Question

Graph each polar equation. $$ r=\frac{3}{\theta} \quad(\text {reciprocal spiral}) $$

Short answer

Plot the points for specific \(\theta\)-values, then connect them to form a spiral, noting the radius decreases as the angle increases.

Step-by-step solution

Understand the Equation

The given polar equation is \( r = \frac{3}{\theta} \). This is a reciprocal spiral, a type of spiral where the radius decreases as the angle increases.

Determine Key Points

Calculate values of \(r\) for specific \(\theta\) values to plot points. For example, \(\theta = \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3}\), and calculate corresponding \(r\) values.

Plot the Points

Using the calculated values, plot the points on polar graph paper. Plot each pair \((r, \theta)\) on the graph.

Sketch the Graph

Join the points smoothly to form the spiral. Notice how the spiral comes closer to the origin as \(\theta\) increases.

Analyze the Behavior

Understand that in the graph of \( r = \frac{3}{\theta} \), as \(\theta\) approaches zero or increases indefinitely, the corresponding radius \(r\) becomes very large or very small, respectively.

Key concepts

reciprocal spiral

The reciprocal spiral is a special type of spiral graph described by the equation \( r = \frac{3}{\theta} \). Unlike other spirals that may evenly spread out, a reciprocal spiral has the unique property where the radius decreases with increasing angle.

This means as you move further around the spiral (increasing \( \theta \)), the points on the spiral get closer to the center of the graph. This gives the graph a tight coiling look around the origin as it progresses.

This behavior is clearly seen by calculating a few specific points: if \( \theta = \frac{\pi}{6} \), then \( r = \frac{3}{\frac{\pi}{6}} = \frac{18}{\pi} \).

As \( \theta \) grows larger, such as \( \theta = \frac{2\pi}{3} \), the equation becomes \( r = \frac{3}{\frac{2\pi}{3}} = \frac{9}{2\pi} \), showing how the radius decreases further.

polar coordinates

Polar coordinates provide an alternative to Cartesian coordinates for plotting points on a graph. Instead of using \( (x, y) \), polar coordinates are defined by \( (r, \theta) \) where:

• \( r \) is the distance from the origin to the point.
• \( \theta \) is the angle measured from the positive x-axis to the point (in counter-clockwise direction).

This coordinate system is particularly useful for graphs like the reciprocal spiral. By plotting \( r \) vs. \( \theta \), you often get clearer insights into the nature and behavior of the graph compared to traditional coordinates.

To plot the reciprocal spiral, you calculate the radius for various angles and plot each \( (r, \theta) \) pair on polar graph paper. Each plot draws you closer to seeing the full structure of the spiral as the angles increase.

theta

In the context of polar coordinates, \( \theta \) is the angle that helps determine the position of a point on the graph. Represented in radians, \( \theta \) measures how far you rotate around the origin from the positive x-axis.

This angle significantly influences the shape of a graph, as seen in the reciprocal spiral. As \( \theta \) increases, the radius \( r \) adjusts according to the equation \( r = \frac{3}{\theta} \).

When choosing values for \( \theta \), you typically pick angles like \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2},\) etc., because they are simpler to work with and offer a clear picture of how the radius changes.

The nature of \( \theta \) helps illustrate the behavior of complex graphs and allows for a thorough analysis of how different angles impact the plotted points.