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Chapter 7: Problem 74

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.

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Most popular questions from this chapter

In Exercises \(77-80,\) convert to polar form and then perform the indicated operations. Express answers in polar and rectangular form. $$ (1+i)(1-i \sqrt{3})(-\sqrt{3}+i) $$

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Explaining the Concepts Describe how to graph a polar equation.

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