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Unformatted text preview: The Polar Coordinate System Alisa Favinger Cozad, Nebraska In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 Polar Coordinate System ~ 1 Representing a position in a twodimensional plane can be done several ways. It is taught early in Algebra how to represent a point in the Cartesian (or rectangular) plane. In this plane a point is represented by the coordinates ( x , y ), where x tells the horizontal distance from the origin and y the vertical distance. The polar coordinate system is an alternative to this rectangular system. In this system, instead of a point being represented by ( x , y ) coordinates, a point is represented by ( r , θ ) where r represents the length of a straight line from the point to the origin and θ represents the angle that straight line makes with the horizontal axis. The r component is commonly referred to as the radial coordinate and θ as the angular coordinate. Just as in the Cartesian plane, the polar plane has a horizontal axis and an origin. In the polar system the origin is called the pole and the horizontal axis, which is a ray that extends horizontally from the pole to the right, is called the polar axis . An illustration of this can be seen in the figure below: r Polar Coordinate System ~ 2 In the figure, the pole is labeled (0, θ ) because the 0 indicates a distance of 0 from the pole, so (0, θ ) will be exactly at the pole regardless of the angle θ . The units of θ can be given in radians or degrees, but generally is given in radians. In this paper we will use both radians and degrees. To translate between radians and degrees, we recall the conversion rules: To convert from radians to degrees, multiply by π 180 To convert from degrees to radians, multiply by 180 π Plotting points on the polar plane and multiple representations For any given point in the polar coordinate plane, there are multiple ways to represent that point (as opposed to Cartesian coordinates, where point representations are unique). To begin understanding this idea, one must consider the process of plotting points in the polar coordinate plane. To do this in the rectangular plane one thinks about moving horizontally and then vertically. However, in the polar coordinate plane, one uses the given distance and angle measure instead. Although the distance is given first, it is easier to use the angle measure before using the given distance. Polar Coordinate System ~ 3 The figure above is a picture of a polar coordinate system with degrees in black and radians in green. This is just one example of what a polar coordinate plane may look like; other examples may have additional or fewer angle measures marked. From this figure it is easy to see why it may be called the polar system; it resembles what one might see looking down on the north (or south) pole with visible longitudinal and latitude lines. see looking down on the north (or south) pole with visible longitudinal and latitude lines....
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 Spring '09
 tapia
 Cartesian Coordinate System, Polar coordinate system, polar coordinate

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