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Chapter 6: Problem 65

Use a graphing utility to graph the polar equation. $$r=4+2 \sin \theta$$

Short Answer

Expert verified
Following the steps above, you can successfully plot the polar equation \(r = 4 + 2 \sin \theta\) on a graphing utility. This results in a lima莽on shape curve.

Step by step solution

01

Setting Up The Graph

First, set your graphing utility to plot in polar coordinates. This might be a setting labeled 'Polar mode' or similar. Make sure to set the \(\theta\) range from 0 to \(2\pi\) (or 0 to 360掳 if your utility uses degrees) to view the full graph.
02

Sketching The Curve

Next, enter the given polar equation \(r = 4 + 2 \sin \theta\) into your graphing utility and sketch the curve. The graph will start at \(\theta = 0\) with a radius of \(r = 4+2\sin(0) = 4\), and it changes as \(\theta\) increases, reaching a maximum radius of 6 when \(\sin(\theta) = 1\), and a minimum radius of 2 when \(\sin(\theta) = -1\). Following these patterns, the curve will form a shape known as a 'lima莽on'.
03

Interpreting The Graph

Finally, observe the graph. Note how the radius changes with respect to \(\theta\). When the value of \(\sin\theta\) is maximum, value of 'r' is maximum and when it's minimum, 'r' is minimum. Therefore, the sine function in the equation causes the radius to oscillate back and forth as \(\theta\) increases.

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