622 CHAPTER 9 Polar Coordinates; Vectors The curve in Figure 36 is called a logarithmic spiral, since its equation may be written as r 5ln θ = and it spirals infinitely both toward the pole and away from it. Classification of Polar Equations The equations of some lines and circles in polar coordinates and their corresponding equations in rectangular coordinates are given in Table 7. Also included are the names and graphs of a few of the more frequently encountered polar equations. Figure 36 = θ r e 5 x u 5 0 u 5 p u 5 p– 2 u 5 3p –– 2 u 5 7p –– 4 u 5 p– 4 u 5 3p –– 4 u 5 5p –– 4 4 y 2 1.17, p– 4 ( ) (3.51, 2p) (1, 0) 1.37, p– 2 ( 2.57, ) 3p –– 2 ( ) (1.87, p) 2 4 (b) (a) With the help of a calculator, the values in Table 6 can be obtained. See Figure 36(a) for the graph drawn by hand. Figure 36(b) shows the graph using Desmos. θ = θ r e 5 π − 3 2 0.39 π− 0.53 π − 2 0.73 π − 4 0.85 0 1 π 4 1.17 π 2 1.37 π 1.87 π3 2 2.57 π2 3.51 Table 6 Lines Description Line passing through the pole making an angle α with the polar axis Vertical line Horizontal line Rectangular equation α ( ) = y x tan = a x = b y Polar equation θ α = θ = a r cos θ = b r sin Typical graph y x a y x y x Circles Description Center at the pole, radius a Passing through the pole, tangent to the line θ π = 2 , center on the polar axis, radius a Passing through the pole, tangent to the polar axis, center on the line θ π = 2 , radius a Rectangular equation + = > a a x y , 0 2 2 2 + = ± > a a x y x 2 , 0 2 2 + = ± > a a x y y 2 , 0 2 2 Polar equation = > a a r , 0 θ = ± > a a r 2 cos , 0 θ = ± > a a r 2 sin , 0 Typical graph y x a a y x a y x Table 7 (continued)
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