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This essential trains for: Math Kangaroo 9-10, SAT-II, AMC-10, AMC-12.

Positions of points in the plane can be described in more than 1 way. Cartesian coordinates are one way, polar coordinates are another.

While Cartesian coordinates specify the position of a point in the plane using two distances, polar coordinates specify the position of a point using a distance and an angle.

The Cartesian coordinates of the point P are xP, yP:

while the polar coordinates of the same point are: the distance r from the origin to the point and the angle between the x-axis and the segment r:

Converting from Cartesian to polar coordinates is done simply, by using the Pythagorean theorem:

and any one of the trigonometric functions from elementary geometry. Usually, the tangent function:

Converting from polar coordinates to Cartesian coordinates is done by using trigonometric relations in the right triangle:

It is important to note that, due to the periodicity of the trigonometric functions, polar representations are not unique (whereas Cartesian ones are). For example:

To make them unique we must restrict the domain for r and θ. Either:

or

are satisfactory. The first one is preferred by physicists who prefer positive lengths for r.