Problem 25 - Precalculus with Limits

Question

In Exercises 23-48, sketch the graph of the polar equation using symmetry, zeros, maximum \(r\)-values, and any other additional points. \(r=\dfrac{\pi}{3}\)

Step-by-Step Solution

Answer

The polar plot of the given equation represents a circle with a radius of \( \pi /3\). There is symmetry about the line of \(θ=0\). The maximum r-value is \( \pi /3\). The graph does not have zero as r always equals \( \pi /3\).

Step 1: Decode the Polar Equation

The given polar equation is \(r = \pi /3\). As can be seen, this equation doesn't include any dependence on the angle θ. In such a case, it means that r has a constant value for all θ. Thus, the polar plot will represent a circle with a radius of \(\pi /3\).

Step 2: Sketch the Polar Graph

Draw a circle centered at the origin. Since the equation doesn't depend on θ, the radius remains the same for all angles. Measure out a distance of \(\pi /3\) along each of the polar axis and draw a circle of this radius.

Step 3: Note down the Characteristics of the Graph

The graph is a perfect circle with a constant radius of \(\pi /3\). The polar graph has symmetry about the line of θ=0 due to its round shape that is identical in all directions. The maximum r-value is \(\pi /3\) as stated in the equation.

Key Concepts

Understanding the Polar Equation

A polar equation like the one given, \(r = \frac{\pi}{3}\), is unique as it describes relationships using polar coordinates. In polar coordinates, a point in the plane is represented as \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angular direction. In our case, the equation is particularly simple because \(\theta\) doesn't appear in the equation. This means that \(r\) is constant, irrespective of the angle. As a result, this equation represents a circle with a fixed radius extending in all directions, forming a circle around the origin.

Identifying Symmetry in Polar Graphs

Symmetry in polar graphs is essential for simplifying graphing of polar equations. There are three main types of symmetry to consider in polar graphs:

• Symmetry about the polar axis (horizontal symmetry).
• Symmetry about the line \(\theta = \frac{\pi}{2}\) (vertical symmetry).
• Symmetry about the pole (origin).

For the polar equation \(r = \frac{\pi}{3}\), the graph exhibits symmetry about all these axes because it forms a perfect circle. Regardless of the direction you examine, the graph remains unchanged. This is a distinct characteristic of a circle in polar coordinates, making the identification of symmetry straightforward in these cases.

Determining the Maximum r-value

In polar graphs, the maximum \(r\)-value is the largest distance from the origin that is plotted in the graph. For our polar equation \(r = \frac{\pi}{3}\), the maximum value of \(r\) is directly provided by the equation. This is because \(r\) does not vary with \(\theta\). Every point on the graph maintains the same distance \(\frac{\pi}{3}\) from the origin, so \(\frac{\pi}{3}\) is both the minimum and maximum \(r\)-value. This contrast from more complex polar equations where \(r\) varies, underlines the simplicity of graphing a circle via polar coordinates.

Graphing Polar Functions

Graphing polar functions involves plotting points for various angles \(\theta\) based on their corresponding \(r\)-values. With polar equations like \(r = \frac{\pi}{3}\), where \(r\) is constant, graphing becomes straightforward. Here’s how you can sketch it:

• Draw the polar axes with origin marked clearly.
• Measure a consistent distance of \(\frac{\pi}{3}\) from the origin in every direction.
• Connect these points with a smooth curve to form a circle.

This graph depicts all possible points at a fixed distance from the origin. Simplifying further, since \(r\) does not depend on \(\theta\), adding more points around the circle will all have the same radial distance, resulting in a circle that uniformly portrays the relationship described by the polar equation.