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Unformatted text preview: UNIT 3 POLAR COORDINATES, GRAPHING INTRO: We will continue our study of polar coordinates. The emphasis will be on graphing. Here are the transformations between Cartesian coordinates and polar coordinates introduced in the last Unit. r = p x 2 + y 2 x = r cos tan = y x y = r sin 1. Graphs of Polar Equations We wish to plot a polar equation of the form r = f ( ), where the right hand side is a function of sin n or cos n for n=1,2,3.... Rather than plotting many points for different s, we will outline a much better method to plot simple polar equations. We first explain the method and then try it on the following four examples: r = cos r = 3sin2 r = 2cos3 r = 1 + 2sin2 However, before beginning the plot, we must explain a peculiar convention. In polar equations and inequalities, r is permitted to be negative as we will see in the examples even though negative r does not make a lot of sense, since r = p x 2 + y 2 is always positive. Here is the convention: if, for a particular value of , r is negative, then the point to be plotted is rotated 180 degrees. For example, if, for = / 6, r = 2, then this really means r = 2 but at the angle = / 6 + = 7 / 6. This convention is usually followed only in equations and inequalities. It would be very abnormal, for example, to refer to the Cartesian point ( x,y ) = (1 , 0), which is r = 1 , = 0, as r = 1 with = . Now for the plotting of a polar equation of the form r = f ( ). If the right hand side of the equation for r contains sin , we place the values of sin at the ends of the axes (We may change these labels very shortly): 0 at the end of the positive xaxis (since sin0 = 0), 1 at the end of the positive yaxis, then 0 and 1 at the ends of the negative axes. Likewise, 1,0, 1,0 would be written on the axes if the equation contained cos (going around counterclockwise, of course, since this is the direction of increasing , and beginning with 1 since cos0 = 1). If, on the other hand, the equation contained sin2 or cos2 , then then we first draw dotted halfaxes at 45 degrees in each quadrant, and label all the axes with the value of the trig function, 0,1,0, 1,0,1,0, 1 on the axes in the case of sin2 and 1,0, 1,0,1,0, 1,0 in the case of cos2 . Similarly for the case of sin3 and cos3 . The axes for this first step are shown below for various possibilities. 1 Minus 1 1 1 Minus 1 Minus 1 1 Minus 1 1 Minus 1 1 Minus 1 r a function of cos r a function of sin2 r a function of cos3 Using these values of the trig function, we then use the equation for r to change these values to the actual values of r at the angles of these axes. Having the values of r already on the graph at the correct angles makes it much easier to evaluate r as we sweep around the origin. You can see these axes labeled with the actual values of r rather than the values of sin and cos in the graphs of Figs....
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This note was uploaded on 03/04/2009 for the course MATH 1224 taught by Professor Dontremember during the Fall '08 term at Virginia Tech.
 Fall '08
 DONTREMEMBER
 Geometry, Transformations, Polar Coordinates

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