Problem 74 Use a graphing utility to graph ... [FREE SOLUTION]

Chapter 6: Problem 74

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

Short Answer

Expert verified

To graph the given polar equation, you need to understand the polar coordinate system, analyze the function's behavior for different values of $\theta$, and then plot the function using a graphing utility, skipping values that make the function undefined.

Step by step solution

Understand the Polar Coordinate System

In the polar coordinate system, the location of a point is determined by its distance $r$ from the origin (or pole) and by the measure of the angle $\theta$ (in radians or degrees) between the line segment from the origin to the point and the positive x-axis. The relation between Cartesian coordinates $(x, y)$ and polar coordinates $(r, \theta)$ is given by $x = r cos \theta$ and $y = r sin \theta$.

Analyze the Polar Equation

Here, $r$ is a function of $\theta$, whereby $r=\frac{1}{1-\sin \theta}$. Before graphing, it is helpful to understand that the $\sin$ function fluctuates between -1 and 1. Because of the 1 in the denominator of the function, we must be mindful that values of $\theta$ that result in $1 - \sin \theta = 0$ will result in undefined points in the graph (divide by zero is undefined). That means when $\sin \theta = 1$ so $\theta = 0.5\pi + 2n\pi$, where $n$ is an integer, the function $r$ becomes undefined.

Plot the Function

Utilizing a graphing utility, you can begin plotting points for different values of $\theta$ (from 0 to $2\pi$ for example). It's important to remember to omit values of $\theta$ where the function is undefined. Make sure to note that since this is a polar graph, points are plotted as a distance from the origin at the specified angle rather than as Cartesian coordinates. For a more accurate graph, it is necessary to plot several points for each quadrant. The utility will then connect these points to create the polar graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Polar Equations

Graphing polar equations can be both fascinating and challenging. The main idea is to visualize the equation by plotting points in a polar coordinate system. In polar equations like $ r=\frac{1}{1-\sin \theta} $, $ r $ represents the distance from the origin, and $ \theta $ is the angle from the positive x-axis.

To graph such an equation, here's what you do:

Select various values for $ \theta $, ideally covering a full circle, from $ 0 $ to $ 2\pi $.
Calculate the corresponding $ r $ for each $ \theta $.
Plot each $ r $ at its respective angle $ \theta $ from the origin.
Note that, in this specific equation, there are points where the equation is undefined because dividing by zero isn't possible (e.g., when $ 1 - \sin \theta $ equals zero).
Use a graphing tool or software for accuracy and to visualize trends more easily.

Remember, in polar graphs, the shapes can be quite diverse, ranging from simple circles to more intricate spirals and loops.

Polar Coordinate System

The polar coordinate system is an alternative to the Cartesian coordinate system. In this system, the position of a point is based on a radius and angle. This system is particularly useful in scenarios where relationships between points are easier to describe with angles and distances instead of standard x and y coordinates.

Here's how the polar coordinate system works:

The first coordinate, $ r $, denotes the distance from the origin (or pole) to the point.
The second coordinate, $ \theta $, measures the angle from the positive x-axis to the point's position vector (the line from the origin to the point).

Polar coordinates offer a unique way of plotting that can make certain shapes or patterns more intuitive to visualize, like spirals or circles, especially when dealing with phenomena that rotate around an origin.

Cartesian Coordinates Conversion

Converting between polar and Cartesian coordinates is a vital skill as each has its advantages in different scenarios. Polar coordinates $(r, \theta)$ can be converted into Cartesian coordinates $(x, y)$ using the following formulas:

$ x = r \cdot \cos \theta $
$ y = r \cdot \sin \theta $

Conversely, if you want to convert from Cartesian coordinates to polar coordinates, you'd use:

$ r = \sqrt{x^2 + y^2} $
$ \theta = \tan^{-1} \left( \frac{y}{x} \right) $

These conversions are particularly useful in calculus and physics, where oftentimes problems are easier to solve when switching between systems. Remember that the choice of coordinate system can simplify problem handling based on symmetry and functional form.

91影视

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

Short Answer

Step by step solution

Understand the Polar Coordinate System

Analyze the Polar Equation

Plot the Function

Key Concepts

Graphing Polar Equations

Polar Coordinate System

Cartesian Coordinates Conversion

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