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Chapter 6: Problem 68

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

Short Answer

Expert verified
The graph of the polar equation \( r = \frac{3}{\sin\theta} \) is a curve that has symmetry with respect to x-axis. The curve reaches a maximum of 3 and minimum of -3 for values of \(\theta\) in the range of \([-90, 90]\) degrees. For other values of \(\theta\), the curve closes upon itself to form identical lobes in the second and fourth quadrants.

Step by step solution

01

Understand the Polar Equation

It's crucial to interpret what this polar equation \( r = \frac{3}{\sin\theta} \) means. It's apparent that the radius varies with the angle (\胃).
02

Identify the Plotting Points

The behavior of this equation can be identified by plotting some key points. Choose values for \(\theta\) and calculate the corresponding values for \(r\). As \(\sin\theta\) varies between -1 and 1, \(r\) will reach from -3 to 3. One could choose values like \(\theta= 0, 30, 45, 60, 90\) (in degrees) and their negatives for a complete coverage.
03

Sketch the Graph

With these points, start sketching the graph. Usually, a graphing tool would automate this process, but understanding how it works is important. To do this, start from the pole (the point where r=0), and draw a line for each point at the corresponding angle (\(\theta\)) from the positive x-axis, with length equal to the absolute value of \(r\). For negative values of \(r\), the line should be drawn in the opposite direction.
04

Complete the Graph using Graphing Utility

Use a graphing utility at this point to confirm the sketched graph. The utility will plot the lines for all values of \(\theta\), giving a complete graph of the polar equation. It should have symmetry with respect to the x-axis

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Key Concepts

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

Graphing Utility
When working on graphing polar equations, a graphing utility becomes an essential tool. A graphing utility is typically a software or an online platform that helps visualize mathematical functions like polar equations.
These tools are especially valuable when the equation involves complex structures or when accuracy is vital.
  • They allow you to see immediate visual feedback as you input the equation.
  • You can easily adjust the parameters to see how the graph changes in real-time.
  • Many graphing utilities provide additional features like zooming, tracing, and even 3D plotting.
Using a graphing utility simplifies the process of accurately determining the nature and shape of polar graphs, which can be challenging with manual sketching.
Polar Equation
Polar equations are a way to represent curves using polar coordinates, where each point on the plane is determined by a distance from a reference point (radius) and an angle from a reference direction. The given polar equation is \(r = \frac{3}{\sin \theta}\).
It defines a relationship where the radius \(r\) depends directly on the angle \(\theta\).
  • In this specific equation, as \(\sin \theta\) approaches 1, \(r\) becomes closer to 3.
  • When \(\sin \theta\) is negative or zero, the calculation for \(r\) becomes much different, as these values can lead to very large positive or negative \(r\) values.
  • The equation has undefined points whenever \(\sin \theta = 0\) because of the division by zero issue.
Understanding the basic shape and behavior of polar equations like these helps in anticipating what a graph will look like and in using tools like graphing utilities more effectively.
Radius and Angle
The relationship between radius and angle in polar coordinates is fundamental for understanding how polar equations work. Each point in polar coordinates is described by \((r, \theta)\), where \(r\) is the radius and \(\theta\) is the angle.
  • The radius \(r\) is the distance from the pole (the origin of the polar coordinate system) to the point.
  • The angle \(\theta\) indicates the direction of the point from the polar axis (usually considered the positive x-axis in Cartesian coordinates).
  • This system allows representation of complex curves more naturally than Cartesian coordinates, especially those with circular or spiral properties.
In the context of the problem \(r = \frac{3}{\sin \theta}\), the radius \(r\) is determined dynamically as the angle \(\theta\) changes, making the curve exhibit behavior that can fluctuate greatly, depending on the angle, which is what makes polar graphing such an interesting subject.

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Most popular questions from this chapter

Find the rectangular coordinates of each pair of points. Then find the distance, in simplified radical form, between the points. $$(6, \pi) \text { and }\left(5, \frac{7 \pi}{4}\right)$$

Suppose that \((r, \theta)\) describes the sailing speed, \(r,\) in knots, at an angle \(\theta\) to a wind blowing at 20 knots. You have a list of all ordered pairs \((r, \theta)\) for integral angles from \(\theta=0^{\circ}\) to \(\theta=180^{\circ} .\) Describe a way to present this information so that a serious sailboat racer can visualize sailing speeds at different sailing angles to the wind.

Convert each rectangular equation to a polar equation that expresses \(r\) in terms of \(\theta\) $$x+5 y=8$$

Show that each statement is true by converting the given polar equation to a rectangular equation. Show that the graph of \(r=a \sin \theta\) is a circle with center at \(\left(0, \frac{a}{2}\right)\) and radius \(\frac{a}{2}\)

Convert each rectangular equation to a polar equation that expresses \(r\) in terms of \(\theta\) $$3 x+y=7$$
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