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To understand which graph is represented by the given table of polar coordinates, let's analyze the points step-by-step:
1. First Point:
[tex]\(\theta = 0\)[/tex], [tex]\(r = 4\)[/tex]
This means we have a point on the polar graph at an angle of [tex]\(0\)[/tex] radians (or [tex]\(0^\circ\)[/tex]), which is directly along the positive x-axis, and a radius (distance from the origin) of 4.
Cartesian coordinates: [tex]\((4, 0)\)[/tex]
2. Second Point:
[tex]\(\theta = \frac{\pi}{6}\)[/tex], [tex]\(r = 2\sqrt{3}\)[/tex]
Here, the angle is [tex]\(\frac{\pi}{6}\)[/tex] radians (or [tex]\(30^\circ\)[/tex]), with a radius of [tex]\(2\sqrt{3}\)[/tex].
Cartesian coordinates: [tex]\((2\sqrt{3}\cos(\frac{\pi}{6}), 2\sqrt{3}\sin(\frac{\pi}{6})) = (3, 1)\)[/tex]
3. Third Point:
[tex]\(\theta = \frac{\pi}{3}\)[/tex], [tex]\(r = 2\)[/tex]
The angle is [tex]\(\frac{\pi}{3}\)[/tex] radians (or [tex]\(60^\circ\)[/tex]), with a radius of 2.
Cartesian coordinates: [tex]\((2\cos(\frac{\pi}{3}), 2\sin(\frac{\pi}{3})) = (1, \sqrt{3}) = (1, 1.732)\)[/tex]
4. Fourth Point:
[tex]\(\theta = \frac{\pi}{2}\)[/tex], [tex]\(r = 0\)[/tex]
The angle [tex]\(\frac{\pi}{2}\)[/tex] radians (or [tex]\(90^\circ\)[/tex]) with a radius of 0 is simply the origin.
Cartesian coordinates: [tex]\((0, 0)\)[/tex]
5. Fifth Point:
[tex]\(\theta = \frac{2 \pi}{3}\)[/tex], [tex]\(r = -2\)[/tex]
Here, the angle is [tex]\(\frac{2 \pi}{3}\)[/tex] radians (or [tex]\(120^\circ\)[/tex]), and since the radius is negative, we move along the line at [tex]\(120^\circ\)[/tex] but opposite the direction.
Cartesian coordinates: [tex]\((-2\cos(\frac{2\pi}{3}), -2\sin(\frac{2\pi}{3})) = (-1, -\sqrt{3}) = (-2, -1.732)\)[/tex]
6. Sixth Point:
[tex]\(\theta = \frac{5 \pi}{6}\)[/tex], [tex]\(r = -2\sqrt{3}\)[/tex]
The angle is [tex]\(\frac{5 \pi}{6}\)[/tex] radians (or [tex]\(150^\circ\)[/tex]), and with a negative radius of [tex]\(-2 \sqrt{3}\)[/tex].
Cartesian coordinates: [tex]\((-2\sqrt{3}\cos(\frac{5\pi}{6}), -2\sqrt{3}\sin(\frac{5\pi}{6})) =\\ (-\sqrt{3}, -2\sqrt{3}\left(\frac{1}{2}\right)) = (-2.598, -1)\)[/tex]
7. Seventh Point:
[tex]\(\theta = \pi\)[/tex], [tex]\(r = -4\)[/tex]
The angle [tex]\(\pi\)[/tex] radians (or [tex]\(180^\circ\)[/tex]) with a radius of [tex]\(-4\)[/tex] is directly opposite along the negative x-axis.
Cartesian coordinates: [tex]\((-4, 0)\)[/tex]
By plotting these points on a polar graph, you can determine which graph represents the given table accurately. Each point should be placed according to its angle and radius.
Understanding the transformation from polar coordinates to Cartesian can help in verifying them:
- The angles [tex]\(\theta\)[/tex] are given in radians and should be measured counter-clockwise from the positive x-axis.
- The radius [tex]\(r\)[/tex] indicates the distance from the origin, with negative values indicating a direction opposite to the angle [tex]\(\theta\)[/tex].
Thus, the representation of the given points on a polar graph aligns with these computed points and their properties. This will help you in choosing the correct graphical representation from the given options.
1. First Point:
[tex]\(\theta = 0\)[/tex], [tex]\(r = 4\)[/tex]
This means we have a point on the polar graph at an angle of [tex]\(0\)[/tex] radians (or [tex]\(0^\circ\)[/tex]), which is directly along the positive x-axis, and a radius (distance from the origin) of 4.
Cartesian coordinates: [tex]\((4, 0)\)[/tex]
2. Second Point:
[tex]\(\theta = \frac{\pi}{6}\)[/tex], [tex]\(r = 2\sqrt{3}\)[/tex]
Here, the angle is [tex]\(\frac{\pi}{6}\)[/tex] radians (or [tex]\(30^\circ\)[/tex]), with a radius of [tex]\(2\sqrt{3}\)[/tex].
Cartesian coordinates: [tex]\((2\sqrt{3}\cos(\frac{\pi}{6}), 2\sqrt{3}\sin(\frac{\pi}{6})) = (3, 1)\)[/tex]
3. Third Point:
[tex]\(\theta = \frac{\pi}{3}\)[/tex], [tex]\(r = 2\)[/tex]
The angle is [tex]\(\frac{\pi}{3}\)[/tex] radians (or [tex]\(60^\circ\)[/tex]), with a radius of 2.
Cartesian coordinates: [tex]\((2\cos(\frac{\pi}{3}), 2\sin(\frac{\pi}{3})) = (1, \sqrt{3}) = (1, 1.732)\)[/tex]
4. Fourth Point:
[tex]\(\theta = \frac{\pi}{2}\)[/tex], [tex]\(r = 0\)[/tex]
The angle [tex]\(\frac{\pi}{2}\)[/tex] radians (or [tex]\(90^\circ\)[/tex]) with a radius of 0 is simply the origin.
Cartesian coordinates: [tex]\((0, 0)\)[/tex]
5. Fifth Point:
[tex]\(\theta = \frac{2 \pi}{3}\)[/tex], [tex]\(r = -2\)[/tex]
Here, the angle is [tex]\(\frac{2 \pi}{3}\)[/tex] radians (or [tex]\(120^\circ\)[/tex]), and since the radius is negative, we move along the line at [tex]\(120^\circ\)[/tex] but opposite the direction.
Cartesian coordinates: [tex]\((-2\cos(\frac{2\pi}{3}), -2\sin(\frac{2\pi}{3})) = (-1, -\sqrt{3}) = (-2, -1.732)\)[/tex]
6. Sixth Point:
[tex]\(\theta = \frac{5 \pi}{6}\)[/tex], [tex]\(r = -2\sqrt{3}\)[/tex]
The angle is [tex]\(\frac{5 \pi}{6}\)[/tex] radians (or [tex]\(150^\circ\)[/tex]), and with a negative radius of [tex]\(-2 \sqrt{3}\)[/tex].
Cartesian coordinates: [tex]\((-2\sqrt{3}\cos(\frac{5\pi}{6}), -2\sqrt{3}\sin(\frac{5\pi}{6})) =\\ (-\sqrt{3}, -2\sqrt{3}\left(\frac{1}{2}\right)) = (-2.598, -1)\)[/tex]
7. Seventh Point:
[tex]\(\theta = \pi\)[/tex], [tex]\(r = -4\)[/tex]
The angle [tex]\(\pi\)[/tex] radians (or [tex]\(180^\circ\)[/tex]) with a radius of [tex]\(-4\)[/tex] is directly opposite along the negative x-axis.
Cartesian coordinates: [tex]\((-4, 0)\)[/tex]
By plotting these points on a polar graph, you can determine which graph represents the given table accurately. Each point should be placed according to its angle and radius.
Understanding the transformation from polar coordinates to Cartesian can help in verifying them:
- The angles [tex]\(\theta\)[/tex] are given in radians and should be measured counter-clockwise from the positive x-axis.
- The radius [tex]\(r\)[/tex] indicates the distance from the origin, with negative values indicating a direction opposite to the angle [tex]\(\theta\)[/tex].
Thus, the representation of the given points on a polar graph aligns with these computed points and their properties. This will help you in choosing the correct graphical representation from the given options.
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