Show that the sets of points satisfying

are equal.

Letting and we plug in to the given Cartesian equation,

where so (hence, we can divide by ) in the second to last line.

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Stumbling Robot

A Fraction of a Dot
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Show an equivalence between given sets in Cartesian and polar coordinates

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Show that the sets of points satisfying

are equal.

Letting and we plug in to the given Cartesian equation,

where so (hence, we can divide by ) in the second to last line.

The sets are not equal.

(x, y) == (0, 0) satisfies the left equation. The origin expressed as polar coordinates is (0, theta), but the equations on the right say that cos(theta)>0. Therefore, r = 2*cos(theta) > 0. Therefore, the Origin is not included in the set defined by the equations on the right.

The set of points on the right is r = 2 sin theta.

The origin is certainly included in that set. At theta = 0, cos theta = 1 which is greater than 0.

Then, r becomes r = 2*cos 0 = 2, but it has to be zero to be the origin.