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/42πr−1−0. 410122. 4132. 41210−0. 41−1The 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 axis0≤≤θ13π[–1, 0]0≤≤θπ[3,]π[–1, 0]polar axis0≤≤θ53π0≤≤2θπpolar axispolar axisFigure 9.4.2Asθincreases from 0 to13π, cosθdecreases from 1 to12andr=1−2 cosθincreasesfrom−1 to 0.Asθincreases from13πtoπ, cosθdecreases from12to−1 andrincreases from 0 to 3.Asθincreases fromπto53π, cosθincreases from−1 to12andrdecreases from 3 to 0.Finally, asθincreases from53πto 2π, cosθincreases from12to 1 andrdecreasesfrom 0 to−1.As we could have read from the equation, the curve is symmetric about thex-axis[r(−θ)=1−2 cos (−θ)=1−2 cosθ=r(θ)].Example 2Sketch the curver=cos 2θ,0≤θ≤2π.