# Parametric Equations and Polar Coordinates

### Polar Coordinates

The polar coordinate system consists of a pole and a polar axis. The pole is a fixed point, and the polar axis is a directed ray whose endpoint is the pole. Every point in the plane of the polar axis can be specified according to two coordinates: r , the distance between the point and the pole, and θ , the angle between the polar axis and the ray containing the point whose endpoint is also the pole.

Figure %: The polar coordinate system
The distance r and the angle θ are both directed--meaning that they represent the distance and angle in a given direction. It is possible, therefore to have negative values for both r and θ . However, we typically avoid points with negative r , since they could just as easily be specified by adding Π (or 180 o ) to θ . Similarly, we typically ask that θ be in the range 0≤θ < 2Π , since there is always some θ in this range corresponding to our point. This doesn't eliminate all ambiguity, however; the pole can still be specified by (0, θ) for any angle θ . But it is true that any other point can be described uniquely with these conventions.

To convert equations between polar coordinates and rectangular coordinates, consider the following diagram:

Figure %: The x and y coordinates in the polar coordinate system
See that sin(θ) = , and cos(θ) = .

To convert from rectangular to polar coordinates, use the following equations: x = r cos(θ), y = r sin(θ) . To convert from polar to rectangular coordinates, use these equations: r = sqrtx 2+y 2, θ = arctan() .

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