Multiple Choice
Given the point with polar coordinates , which of the following polar coordinate pairs labels the same point?
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9. Polar Equations
Polar Coordinate System
Multiple Choice
Which of the following correctly expresses the conversion from rectangular coordinates to cylindrical coordinates ?
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Verified step by step guidance1
Recall that cylindrical coordinates \((r, \theta, z)\) relate to rectangular coordinates \((x, y, z)\) by expressing the position in terms of a radius \(r\) in the \(xy\)-plane, an angle \(\theta\) from the positive \(x\)-axis, and the same height \(z\).
Calculate the radius \(r\) as the distance from the origin to the projection of the point onto the \(xy\)-plane using the Pythagorean theorem:
\[r = \sqrt{x^{2} + y^{2}}\]
Determine the angle \(\theta\) as the angle between the positive \(x\)-axis and the line connecting the origin to the point's projection in the \(xy\)-plane. This is given by the arctangent of \(y/x\):
\[\theta = \arctan\left(\frac{y}{x}\right)\]
Note that the height \(z\) remains the same in both coordinate systems, so:
\[z = z\]
Combine these results to express the conversion from rectangular to cylindrical coordinates as:
\[\left(r, \theta, z\right) = \left(\sqrt{x^{2} + y^{2}}, \arctan\left(\frac{y}{x}\right), z\right)\]
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