(a) In Example 11 the graphs suggest that the limaon r = 1 + c sin has an inner loop when |c| > 1.

(a) In Example 11 the graphs suggest that the limaçon r = 1 + c sin θ has an inner loop when |c| > 1. Prove that this is true, and find the values of θ that correspond to the inner loop.
(b) From Figure 19 it appears that the limaçon loses its dimple when c = 1/2. Prove this.