SECTION 9.2 Polar Equations and Graphs 617 The curve in Figure 31 is an example of a cardioid (a heart-shaped curve). Figure 31 θ = − r 1 sin x y 1 2 (1, 0) (0, ) (2, ) 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 p– 62 p– 22 (1.87, )p– 3 2 3 – 2 p– 6 p– 2 ( , ) 1 – 2 (0.13, )p– 3 ( , ) (2, ) (a) (b) 22 22 2 0.5 The Line θ π = 2 : Replace θ by . π θ − The result is r 1 sin 1 sin cos cos sin 1 0 cos 1 sin 1 sin π θ π θ π θ θ θ θ ( ) ( ) ( ) [ ] = − − = − − = − ⋅ − − = − The test is satisfied, so the graph is symmetric with respect to the line 2 . θ π = The Pole: Replace r by r. − Then the result is r 1 sin ,θ − = − so r 1 sin .θ = − + The test fails. Replace θ by . θ π + The result is r 1 sin 1 sin cos cos sin 1 sin 1 cos 0 1 sin θ π θ π θ π θ θ θ ( ) [ ] ( ) [ ] = − + = − + = − ⋅ − + ⋅ = + This test also fails, so the graph may or may not be symmetric with respect to the pole. Next, identify points on the graph by assigning values to the angle θ and calculating the corresponding values of r. Due to the periodicity of the sine function and the symmetry with respect to the line 2 , θ π = just assign values to θ from 2 π − to 2 , π as given in Table 1. Now plot the points r, θ ( ) from Table 1 and trace out the graph, beginning at the point 2, 2 π ( ) − and ending at the point 0, 2 . π ( ) Then reflect this portion of the graph about the line 2 θ π = (the y -axis) to obtain the complete graph. Figure 31(a) shows the graph drawn by hand. Figure 31(b) shows the graph using a TI-84 Plus CE with θ θ π θ π = = = min 0, max 2 , and step 24 . NOTE For each symmetry test, if one condition is satisfied, then the symmetry is guaranteed and we do not need to check the other condition. j Exploration Graph θ = + r 1 sin . 1 Clear the screen and graph θ = − r 1 cos . 1 Clear the screen and graph θ = + r 1 cos . 1 Do you see a pattern? θ θ = − r 1 sin π − 2 ( ) − − = 1 1 2 π − 3 − − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ ≈ 1 3 2 1.87 π − 6 ( ) − − = 1 1 2 3 2 0 − = 1 0 1 π 6 − = 1 1 2 1 2 π 3 − ≈ 1 3 2 0.13 π 2 − = 1 1 0 Table 1 DEFINITION Cardioids Cardioids are characterized by equations of the form • r a 1 cosθ ( ) = + • r a 1 cosθ ( ) = − • r a 1 sinθ ( ) = + • r a 1 sinθ ( ) = − where a 0. > The graph of a cardioid passes through the pole. Now Work PROBLEM 39
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