Problem 75 - Algebra and Trigonometry

Question

Use a graphing utility to graph the polar equation. $$r=\frac{1}{3-2 \sin \theta}$$

Step-by-Step Solution

Answer

The shape rendered by the graph of the equation \(r=1/(3-2*\sin(\theta))\) can only be ascertained by correctly inputting it into a graphing utility, properly adjusting the settings, and then generating the graph. It's recommended to use a digital tool for this task as they are specifically designed for such calculations and visualizations.

Step 1: Understand the Polar Coordinates

Before graphing, make sure you are familiar with polar coordinates. This system is different from the Cartesian coordinate system you may be more accustomed to. In polar coordinates, points are located by determining the distance \(r\) from the origin and the angle \(\theta\) measured counterclockwise from the positive x-axis.

Step 2: Input the Polar Equation

Input your polar equation into the graphing utility. If the utility uses 'theta' for the angle, your input will look something like this: \(r=1/(3-2*\sin(\theta))\). Make sure you enter the equation correctly.

Step 3: Adjust the Graph Settings

Before the graph is displayed, check the graph settings. Ensure that you are in polar mode, not rectangular mode. Adjust the window to fit the graph, and choose an appropriate range for \(\theta\), such as from 0 to 2π.

Step 4: Display the Graph

Now you can display the graph. If you've entered everything correctly, you will see a curve that perfectly fits the polar equation. Keep in mind that the looks of these graphs can be quite different from those of Cartesian coordinate equations.