Multiple Choice
Given the point with polar coordinates , which of the following polar coordinate pairs labels the same point?
9. Polar Equations
Polar Coordinate System
Multiple Choice
Which of the following correctly expresses the conversion from rectangular coordinates to cylindrical coordinates ?
A
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D
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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