Examples for Section 9.7
Graphing in Polar Coordinates

p.763

<< Graphics`Graphics`

Cardioids and Limacons

PolarPlot[1 + Cos[t], {t, 0, 2  π}]

Note: This (and all polar coordinate plots) could be done with ParametricPlot[], by converting to rectangular coordinates: \(TraditionalForm
      \`x\  = \ \(r\ \(cos(θ)\)\  = \ 
          \((1 + cos(θ))\) \(cos(θ)\)\)\), \(TraditionalForm
      \`y\  = \ \(r\ \(sin(θ)\) = \ 
          \((1 + cos(θ))\) \(sin(θ)\)\)\).

ParametricPlot[{(1 + Cos[t]) Cos[t],  (1 + Cos[t]) Sin[t]}, {
        t, 0, 2  π},  AspectRatio -> Automatic]

PolarPlot[1 - Cos[t], {t, 0, 2  π}]

PolarPlot[1 + 2  Cos[t], {t, 0, 2  π}]

PolarPlot[1 + Sin[t], {t, 0, 2  π}]

PolarPlot[(1/3) - Sin[t], {t, 0, 2  π}]

PolarPlot[2 + Sin[t], {t, 0, 2  π}]

Intersections

31.

PolarPlot[{1 + Cos[t], 1 - Cos[t]}, {t, 0, 2  π}]

To solve, set 1 + cos(θ)  =  1 - cos(θ), which gives cos(θ) = 0 or θ = ±π/2. The corresponding rectangular points are (0,1) and (0,-1). These are the points where the curve intersect "simultaneously," that is, for the same value of θ. The other point of intersection is at the origin. The first curve is at the origin when θ = π, while the second curve is at the origin when θ = 0 (so it is not a "simultaneous" intersection, but still a point of intersection of the curves).

33.

PolarPlot[{2  Sin[t], 2  Sin[2  t]}, {t, 0, 2  π}]

Setting \(TraditionalForm
      \`2 \( sin(θ)\) = \ 
        \(2 \( sin(2  θ)\) = 
          4 \( sin(θ)\) \(cos(θ)\)\)\) gives sin(θ) = 0 or cos(θ) = (1/2), yielding θ = 0, θ = π/3 or θ=2π/3. These are the simultaneous points of intersection and have corresponding rectangular coordinates \(TraditionalForm
      \`\((x, y)\) = 
        \(\((2 \( sin(0)\) \(cos(0)\), 2 \( sin(0)\) \(sin(0)\))\) = 
          \((0, 0)\)\)\), \(TraditionalForm
      \`\((x, y)\) = 
        \(\((2 \( sin(π/3)\) \(cos(π/3)\), 
            2 \( sin(π/3)\) \(sin(π/3)\))\) = 
          \((\@3\/2, 3\/2)\)\)\), and \(TraditionalForm
      \`\((x, y)\) = 
        \(\((2 \( sin(2  π/3)\) \(cos(2  π/3)\), 
            2 \( sin(2  π/3)\) \(sin(2  π/3)\))\) = 
          \((\(-\(\@3\/2\)\), 3\/2)\)\)\).
 Note that the second curve passes through the origin for other values of θ, θ = π/2, θ = π, and  θ = 3  π/2.

41.

PolarPlot[{1, 2  Sin[2  t]}, {t, 0, 2  π}]

There are 8 points of intersection. Since the first equation does not involve θ, we don't have to worry about whether the intersetions are "simultaneous" or not. We simply need to solve 2  sin(2  θ) = 1. This gives \(TraditionalForm
      \`2  θ = 
        \(\(sin\^\(-1\)\)(1/2) = π\/6 + \ 2\ n\ π\)\), or \(TraditionalForm
      \`2  θ = 
        \(\(sin\^\(-1\)\)(1/2) = \(5  π\)\/6 + \ 2\ n\ π\)\) for any integer n. Thus, θ =  (π/12) + n π, or θ =  ((5  π)/12) + n π. The corresponding points in rectangular coordinates are: \(TraditionalForm
      \`\((cos(π\/12 + n\ π), sin(π\/12 + n\ π))\)\), \(TraditionalForm
      \`\((cos(\(5  π\)\/12 + n\ π), sin(\(5  π\)\/12 + n\ π))
        \)\).

{Cos[(π/12)], Sin[(π/12)]}{Cos[(π/12) + π], Sin[(π/12) +  π]}{Cos[((5  π)/12)], Sin[((5  π)/12)]}{Cos[((5  π)/12) + π], Sin[((5  π)/12) +  π]}