# 15. Polar Coordinates

## b. Graphs of Polar Equations

## 2. Plotting Polar Equations

Polar equations often contain trigonometric functions. So you need to understand the graphs of the basic trig functions and how to shift and scale them. You can review by practicing with the following Maplets (requires Maple on the computer where this is executed):

Basic 6 Trigonometric Functions Rate It

Shifting Trigonometric Functions Rate It

Properties of Sine and Cosine Curves Rate It

We have seen the graphs of a few simple polar equations. We want to be able to graph more complicated polar equations. So we'll do it by examples. The procedure is to make a table of points and/or make a rectangular plot of the polar equation and use them to draw the polar plot.

Plot the cardioid \(r=1+\sin\theta\).

Frequently when plotting polar functions we allow \(r\) to be negative by measuring backward. In particular, if the direction is \(\theta\), then

- when \(r\) is positive, we move in the direction \(\theta\), (and plot it in blue), but
- when \(r\) is negative, we move in the direction \(\theta+\pi\), (and plot it in red).

Plot the limaçon \(r=1+2\sin\theta\)

When there is a multiple of \(\theta\) inside the sine or cosine, the horizontal scale of the rectangular plot stretches or shrinks, changing the period.

Plot the curve \(r=\sin^2\theta\).

Now let's combine a change of period with negative values of \(r\).

Plot the 3-leaf rose \(r=\cos{3\theta}\)

Here is a table of values and the rectangular plot:

\(\theta=\) | \(0\) | \(\dfrac{\pi }{6}\) | \(\dfrac{\pi }{3}\) | \(\dfrac{\pi }{2}\) | \(\dfrac{2\pi }{3}\) | \(\dfrac{5\pi }{6}\) | \(\pi\) |

\(r=\) | \(1\) | \(0\) | \(-1\) | \(0\) | \(1\) | \(0\) | \(-1\) |

There are \(3\) positive bumps and \(3\) negative bumps in the interval \([0,2\pi]\). Here is the polar plot:

In the animation, the radial line is blue when \(r\) is positive and red when \(r\) is negative. Notice that each leaf of the rose is traced out twice, once when \(r\) is positive and once when \(r\) is negative.

How many leaves are there on the rose \(r=\cos(n\theta)\) if \(n\) is even? Why?

The rose \(r=\cos(n\theta)\) with even \(n\) has \(2n\) leaves.

For example, the rose \(r=\cos(4\theta)\) has \(8\) leaves:

Looking at the above example, we see that when \(n\) is even the polar plot has \(2n\) leaves because \(r\) becomes negative at \(r=\dfrac{\pi}{2n}\) and traces a leaf in the opposite quadrant before becoming positive again at \(r=\dfrac{3\pi}{2n}\), and this repeats for \(n\) cycles. So \(r\) traces \(2n\) leaves from \(\theta=0\) to \(\theta=2\pi\). For further evidence, the polar plot of \(r=\cos(4\theta)\) is below, which has \(8\) leaves:

How many leaves are there on the rose \(r=\cos(n\theta)\) if \(n\) is odd? Why is this not the same answer as in the even case?

The rose \(r=\cos(n\theta)\) with odd \(n\) has \(n\) leaves.

For example, the rose \(r=\cos(5\theta)\) has \(5\) leaves:

When \(n\) is odd the polar plot has \(n\) leaves because when \(r\) becomes negative it overwrites a piece of the graph in the opposite quadrant. For example, in the rectangular plot of \(r=\cos(5\theta)\) below, there are \(5\) positive bumps and \(5\) negative bumps. Each time \(r\) becomes negative, the graph retraces a leaf in the opposite quadrant. So the polar plot has only \(5\) leaves.

You can review the graphs and equations of polar curves and practice identifying polar curves from their graphs by using the following Maplets (requires Maple on the computer where this is executed):

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