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Chapter 10: Problem 74

Sketch the graph of the polar equation. $$r \theta=1, \theta>0 \quad \text { (spiral) }$$

Short Answer

Expert verified
The graph is an inward spiral in the polar coordinate system, getting closer to the origin as \(\theta\) increases.

Step by step solution

01

Understand the Format of the Polar Equation

The equation given is in polar form, where the variables are in terms of \(r\) (the radius from the origin) and \(\theta\) (the angle from the positive x-axis). In this equation, we have \(r \theta = 1\), meaning that \(r\) and \(\theta\) are inversely proportional.
02

Express \(r\) in Terms of \(\theta\)

Rearrange the equation to express \(r\) as a function of \(\theta\). Divide both sides by \(\theta\) to get \(r = \frac{1}{\theta}\). This equation will help determine how \(r\) changes as \(\theta\) increases.
03

Identify Key Characteristics of the Spiral

With \(r = \frac{1}{\theta}\), notice that as \(\theta\) increases from a small positive value, \(r\) decreases. This indicates an inward spiral that comes closer to the origin as \(\theta\) increases, maintaining \(\theta > 0\).
04

Sketch the Graph

Draw a polar coordinate system. For small \(\theta\), \(r\) is large. As \(\theta\) increases, \(r\) decreases, resulting in a spiral that winds closer to the origin. The spiral continues infinitely, but getting tighter and closer to the origin.

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Key Concepts

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

Polar Coordinates
Polar coordinates offer a unique way to represent points in a plane. Unlike Cartesian coordinates, which use
  • a pair of values
    • horizontal position (x)
    • vertical position (y)
polar coordinates use a different approach:
  • The distance from a fixed point (the origin), denoted as \( r \).
  • The angle from a fixed direction (positive x-axis), denoted as \( \theta \).
Understanding polar coordinates involves visualizing locations as radiating out from a central point, similar to how spokes extend from the center of a wheel.
This format is especially useful in scenarios where direction and distance from a central point are more intuitive, such as in navigation or when describing curved paths like spirals.
Graphing Polar Equations
Graphing polar equations involves plotting points in the polar coordinate system.
For any given polar equation, the main task is to determine how \(r\) varies with respect to \(\theta\). In our exercise, the equation is \( r\theta = 1 \).
To graph this:
  • Rearrange the equation to find \(r\) in terms of \(\theta\), giving us \(r = \frac{1}{\theta}\).
  • Choose values for \(\theta\) and calculate corresponding \(r\) values.
  • Plot these (\(r, \theta\)) pairs by measuring \(r\) units from the origin at the angle \(\theta\).
As you plot more points, you can begin to see the shape forming. For this particular equation, the plot reveals a spiral that tightens as it moves closer to the origin, due to the decrease in \(r\) as \(\theta\) increases.
Spiral Graphs
Spiral graphs are captivating due to their smooth, continuous curves that loop endlessly towards or away from a central point.
In the given polar equation, \( r\theta = 1 \), we identified that \(r = \frac{1}{\theta}\).
This feature suggests a specific type of spiral:
  • An inward spiral moving closer to the origin as \(\theta\) increases.
  • Each loop becomes tighter than the previous one.
This behavior is typical of inversely proportional relationships in polar equations.
Spirals like these can be seen in nature, such as in shells or galaxies, where the constant change in angle represents a diversity of paths around a focal point.
By sketching these graphs, you can gain insight into the elegant relationships and patterns that arise from simple mathematical equations.

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Most popular questions from this chapter

Sketch the graph of the polar equation. $$r=1+2 \cos \theta$$

Graph the ellipses on the same coordinate plane, and estimate their points of intersection. $$\frac{(x+0.1)^{2}}{1.7}+\frac{y^{2}}{0.9}=1 ; \quad \frac{x^{2}}{0.9}+\frac{(y-0.25)^{2}}{1.8}=1$$

Graph the Lissajous figure in the viewing rectangle \([-1,1]\) by \([-1,1]\) for the specified range of \(t\). $$x(t)=\sin (4 t), \quad y(t)=\sin (3 t+\pi / 6) ; \quad 0 \leq t \leq 6.5$$

Sketch the graph of the polar equation. $$r=8 \cos 5 \theta$$

Sketch the graph of the polar equation. $$r=-6(1+\cos \theta)$$
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