8.4 Polar Coordinates: Graphs

Polar Coordinate System

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Symmetry in polar equations

Symmetry tests let you predict the shape of a polar graph before plotting every single point. If you can identify symmetry early, you only need to plot half (or even a quarter) of the curve and then mirror it.

There are three main symmetry tests you should know:

• Symmetry about the polar axis (the horizontal line $\theta = 0$, equivalent to the positive x-axis)
  • Replace $\theta$ with $-\theta$. If the equation is unchanged, the graph is symmetric across the polar axis.
  • Example: $r = 2\cos\theta$ works because $\cos(-\theta) = \cos\theta$
• Symmetry about the line $\theta = \frac{\pi}{2}$ (the vertical axis, equivalent to the positive y-axis)
  • Replace $\theta$ with $\pi - \theta$. If the equation is unchanged, the graph is symmetric across this vertical line.
  • Example: $r = 2\sin\theta$ works because $\sin(\pi - \theta) = \sin\theta$
• Symmetry about the pole (the origin)
  • Replace $r$ with $-r$. If the equation is unchanged, the graph is symmetric about the origin.
  • Alternatively, replace $\theta$ with $\theta + \pi$ and check if the equation still holds.
  • Example: $r^2 = 4\cos(2\theta)$ (a lemniscate) has pole symmetry

A couple of things to watch out for: failing a symmetry test doesn't prove the graph lacks that symmetry. Polar equations can have multiple representations of the same point, so a curve might still be symmetric even if the algebraic test doesn't confirm it. However, passing a test does guarantee symmetry.

Graphing techniques for polar equations

When you need to sketch a polar curve by hand, follow these steps:

1. Check for symmetry using the tests above. This tells you how much of the curve you actually need to plot.
2. Determine the domain. Most polar equations use $0 \leq \theta \leq 2\pi$, but some (like rose curves with odd $n$) trace out completely over $0 \leq \theta \leq \pi$.
3. Build a table of values. Evaluate $r$ at key angles: $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{3\pi}{4}, \frac{5\pi}{6}, \pi$, and continue through $2\pi$ as needed.
4. Plot the points $(r, \theta)$ on polar graph paper. If $r$ is negative, plot the point in the opposite direction (add $\pi$ to the angle).
5. Connect the points with a smooth curve, using symmetry to fill in the rest.

If you need to convert a polar point to rectangular coordinates for any reason, use:

• $x = r\cos\theta$
• $y = r\sin\theta$

Classic polar curve identification

Recognizing the standard forms saves you a lot of time. Here are the curves you need to know:

Cardioids: $r = a(1 \pm \cos\theta)$ or $r = a(1 \pm \sin\theta)$

• Heart-shaped curves that pass through the pole exactly once
• The cosine versions are symmetric about the polar axis; the sine versions are symmetric about $\theta = \frac{\pi}{2}$
• Example: $r = 2(1 + \cos\theta)$ produces a cardioid that extends to $r = 4$ along the polar axis

Limaçons: $r = a \pm b\cos\theta$ or $r = a \pm b\sin\theta$

The ratio $\frac{a}{b}$ determines the shape:

• $\frac{a}{b} < 1$: inner loop (Example: $r = 1 + 2\cos\theta$)
• $\frac{a}{b} = 1$: cardioid (this is the special case above)
• $1 < \frac{a}{b} < 2$: dimpled limaçon (Example: $r = 3 + 2\cos\theta$)
• $\frac{a}{b} \geq 2$: convex limaçon, no dimple (Example: $r = 4 + \cos\theta$)

Rose curves: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$

The number of petals depends on whether $n$ is odd or even:

• If $n$ is odd: the rose has $n$ petals. Example: $r = \cos(3\theta)$ has 3 petals.
• If $n$ is even: the rose has $2n$ petals. Example: $r = \sin(4\theta)$ has 8 petals.

Each petal has length $|a|$ (the maximum $r$-value). Cosine roses have a petal along the polar axis; sine roses are rotated by $\frac{\pi}{2n}$.

Additional Concepts in Polar Coordinates

Periodicity plays a big role in polar graphing. Because trig functions repeat, many polar curves trace out completely before $\theta$ reaches $2\pi$. For example, $r = \cos(3\theta)$ completes its full graph over just $0 \leq \theta \leq \pi$. Recognizing the period helps you avoid plotting redundant points.

Negative $r$-values can be confusing at first. When your equation gives a negative $r$ for some angle $\theta$, you plot the point at distance $|r|$ in the direction $\theta + \pi$. This is how inner loops on limaçons form: the curve passes through the pole and loops back through negative $r$-values.

Polar and complex number connections: Polar form is the same framework used to represent complex numbers as $r(\cos\theta + i\sin\theta)$. If you've seen that notation in class, the graphing intuition you build here carries over directly.