Unit_3 - UNIT 3 POLAR COORDINATES GRAPHING INTRO We will...

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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 θ tan θ = y x y = r sin θ 1. Graphs of Polar Equations We wish to plot a polar equation of the form r = f ( θ ), where the right hand side is a function of sin or cos for n=1,2,3 . . . . Rather than plotting many points for different θ ’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 θ r = 3 sin 2 θ r = 2 cos 3 θ r = 1 + 2 sin 2 θ 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 θ , r is negative, then the point to be plotted is rotated 180 degrees. For example, if, for θ = π/ 6, r = - 2, then this really means r = 2 but at the angle θ = π/ 6 + π = 7 π/ 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 , θ = 0, as r = - 1 with θ = π . Now for the plotting of a polar equation of the form r = f ( θ ). If the right hand side of the equation for r contains sin θ , we place the values of sin θ at the ends of the axes (We may change these labels very shortly): 0 at the end of the positive x -axis (since sin 0 = 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 θ (going around counter-clockwise, of course, since this is the direction of increasing θ , and beginning with 1 since cos 0 = 1). If, on the other hand, the equation contained sin 2 θ or cos 2 θ , then then we first draw dotted half-axes at 45 degrees in each
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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 sin 2 θ and 1,0, - 1,0,1,0, - 1,0 in the case of cos 2 θ . Similarly for the case of sin 3 θ and cos 3 θ . The axes for this first step are shown below for various possibilities. 0 0 1 Minus 1 1 1 Minus 1 Minus 1 0 0 0 0 0 0 1 Minus 1 1 Minus 1 0 0 0 1 Minus 1 0 r a function of cos θ r a function of sin 2 θ r a function of cos 3 θ 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 θ and cos θ in the graphs of Figs.
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