Problem 74 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph of the polar equation \(r=\frac{1}{1-\sin \theta}\) depicts a conic section. By substituting the polar function into Cartesian coordinates and graphing the resulting equations \(x = \frac{\cos(\theta)}{1-\sin(\theta)}\) and \(y =\frac{\sin(\theta)}{1-\sin(\theta)}\), the graph of the original function can be visualized.

Step 1: Understanding Polar Functions

Polar coordinates describe a point's distance from the origin 'r' and the counterclockwise angle from the x-axis '\(\theta\)'. In the polar equation \(r=\frac{1}{1-\sin \theta}\), 'r' represents the distance from the origin and '\(\theta\)' is the angle from the positive x-axis.

Step 2: Converting Polar to Cartesian Coordinates

Polar coordinates can be converted to Cartesian coordinates using the following transformations: \(x = r*\cos(\theta)\) and \(y = r*\sin(\theta)\). Substitute \(r=\frac{1}{1-\sin \theta}\) into these equations to get \(x = \frac{\cos(\theta)}{1-\sin(\theta)}\) and \(y =\frac{\sin(\theta)}{1-\sin(\theta)}\). Now, these equations can be graphed on a Cartesian coordinate system.

Step 3: Using a Graphing Tool

Plug in the equations \(x = \frac{\cos(\theta)}{1-\sin(\theta)}\) and \(y =\frac{\sin(\theta)}{1-\sin(\theta)}\) into a graphing tool (like Desmos, GeoGebra etc.). The graph drawn is the graph of the given polar equation.