845 8.7 Polar Equations and Graphs S Now Try Exercise 59. EXAMPLE 6 Graphing a Polar Equation (Lemniscate) Graph r2 = cos 2u. ALGEBRAIC SOLUTION Complete a table of ordered pairs, and sketch the graph, as in Figure 69. The point 1-1, 0°2, with r negative, may be plotted as 11, 180°2. Also, 1-0.7, 30°2 may be plotted as 10.7, 210°2, and so on. Values of u for 45° 6u 6135° are not included in the table because the corresponding values of cos 2u are negative (quadrants II and III) and so do not have real square roots. Values of u greater than 180° give 2u greater than 360° and would repeat the points already found. This curve is called a lemniscate. 180° 0° 90° 270° r2 = cos 2 r2 = cos 2U 1 Figure 69 Using angle measures in radians will produce the same graph. GRAPHING CALCULATOR SOLUTION To graph r2 = cos 2u with a graphing calculator, first solve for r by considering both square roots. Enter the two polar equations as r1 = 2cos 2u and r2 = -2cos 2u. See Figures 70(a) and (b). U 0° 30° 45° 135° 150° 180° 2U 0° 60° 90° 270° 300° 360° cos 2U 1 0.5 0 0 0.5 1 r = t!cos 2U {1 {0.7 0 0 {0.7 {1 EXAMPLE 7 Graphing a Polar Equation (Spiral of Archimedes) Graph r = 2u (with u measured in radians). SOLUTION Some ordered pairs are shown in the table. Because r = 2u does not involve a trigonometric function of u, we must also consider negative values of u. The graph in Figure 71(a) on the next page is a spiral of Archimedes. Figure 71(b) shows a calculator graph of this spiral. Radian measures have been rounded. U (radians) r =2U U (radians) r =2U -p -6.3 p 3 2.1 - p 2 -3.1 p 2 3.1 - p 4 -1.6 p 6.3 0 0 3p 2 9.4 p 6 1 2p 12.6 −1.4 −2.2 1.4 2.2 r1 r2 r2 = -2cos 2u is in red. (b) Figure 70 Settings for the graph below (a)
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