
(-r, \theta + 3\pi) \equiv (-r, \theta + \pi)
\]
This also represents the point where \(r\) is negated, which reflects it. Thus, this option **does not** represent the same point.
3. **\((r, \theta + 2\pi)\)**:
This maintains the same radial distance but adds \(2\pi\) to the angle, thus representing the same point due to periodicity. This option **does** represent the same point.
4. **\((r, \theta - \pi)\)**:
This reflects the point since it keeps \(r\) and shifts the angle by \(-\pi\). This effectively negates the radial distance resulting in the same direction but at a distance of \(-r\). Thus, this option **does** represent the same point.
The correct options that represent the same point as \((r, \theta)\) are:
- **\((r, \theta + 2\pi)\)**
- **\((r, \theta - \pi)\)**