Question
Given the rectangular coordinates left-parenthesis 2 comma negative 2 right-parenthesis, which of the following polar coordinate pairs, in radians, represents the same point?
(1 point)
Responses
left-parenthesis 2 Start Root 2 End Root comma Start Fraction 7 pi over 4 End Fraction right-parenthesis
Image with alt text: left-parenthesis 2 Start Root 2 End Root comma Start Fraction 7 pi over 4 End Fraction right-parenthesis
left-parenthesis 2 Start Root 2 End Root comma negative Start Fraction 7 pi over 4 End Fraction right-parenthesis
Image with alt text: left-parenthesis 2 Start Root 2 End Root comma negative Start Fraction 7 pi over 4 End Fraction right-parenthesis
left-parenthesis negative 2 Start Root 2 End Root comma negative Start Fraction pi over 4 End Fraction right-parenthesis
Image with alt text: left-parenthesis negative 2 Start Root 2 End Root comma negative Start Fraction pi over 4 End Fraction right-parenthesis
left-parenthesis 2 Start Root 2 End Root comma Start Fraction pi over 4 End Fraction right-parenthesis
Image with alt text: left-parenthesis 2 Start Root 2 End Root comma Start Fraction pi over 4 End Fraction right-parenthesis
(1 point)
Responses
left-parenthesis 2 Start Root 2 End Root comma Start Fraction 7 pi over 4 End Fraction right-parenthesis
Image with alt text: left-parenthesis 2 Start Root 2 End Root comma Start Fraction 7 pi over 4 End Fraction right-parenthesis
left-parenthesis 2 Start Root 2 End Root comma negative Start Fraction 7 pi over 4 End Fraction right-parenthesis
Image with alt text: left-parenthesis 2 Start Root 2 End Root comma negative Start Fraction 7 pi over 4 End Fraction right-parenthesis
left-parenthesis negative 2 Start Root 2 End Root comma negative Start Fraction pi over 4 End Fraction right-parenthesis
Image with alt text: left-parenthesis negative 2 Start Root 2 End Root comma negative Start Fraction pi over 4 End Fraction right-parenthesis
left-parenthesis 2 Start Root 2 End Root comma Start Fraction pi over 4 End Fraction right-parenthesis
Image with alt text: left-parenthesis 2 Start Root 2 End Root comma Start Fraction pi over 4 End Fraction right-parenthesis
Answers
GPT-4o mini
To convert the rectangular coordinates \((2, -2)\) into polar coordinates, we use the formulas for polar coordinates:
1. \( r = \sqrt{x^2 + y^2} \)
2. \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \)
where \( (x, y) = (2, -2) \).
First, let's find \( r \):
\[
r = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
Next, we'll calculate \( \theta \):
\[
\theta = \tan^{-1} \left( \frac{-2}{2} \right) = \tan^{-1}(-1)
\]
The angle \(\tan^{-1}(-1)\) corresponds to \(-\frac{\pi}{4}\) radians. However, we need to determine which quadrant the point \( (2, -2) \) is located in. Since \(x\) is positive and \(y\) is negative, the point is in the fourth quadrant.
The angle \(-\frac{\pi}{4}\) is indeed in the fourth quadrant, but polar coordinates can also represent angles by adding \(2\pi\). Therefore, we can express the angle in a positive direction within a full circle:
\[
\theta = -\frac{\pi}{4} + 2\pi = 2\pi - \frac{\pi}{4} = \frac{8\pi}{4} - \frac{\pi}{4} = \frac{7\pi}{4}
\]
Now, the polar coordinates corresponding to the rectangular coordinates \((2, -2)\) are \((2\sqrt{2}, \frac{7\pi}{4})\).
So, checking the options provided:
1. \( \left( 2\sqrt{2}, \frac{7\pi}{4} \right) \)
2. \( \left( 2\sqrt{2}, -\frac{7\pi}{4} \right) \)
3. \( \left( -2\sqrt{2}, -\frac{\pi}{4} \right) \)
4. \( \left( 2\sqrt{2}, \frac{\pi}{4} \right) \)
The correct answer that represents the same point as \((2, -2)\) in polar coordinates is:
**\( \left( 2\sqrt{2}, \frac{7\pi}{4} \right) \)**.
1. \( r = \sqrt{x^2 + y^2} \)
2. \( \theta = \tan^{-1} \left( \frac{y}{x} \right) \)
where \( (x, y) = (2, -2) \).
First, let's find \( r \):
\[
r = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
Next, we'll calculate \( \theta \):
\[
\theta = \tan^{-1} \left( \frac{-2}{2} \right) = \tan^{-1}(-1)
\]
The angle \(\tan^{-1}(-1)\) corresponds to \(-\frac{\pi}{4}\) radians. However, we need to determine which quadrant the point \( (2, -2) \) is located in. Since \(x\) is positive and \(y\) is negative, the point is in the fourth quadrant.
The angle \(-\frac{\pi}{4}\) is indeed in the fourth quadrant, but polar coordinates can also represent angles by adding \(2\pi\). Therefore, we can express the angle in a positive direction within a full circle:
\[
\theta = -\frac{\pi}{4} + 2\pi = 2\pi - \frac{\pi}{4} = \frac{8\pi}{4} - \frac{\pi}{4} = \frac{7\pi}{4}
\]
Now, the polar coordinates corresponding to the rectangular coordinates \((2, -2)\) are \((2\sqrt{2}, \frac{7\pi}{4})\).
So, checking the options provided:
1. \( \left( 2\sqrt{2}, \frac{7\pi}{4} \right) \)
2. \( \left( 2\sqrt{2}, -\frac{7\pi}{4} \right) \)
3. \( \left( -2\sqrt{2}, -\frac{\pi}{4} \right) \)
4. \( \left( 2\sqrt{2}, \frac{\pi}{4} \right) \)
The correct answer that represents the same point as \((2, -2)\) in polar coordinates is:
**\( \left( 2\sqrt{2}, \frac{7\pi}{4} \right) \)**.