SalasSV_09_04 - 9.4 GRAPHING IN POLAR COORDINATES 539 25...

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9.4 GRAPHING IN POLAR COORDINATES We begin with the curve r = θ , θ 0. The graph is a nonending spiral, part of the famous spiral of Archimedes . The curve is shown in detail from θ = 0 to θ = 2 π in Figure 9.4.1. At θ = 0, r = 0; at θ = 1 4 π , r = 1 4 π ; at θ = 1 2 π , r = 1 2 π ; and so on. The next examples involve trigonometric functions. Example 1 Sketch the curve r = 1 2 cos θ . SOLUTION Since the cosine function is periodic with period 2 π , the curve r = 1 2 cos θ is a closed curve. We will draw it from θ = 0 to θ = 2 π . The curve just repeats itself for values of θ outside the interval [0, 2 π ].
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540 CHAPTER 9 THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS x O r = θ θ 0 spiral of Archimedes , , 4 3 π 4 3 π , π π , 4 5 π 4 5 π , 2 3 π 2 3 π Polar axis [2 , 2 ] π π [ ] , 4 1 π 4 1 π , 2 1 π 2 1 π , 4 7 π 4 7 π Figure 9.4.1 We begin by compiling a table of values: θ 0 π/ 4 π/ 3 π/ 2 2 π/ 3 3 π/ 4 π 5 π/ 4 4 π/ 3 3 π/ 2 5 π/ 3 7 π/ 4 2 π r 1 0. 41 0 1 2 2. 41 3 2. 41 2 1 0 0. 41 1 The values of θ for which r = 0 or | r | is a local maximum are as follows: r = 0 at θ = 1 3 π , 5 3 π for then cos θ = 1 2 ; | r | is a local maximum at θ = 0, π , 2 π . These five values of θ generate four intervals: [0, 1 3 π ], [ 1 3 π , π ], [ π , 5 3 π ], [ 5 3 π , 2 π ]. We sketch the curve in four stages. These stages are shown in Figure 9.4.2. [ 1, 0] polar axis 0 θ 1 3 π [ 1, 0] 0 θ π [3, ] π [ 1, 0] polar axis 0 θ 5 3 π 0 2 θ π polar axis polar axis Figure 9.4.2 As θ increases from 0 to 1 3 π , cos θ decreases from 1 to 1 2 and r = 1 2 cos θ increases from 1 to 0. As θ increases from 1 3 π to π , cos θ decreases from 1 2 to 1 and r increases from 0 to 3. As θ increases from π to 5 3 π , cos θ increases from 1 to 1 2 and r decreases from 3 to 0. Finally, as θ increases from 5 3 π to 2 π , cos θ increases from 1 2 to 1 and r decreases from 0 to 1. As we could have read from the equation, the curve is symmetric about the x -axis [ r ( θ ) = 1 2 cos ( θ ) = 1 2 cos θ = r ( θ )]. Example 2 Sketch the curve r = cos 2 θ , 0 θ 2 π .
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9.4 GRAPHING IN POLAR COORDINATES 541 SOLUTION As an alternative to compiling a table of values as we did in Example 1, we refer to the graph of cos 2 θ in rectangular coordinates for the values of r . See Figure 9.4.3. The values of θ for which r is zero or has an extreme value are as follows: r = 0 at θ = π 4 , 3 π 4 , 5 π 4 , 7 π 4 ; local maxima/minima at θ = 0, π 2 , π , 3 π 2 , 2 π . θ cos 2 1 1 π θ π 2 4 π 2 π 4 π 3 4 π 5 4 π 7 2 π 3 Figure 9.4.3 In Figure 9.4.4 we sketch the curve in eight stages. 0 θ 1 4 π θ 1 4 π = 0 θ 1 2 π 0 θ π 0 θ 3 2 π 0 θ π 2 0 θ 5 4 π 0 θ 3 4 π 0 θ 7 4 π Figure 9.4.4 Example 3 Figure 9.4.5 shows four cardioids , heart-shaped curves. Rotation of r = 1 + cos θ by 1 2 π radians, measured in the counterclockwise direction, gives r = 1 + cos ( θ 1 2 π ) = 1 + sin θ .
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