Question

Tabulate and plot enough points to sketch a graph of the following equations. $$r=8 \cos \theta$$

Short answer

The points obtained for the values of \(\theta\) are as follows: | \(\theta\) | \(r\) | |---------|-----| | 0 | 8 | | \(\frac{\pi}{4}\) | \(4\sqrt{2}\) | | \(\frac{\pi}{2}\) | 0 | | \(\frac{3\pi}{4}\) | \(-4\sqrt{2}\) | | \(\pi\) | -8 | | \(\frac{5\pi}{4}\) | \(-4\sqrt{2}\) | | \(\frac{3\pi}{2}\) | 0 | | \(\frac{7\pi}{4}\) | \(4\sqrt{2}\) | | \(2\pi\) | 8 | The shape of the graph is a circle centered at \((4,0)\) with a radius of 4.

Step-by-step solution

Analyze the Equation

Analyze the given equation: \(r=8\cos\theta\). This is a polar coordinate equation where \(r\) is the distance of each point from the origin and \(\theta\) is the angle each point makes with the positive x-axis.

Choose Values for \(\theta\) and Calculate \(r\)

Choose a few values for \(\theta\) to calculate \(r\) for different points. We can use angles from 0 to \(2\pi\). Remember to work in radians. For this example, we can choose the following values for \(\theta\): $$\theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}, 2\pi$$ Now, calculate the corresponding value of \(r\) for each angle: \(r = 8\cos(\theta)\)

Tabulate the Data

Create a data table to organize the values of \(\theta\) and their corresponding value of \(r\): | \(\theta\) | \(r\) | |---------|-----| | 0 | 8 | | \(\frac{\pi}{4}\) | \(4\sqrt{2}\) | | \(\frac{\pi}{2}\) | 0 | | \(\frac{3\pi}{4}\) | \(-4\sqrt{2}\) | | \(\pi\) | -8 | | \(\frac{5\pi}{4}\) | \(-4\sqrt{2}\) | | \(\frac{3\pi}{2}\) | 0 | | \(\frac{7\pi}{4}\) | \(4\sqrt{2}\) | | \(2\pi\) | 8 |

Sketch the Graph

Plot the points from the table on a polar-coordinate grid using polar axis or by converting them to Cartesian coordinates if needed. After plotting the points and connecting them, the sketch of the graph should look like a circle centered at \((4,0)\) with a radius of 4.