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Certainly! Here is a detailed step-by-step solution for converting the given Cartesian coordinates [tex]\((3, -5)\)[/tex] to polar coordinates [tex]\((r, \theta)\)[/tex], with [tex]\(r > 0\)[/tex] and [tex]\(0 \leq \theta \leq 2\pi\)[/tex]:
1. Identify the Cartesian coordinates:
The given coordinates are [tex]\( (x, y) = (3, -5) \)[/tex].
2. Calculate the radius [tex]\(r\)[/tex]:
The radius [tex]\(r\)[/tex] in polar coordinates is given by the distance from the origin to the point [tex]\((x, y)\)[/tex], which can be calculated using the Pythagorean theorem:
[tex]\[ r = \sqrt{x^2 + y^2} \][/tex]
- [tex]\(x = 3\)[/tex]
- [tex]\(y = -5\)[/tex]
Substituting the values in:
[tex]\[ r = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \][/tex]
Thus, the radius [tex]\(r\)[/tex] is:
[tex]\[ r \approx 5.830951894845301 \][/tex]
3. Determine the angle [tex]\(\theta\)[/tex]:
The angle [tex]\(\theta\)[/tex] in polar coordinates is the angle formed by the line connecting the origin to the point [tex]\((x, y)\)[/tex] with the positive [tex]\(x\)[/tex]-axis. This can be determined using the arctangent function, which considers the ratio [tex]\(\frac{y}{x}\)[/tex]:
[tex]\[ \theta = \arctan\left(\frac{y}{x}\right) \][/tex]
where:
- [tex]\(x = 3\)[/tex]
- [tex]\(y = -5\)[/tex]
[tex]\[ \theta = \arctan\left(\frac{-5}{3}\right) \][/tex]
4. Adjust for the correct quadrant:
Since the point [tex]\((3, -5)\)[/tex] is in the fourth quadrant (where [tex]\(x > 0\)[/tex] and [tex]\(y < 0\)[/tex]), the angle [tex]\(\theta\)[/tex] determined by [tex]\(\arctan\left(\frac{-5}{3}\right)\)[/tex] will be negative. To convert it to an angle in the range [tex]\([0, 2\pi]\)[/tex], we add [tex]\(2\pi\)[/tex] to the negative angle:
[tex]\[ \theta + 2\pi \][/tex]
Thus, after this adjustment, [tex]\(\theta\)[/tex] is:
[tex]\[ \theta \approx 5.252808480655274 \][/tex]
5. Conclusion:
The polar coordinates [tex]\((r, \theta)\)[/tex] for the Cartesian point [tex]\((3, -5)\)[/tex] are:
[tex]\[ (r, \theta) \approx (5.830951894845301, 5.252808480655274) \][/tex]
These are the polar coordinates corresponding to the given Cartesian coordinates with the specified conditions [tex]\(r > 0\)[/tex] and [tex]\(0 \leq \theta \leq 2\pi\)[/tex].
1. Identify the Cartesian coordinates:
The given coordinates are [tex]\( (x, y) = (3, -5) \)[/tex].
2. Calculate the radius [tex]\(r\)[/tex]:
The radius [tex]\(r\)[/tex] in polar coordinates is given by the distance from the origin to the point [tex]\((x, y)\)[/tex], which can be calculated using the Pythagorean theorem:
[tex]\[ r = \sqrt{x^2 + y^2} \][/tex]
- [tex]\(x = 3\)[/tex]
- [tex]\(y = -5\)[/tex]
Substituting the values in:
[tex]\[ r = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \][/tex]
Thus, the radius [tex]\(r\)[/tex] is:
[tex]\[ r \approx 5.830951894845301 \][/tex]
3. Determine the angle [tex]\(\theta\)[/tex]:
The angle [tex]\(\theta\)[/tex] in polar coordinates is the angle formed by the line connecting the origin to the point [tex]\((x, y)\)[/tex] with the positive [tex]\(x\)[/tex]-axis. This can be determined using the arctangent function, which considers the ratio [tex]\(\frac{y}{x}\)[/tex]:
[tex]\[ \theta = \arctan\left(\frac{y}{x}\right) \][/tex]
where:
- [tex]\(x = 3\)[/tex]
- [tex]\(y = -5\)[/tex]
[tex]\[ \theta = \arctan\left(\frac{-5}{3}\right) \][/tex]
4. Adjust for the correct quadrant:
Since the point [tex]\((3, -5)\)[/tex] is in the fourth quadrant (where [tex]\(x > 0\)[/tex] and [tex]\(y < 0\)[/tex]), the angle [tex]\(\theta\)[/tex] determined by [tex]\(\arctan\left(\frac{-5}{3}\right)\)[/tex] will be negative. To convert it to an angle in the range [tex]\([0, 2\pi]\)[/tex], we add [tex]\(2\pi\)[/tex] to the negative angle:
[tex]\[ \theta + 2\pi \][/tex]
Thus, after this adjustment, [tex]\(\theta\)[/tex] is:
[tex]\[ \theta \approx 5.252808480655274 \][/tex]
5. Conclusion:
The polar coordinates [tex]\((r, \theta)\)[/tex] for the Cartesian point [tex]\((3, -5)\)[/tex] are:
[tex]\[ (r, \theta) \approx (5.830951894845301, 5.252808480655274) \][/tex]
These are the polar coordinates corresponding to the given Cartesian coordinates with the specified conditions [tex]\(r > 0\)[/tex] and [tex]\(0 \leq \theta \leq 2\pi\)[/tex].
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