846 CHAPTER 8 Applications of Trigonometry Conversion from Polar to Rectangular Equations Figure 71 −50 −50 50 50 More of the spiral is shown in this calculator graph, with -8p … u … 8p. (b) 4 8 r = 2U p 0 p 2 3p 2 (a) S Now Try Exercise 65. y x y = – x2 1 + 2 8 2 4 –4 0 Figure 72 −10 −10 10 10 0° … u … 360° Figure 73 Classification of Polar Equations The table on the next page summarizes common polar graphs and forms of their equations. In addition to circles, lemniscates, and roses, we include limaçons. Cardioids are a special case of limaçons, where 0 a b 0 = 1. NOTE Some other polar curves are the cissoid, kappa curve, conchoid, trisectrix, cruciform, strophoid, and lituus. Refer to older texts on analytic geometry or the Internet to investigate them. EXAMPLE 8 Converting a Polar Equation to a Rectangular Equation For the equation r = 4 1 + sin u , write an equivalent equation in rectangular coordinates, and graph. SOLUTION r = 4 1 + sin u Polar equation r 11 + sin u2 = 4 Multiply by 1 + sin u. r + r sin u = 4 Distributive property 2 x2 + y2 + y = 4 Let r = 2x2 + y2 and r sin u = y. 2 x2 + y2 = 4 - y Subtract y. x2 + y2 = 14 - y22 Square each side. x2 + y2 = 16 - 8y + y2 x2 - 16 = -8y Subtract y2, and subtract 16. y = - 1 8 x2 + 2 Divide by -8 and rewrite. The final equation represents a parabola and is graphed in Figure 72. S Now Try Exercise 69. Expand the right side; 1a - b22 = a2 - 2ab + b2 Rectangular equation The conversion in Example 8 is not necessary when using a graphing calculator. Figure 73 shows the graph of r = 4 1 + sin u , graphed directly with the calculator in polar mode. 7
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