Step 1

dr/dθ = i sec² (£θ).

y = r sin θ = sin ½ θ sin θ/cos ½θ = sin ½θ(2 sin ½θ cos ½θ)/cos ½θ = 2 sin²(½θ).

Step 2

r = 0 at θ = 0. r is undefined at θ = π. dr/dθ is never 0.

Step 3

See Figure 7.8.10.

Figure 7.8.10

Step 5

The curve crosses itself at the point x = 0, y = 1, because this point has both polar coordinates

(r =1,0 = π/2), (r = -1,0 = 3π/2).

Figure 7.8.11 shows the graph for various stages of development.