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

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9.4GRAPHING IN POLAR COORDINATESWe begin with the curver=θ,θ0.The graph is a nonending spiral, part of the famousspiral of Archimedes. The curveis shown in detail fromθ=0toθ=2πin Figure 9.4.1. Atθ=0,r=0; atθ=14π,r=14π; atθ=12π,r=12π; and so on.The next examples involve trigonometric functions.Example 1Sketch the curver=12 cosθ.SOLUTIONSince the cosine function is periodic with period 2π, the curver=12 cosθis a closed curve. We will draw it fromθ=0 toθ=2π. The curvejust repeats itself for values ofθoutside the interval [0, 2π].
540CHAPTER 9THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONSxOr=θθ0spiral of Archimedes,,43π43π,ππ,45π45π,23π23πPolar axis[2,2]ππ[],41π41π,21π21π,47π47πFigure 9.4.1We begin by compiling a table of values:θ0π/4π/3π/2 2π/3 3π/4π5π/4 4π/3 3π/2 5π/37π/42πr10. 410122. 4132. 412100. 411The values ofθfor whichr=0 or|r|is a local maximum are as follows:r=0 atθ=13π,53πfor then cosθ=12;|r|is a local maximum atθ=0,π, 2π.These five values ofθgenerate four intervals:[0,13π],[13π,π],[π,53π],[53π, 2π].We sketch the curve in four stages. These stages are shown in Figure 9.4.2.[1, 0]polar axis0θ13π[1, 0]0θπ[3,]π[1, 0]polar axis0θ53π02θπpolar axispolar axisFigure 9.4.2Asθincreases from 0 to13π, cosθdecreases from 1 to12andr=12 cosθincreasesfrom1 to 0.Asθincreases from13πtoπ, cosθdecreases from12to1 andrincreases from 0 to 3.Asθincreases fromπto53π, cosθincreases from1 to12andrdecreases from 3 to 0.Finally, asθincreases from53πto 2π, cosθincreases from12to 1 andrdecreasesfrom 0 to1.As we could have read from the equation, the curve is symmetric about thex-axis[r(θ)=12 cos (θ)=12 cosθ=r(θ)].Example 2Sketch the curver=cos 2θ,0θ2π.
9.4GRAPHING IN POLAR COORDINATES541SOLUTIONAs an alternative to compiling a table of values as we did in Example 1, werefer to the graph of cos 2θin rectangular coordinates for the values ofr. See Figure9.4.3. The values ofθfor whichris zero or has an extreme value are as follows:r=0 atθ=π4,3π4,5π4,7π4;local maxima/minima atθ=0,π2,π,3π2, 2π.θcos 211πθπ24π2π4π34π54π72π3Figure 9.4.3In Figure 9.4.4 we sketch the curve in eight stages.0θ14πθ14π=0θ12π0θπ0θ32π0θπ20θ54π0θ34π0θ74πFigure 9.4.4Example 3Figure 9.4.5 shows fourcardioids, heart-shaped curves. Rotation ofr=1+cosθby12πradians, measured in the counterclockwise direction, givesr=1+cos (θ12π)=1+sinθ.r= 1 + cosθr= 1 + sinθr= 1cosθr= 1sinθFigure 9.4.5Rotation by another12πradians givesr=1+cos (θπ)=1cosθ.
542CHAPTER 9THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONSRotation by yet another12πradians givesr=1+cos (θ32π)=1sinθ.Notice how easy it is to rotate axes in polar coordinates: each changecossincossinθθθθ→ −→ −represents a counterclockwise rotation by12πradians.

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Term
Spring
Professor
SMITH
Tags
Distance Formula, Cartesian Coordinate System, Law Of Cosines, Polar Coordinates, Cos, Polar coordinate system, Conic section

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