Chapter 9.2, Problem 52AYU

To determine

To find: Identify the polar equation $r = 3 \text{cos}\left(4\theta \right)$ and graph it.

Answer to Problem 52AYU

Solution

The graph of $r = 3 \text{cos}\left(4\theta \right)$ is an equation of rose with eight petals.

Given:

It is asked to identify and graph the polar equation $r = 3 \text{cos}\left(4\theta \right)$.

Explanation of Solution

Check for symmetry test:

Polar axis:

Replace $\theta$ by $- \theta$. The result is $r = r = 3 \text{cos}\left(- 4\theta \right) = 3 \text{cos}\left(4\theta \right)$.

The test satisfied. So the graph is symmetric with respect to the polar axis.

The line

$\theta = \frac{\pi }{2}$: Replace $\theta$ by $\pi - \theta$. The result is

$r = 3 \text{cos}\left(\pi - 4\theta \right) = 3\left(\text{cos}\pi \text{cos} 4\theta + \text{sin}\pi \text{sin} 4\theta \right) = 3 (\left(- 1\right) \text{cos}4 \theta + \left(0\right) \text{sin}4 \theta ) = - 3 \text{cos}4 \theta$

The test fails, so the graph may or may not symmetric with respect to the line $\theta = \frac{\pi }{2}$.

The Pole:

Replace $r$ by $- r$. Then the result is $- r = 3 \text{cos}\left(4\theta \right)$, so $r = - 3 \text{cos}\left(4\theta \right)$. This test fails. Replace $\theta$ by $\theta + \pi$. The result is

$r = 3 \text{cos}\left(4\theta + \pi \right) = 3\left(\text{cos}\pi \text{cos} 4\theta - \text{sin}\pi \text{sin} 4\theta \right) = 3\left(- 1\right) \text{cos}4 \theta = - 3 \text{cos} 4\theta .$

This test also fails, so the graph may or may not be symmetric with respect to the pole.

Let’s identify points on the graph by assigning values to the angle $\theta$ and calculating the corresponding values of $r$.

As the polar equation is symmetric about the polar axis, it is enough to find the values from $0$ to $\pi$.

$\theta$	$r = 3 \text{cos} 4\theta$
$0$	$3\left(1\right) = 3$
$\frac{\pi }{6}$	$3 \times - \frac{1}{2} = - \frac{3}{2}$
$\frac{\pi }{4}$	$3 \times - 1 = - 3$
$\frac{\pi }{3}$	$3 \times - \frac{1}{2} = - \frac{3}{2}$
$\frac{\pi }{2}$	$3\left(1\right) = 1$
$\frac{2\pi }{3}$	$3 \times - \frac{1}{2} = - \frac{3}{2}$
$\frac{3\pi }{4}$	$3 \times 0 = - 1$
$\frac{5\pi }{6}$	$3 \times - \frac{1}{2} = - \frac{3}{2}$
$\pi$	$3\left(1\right) = 3$

To sketch:

$r = 3 \text{cos}\left(4\theta \right)$