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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 x-axis (since sin0 = 0), 1 at the end of the positive y-axis, 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 counter-clockwise, 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 half-axes 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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