Problem 34 - Precalculus

Question

Use a graphing utility to graph the polar equation. Identify the graph. $$r=\frac{12}{2-\cos \theta}$$

Step-by-Step Solution

Answer

The graph of the equation \(r=\frac{12}{2-\cos \theta}\) is a special curve known as a 'Conchoid of De Sluze'.

Step 1: Enter the Polar Equation into a Graphing Utility

Find a graphing utility that supports polar coordinates and enter the given polar equation \(r=\frac{12}{2-\cos \theta}\). Remember the angle \(\theta\) is measured in radians.

Step 2: Graph the Equation

Graph the equation in the utility. The graph should reveal a curve that possibly rotates about the origin.

Step 3: Identify the Graph

After graphing, try to identify the graph. Your graph is a special curve known as a 'Conchoid of De Sluze'.

Key Concepts

Graphing Utility

Using a graphing utility is a helpful way to visually explore mathematical equations. It allows you to input equations and visualize their graphs, which can provide insight into their behavior.
When dealing with polar coordinates, a suitable graphing utility will enable you to visualize equations that use the \(r\) and \(\theta\) format. Calculators with this feature can plot the points by considering \(\theta\) as the angle and \(r\) as the radius from the origin.

To graph our specific polar equation \(r=\frac{12}{2-\cos \theta}\), input the equation into your polar plotting utility. Ensure the mode is set to radians, as this is the standard measurement for angles in polar coordinates. This graph will help create a bridge between the abstract equation and its visual representation, allowing you to identify patterns or special characteristics.

Polar Equation

Polar equations express relationships using polar coordinates rather than Cartesian coordinates. In a polar coordinate system, each point is defined by an angle \(\theta\) and a radius \(r\).

The equation \(r=\frac{12}{2-\cos \theta}\) is a polar equation. It signifies a relationship where the distance from the origin (r) changes depending on the angle (\(\theta\)).

• The numerator, 12, affects the size and shape of the curve.
• The expression \(2 - \cos \theta\) in the denominator influences the behavior, particularly the vertical asymptotes and direction.

As the equation is graphed, look for how these components alter the curve's appearance. Polar equations can create diverse and unique shapes that are not as straightforward in Cartesian systems.

Conchoid of De Sluze

The Conchoid of De Sluze is a fascinating type of curve with rich historical importance in mathematics. Named after René de Sluze, this curve is defined by a polar equation characteristic of the generalized conchoids.

For the equation \(r=\frac{12}{2-\cos \theta}\), the resulting graph is a Conchoid of De Sluze. These curves have distinct properties:

• They often exhibit loops or asymmetric patterns in their form.
• The graph can reveal sections where it seems to wrap around or shift toward the origin or away based on the angular movement of \(\theta\).

Understanding these curves involves observing their oscillations and how they interact with the polar axes. By graphing the Conchoid of De Sluze, you’ll gain insight into how coefficients and function operations like cosine can profoundly affect the curve's nature.