SECTION 9.2 Polar Equations and Graphs 619 The curve in Figure 33(b) or (c) is an example of a limaçon with an inner loop . Graphing a Polar Equation (Limaçon with an Inner Loop) Graph the equation: r 1 2cosθ = + EXAMPLE 10 θ θ = + r 1 2cos 0 + ⋅ = 1 2 1 3 π 6 + ⋅ ≈ 1 2 3 2 2.73 π 3 + ⋅ = 1 2 1 2 2 π 2 + ⋅ = 1 2 0 1 π2 3 ( ) + − = 1 2 1 2 0 π5 6 + − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ ≈ − 1 2 3 2 0.73 π ( ) + − = − 1 2 1 1 Table 3 Figure 33 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 2 4 y 2, p– 3 1, (–1, p) p– 2 20.73, 5p –– 6 0,2p –– 3 2.73, (3, 0) p– 6 (a) ( ) ( ) ( ) ( ) ( ) (b) r 5 1 1 2 cos u 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 2 4 y 2, p– 3 1, (21, p) p– 2 20.73, 5p –– 6 2.73, (3, 0) p– 6 ( ) ( ) ( ) ( ) 0,2p –– 3 ( ) (c) Solution First, check for symmetry. Polar Axis : Replace θ by .θ− The result is θ θ ( ) = + − = + r 1 2 cos 1 2 cos The test is satisfied, so the graph is symmetric with respect to the polar axis. The Line θ π = 2 : Replace θ by . π θ − The result is π θ π θ π θ θ ( ) ( ) = + − = + + = − r 1 2 cos 1 2 cos cos sin sin 1 2 cos The test fails. Replace r by r− and θ by .θ− The result is r r r 1 2 cos 1 2 cos 1 2 cos θ θ θ ( ) − = + − − = + = − − cos cos θ θ ( ) − = This test also fails, so the graph may or may not be symmetric with respect to the line 2 . θ π = The Pole : Replace r by r. − The test fails. Replace θ by . θ π + This test also fails, so the graph may or may not be symmetric with respect to the pole. Next, identify points on the graph of θ = + r 1 2cos by assigning values to the angle θ and calculating the corresponding values of r. Due to the periodicity of the cosine function and the symmetry with respect to the polar axis, just assign values to θ from 0 to ,π as given in Table 3. Now plot the points r, θ ( ) from Table 3, beginning at 3, 0 ( ) and ending at 1, . π ( ) − See Figure 33(a). Finally, reflect this portion of the graph about the polar axis (the x -axis) to obtain the complete graph. See Figure 33(b). Figure 33(c) shows the graph using Desmos. Exploration Graph θ = − r 1 2cos . 1 Clear the screen and graph θ = + r 1 2sin . 1 Clear the screen and graph θ = − r 1 2sin . 1 Do you see a pattern? Now Work PROBLEM 47 DEFINITION Limaçons with an Inner Loop Limaçons with an inner loop are characterized by equations of the form • θ = + r a bcos • θ = + r a bsin • θ = − r a bcos • θ = − r a bsin where b a 0. > > The graph of a limaçon with an inner loop passes through the pole twice.
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