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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:

figure

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:

figure

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

equation

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

equation

equation

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

equation

equation

equation

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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:

equation

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

equation

or

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