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Chapter 6: Problem 20

Identify the conic and sketch its graph. \(r=\frac{4}{4+\sin \theta}\)

Short Answer

Expert verified
The given polar equation \(r=\frac{4}{4+\sin \theta}\) represents a lima莽on with a loop.

Step by step solution

01

Identify the form of the conic section

Examining the given equation \(r=\frac{4}{4+\sin \theta}\), it fits the general form of a lima莽on with a loop since it is in the form of \(r=a \pm b\cos(\theta)\) or \(r=a \pm b\sin(\theta)\) where a>b in this case. Therefore, the equation represents a lima莽on with a loop.
02

Plot the polar points

Begin by plotting polar points for values of theta ranging from 0 to 2*pi. These points can be found by substituting the value of theta into the equation, \[r=\frac{4}{4+\sin \theta}\], and finding the corresponding r-value. For example, when theta is 0, r is 1. When theta is pi/2, r is 4. Continue this step with various theta values like pi, 3*pi/2 and 2*pi to get different points.
03

Connect the points and sketch the graph

Once a set of suitable points are plotted, connect them in the order of increasing theta. Make sure to capture the curvature and direction of the graph, especially the loop region in the first quadrant, which should be accurate due to the unique pointed 'loop' shape that characterises lima莽on polar equations with this specific setup.

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