Families of Polar Curves

# Families of Polar Curves

We will now look at the following families of polar curves:

• $r = \sin (b \theta)$.
• $r = \cos (b \theta)$.

## Family of r = sin(bΘ)

We're now going to look at the family of curves in the form of $r = \sin(b\theta)$ where $b > 0$.

$r = \sin(\theta)$ $r = \sin(2\theta)$ $r = \sin(3\theta)$ $r = \sin(4\theta)$

Notice that that if b is an even integer, then the curve $r = \sin (b\theta)$ has $2b$ petals. If $b$ is an odd integer, then the graph $r = \sin (b\theta)$ has exactly $b$ petals. Also notice that the only curve that lies above the $x$-axis completely is $r = \sin (b\theta)$, so for the family of curves $r = \sin (b\theta)$, if $b$ is an integer and $b > 1$, then $r = \sin (b\theta)$ has parts of the curve that lie both above the $x$-axis and below the $x$-axis.

Now let's look at where $b < 0$.

$r = \sin(-\theta)$ $r = \sin(-2\theta)$ $r = \sin(-3\theta)$ $r = \sin(-4\theta)$

Once again the number of petals of the curves are determined entirely by $b$ and whether $b$ is an even or odd integer. However, only odd integer values of $b$ reproduce graphs that are reflected along the $x$-axis or polar axis. If $b$ is an even integer such that $b ≤ -2$, then the polar curve is $r = \sin (-b \theta)$ is equal to the curve $r = \sin (b \theta)$.

## Families of r = cos(bθ)

Now let's look at the family of curves in the form of $r = \cos(b\theta)$ where $b > 0$.

$r = \cos(\theta)$ $r = \cos(2\theta)$ $r = \cos(3\theta)$ $r = cos(4\theta)$

These curves are practically identical to those of $r = \sin (b \theta)$ for $b > 0$ with the exception that each graph is rotated $90^{\circ}$. This should make sense though since $\sin (\theta) = \cos \left (\theta - \frac{\pi}{2} \right)$. Once again, the number of petals is determined in the same fashion for $b$ being an integer.

And now for $b < 0$.

$r = \cos(-\theta)$ $r = \cos(-2\theta)$ $r = \cos(-3\theta)$ $r = cos(-4\theta)$

The graphs are identical! This is because $\cos \theta$ is an even function. Recall that a function is even if flipping the function across the $y$-axis results in the exact same graph. Hence, $\cos(2 \theta)$, $\cos (3 \theta)$, etc… are also even functions.