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

a We see that the curve r 1 c sin crosses itself at the origin where r 0 in fact the inner loop corr ...

(a) In Example 11 the graphs suggest that the limaon r = 1 + c sin ...

Question:

(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.