In this video I go over further into Polar Coordinates and this time look at Example 12 Question A which revisits the Limaçons curves from Example 11 but now I take an analytical approach as opposed to the graphical approach from the previous example. Question A looks at the “loop” which I showed occurred for values of c that are larger than 1 or less than -1, i.e. the absolute value of c is greater than 1, in the Limaçon formula r = 1 + c sin ϴ. We are asked to prove this is the case, and thus I go over an VERY extensive analytical proof by first showing that the requirement for the loop is to have two angles where the coordinates are at the origin, and between them the r-values are negative. I show that this is only possible when |c| is greater than 1. This is an extremely detailed proof video, but if you follow along the whole thing you will be that much more knowledgeable about mathematics in general, so make sure to watch this video! I will be going over Question B in the next video so #StayTuned!
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Polar Coordinates: Example 12: Limaçons Analytical Proof: Part 1: Question A
Example 12
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) Also from Example 11, it appears that the Limaçon loses its dimple when c = 1/2.
Prove this.
Recall from Example 11
r = 1 + c sin θ
Solution to a)
We see that the curve crosses itself at the origin, r = 0, in fact the inner loop corresponds to negative r values!
Thus we first solve the equation of the Limaçon for r = 0:
Let's consider the following cases for the values of c:
Let's first check c = +/- 1:
Thus since r is not negative for c = +/- 1 we don't get a loop.
Now if |c| < 1, then this equation has no solution and hence there is no inner loop.
But if c > 1, then on the interval (0 , 2π) the equation has 2 solutions:
Note that since r is continuous for all values of θ, and r(θ1) = r(θ2) = 0, we must have r < 0 because |sinθ| is larger between θ1 and θ2 than at θ1 and θ2.
Thus for c > 1, we have 2 angles in which r = 0 and between them r < 0, thus we have a loop!
Similarly for c < -1, the solutions are:
Also similarly, since r is continuous for all values of θ, and r(θ1) = r(θ2) = 0, we must have r < 0 because sinθ is larger between θ1 and θ2 than at θ1 and θ2.
Thus for c < -1, we have 2 angles in which r = 0 and between them r < 0, thus we have a loop!
Thus for |c| > 1, the Limaçon has a loop, and the curves for positive and negative values of c are mirrored about the horizontal axis.
