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**Unformatted text preview: **10 Polar Coordinates, Parametric Equations aC d iae Coordinate systems are tools that let us use algebraic methods to understand geometry. While the rectangular (also called Cartesian ) coordinates that we have been using are the most common, some problems are easier to analyze in alternate coordinate systems. A coordinate system is a scheme that allows us to identify any point in the plane or in three-dimensional space by a set of numbers. In rectangular coordinates these numbers are interpreted, roughly speaking, as the lengths of the sides of a rectangle. In polar coordinates a point in the plane is identified by a pair of numbers ( r, ). The number measures the angle between the positive x-axis and a ray that goes through the point, as shown in figure 10.1; the number r measures the distance from the origin to the point. Figure 10.1 shows the point with rectangular coordinates (1 , 3) and polar coordinates (2 , / 3), 2 units from the origin and / 3 radians from the positive x-axis. 3 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 , / 3) . . . . . . . Figure 10.1 Polar coordinates of the point (1 , 3). 217 218 Chapter 10 Polar Coordinates, Parametric Equations Just as we describe curves in the plane using equations involving x and y , so can we describe curves using equations involving r and . Most common are equations of the form r = f ( ). EXAMPLE 10.1 Graph the curve given by r = 2. All points with r = 2 are at distance 2 from the origin, so r = 2 describes the circle of radius 2 with center at the origin. EXAMPLE 10.2 Graph the curve given by r = 1 + cos . We first consider y = 1 + cos x , as in figure 10.2. As goes through the values in [0 , 2 ], the value of r tracks the value of y , forming the cardioid shape of figure 10.2. For example, when = / 2, r = 1 + cos( / 2) = 1, so we graph the point at distance 1 from the origin along the positive y-axis, which is at an angle of / 2 from the positive x-axis. When = 7 / 4, r = 1 + cos(7 / 4) = 1 + 2 / 2 1 . 71, and the corresponding point appears in the fourth quadrant. This illustrates one of the potential benefits of using polar coordinates: the equation for this curve in rectangular coordinates would be quite complicated. 1 2 / 2 3 / 2 2 ................................... .. .. .. . . . . . . . . .. .. ... ............................................................... . . .. .. . . .. . .. . . . .. .. .. .... ............................... . . . . . . . . . . . ....

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