Problem 51 - Precalculus Mathematics for Calculus

Question

A polar equation is given. (a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$r=\frac{4}{2-\cos \theta}$$

Step-by-Step Solution

Answer

Parametrically: \(x(\theta) = \frac{4 \cos \theta}{2 - \cos \theta}\), \(y(\theta) = \frac{4 \sin \theta}{2 - \cos \theta}\). Graph it for \(\theta\) from 0 to \(2\pi\).

Step 1: Understand Polar to Parametric Conversion

To express a polar equation in parametric form, we need to use the relationships between polar coordinates \((r, \theta)\) and Cartesian coordinates \((x, y)\). For any polar coordinate, we have the following conversions:- \(x = r \cos \theta\)- \(y = r \sin \theta\)Start by solving these equations using the given polar equation.

Step 2: Express \(r\) in Terms of \(\cos \theta\)

From the given polar equation \(r = \frac{4}{2 - \cos \theta}\), let's substitute this expression of \(r\) into both the conversions for \(x\) and \(y\).

Step 3: Substitute \(r\) in Parametric Equations

Use the expression for \(r\) in both equations:\[x = \left(\frac{4}{2 - \cos \theta}\right) \cos \theta\]\[y = \left(\frac{4}{2 - \cos \theta}\right) \sin \theta\]Here, the parametric equations for \(x\) and \(y\) are in terms of \(\theta\).

Step 4: Simplify the Parametric Equations

Simplify the equations obtained:\[x = \frac{4 \cos \theta}{2 - \cos \theta}\]\[y = \frac{4 \sin \theta}{2 - \cos \theta}\]Now, these are the parametric equations. Note that \(\theta\) serves as the parameter.

Step 5: Graph Parametric Equations

Use a graphing device or software to input the parametric equations \(x(\theta) = \frac{4 \cos \theta}{2 - \cos \theta}\) and \(y(\theta) = \frac{4 \sin \theta}{2 - \cos \theta}\). Set \(\theta\) within an appropriate range to see the complete graph, for instance, from 0 to \(2\pi\).

Step 6: Interpret the Graph

By graphing these parametric equations, you should observe the shape described by the original polar equation. The graph will showcase the path traced by the parametric form, reproducing the polar plot in Cartesian coordinates.

Key Concepts

Parametric Equations

Parametric equations are a useful way to represent curves by linking each point on the curve to a parameter, typically denoted as \( t \). This approach is common in physics and engineering, where complex motion paths are involved. In the context of polar coordinates, the parameter that we use is often the angle \( \theta \).

To convert a polar equation like \( r=\frac{4}{2-\cos \theta} \) into parametric form, we utilize the trigonometric relationships between the polar and Cartesian systems:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

These equations help link the polar angle \( \theta \) with the Cartesian coordinates \( x \) and \( y \). By substituting the expression for \( r \) into these conversion formulas, we transform to parametric mode, where \( \theta \) is treated as the parameter.

Conversion to Cartesian Coordinates

The process of converting polar equations to Cartesian coordinates involves expressing \( x \) and \( y \) in terms of known polar quantities. This transformation is vital since it translates data into a more familiar x-y coordinate system.

For the given polar equation \( r = \frac{4}{2 - \cos \theta} \), once expressed in parametric equations, the transformation becomes:

• \( x = \frac{4 \cos \theta}{2 - \cos \theta} \)
• \( y = \frac{4 \sin \theta}{2 - \cos \theta} \)

Each equation is now in terms of the Cartesian coordinates and the polar angle \( \theta \). These parametric forms enable us to explore and graph the curve using graphing software or devices, making analysis and visualization easier.

Graphing Polar Equations

Graphing polar equations helps visualize the geometric path described by the relationship between \( r \) and \( \theta \). Using parametric forms to graph these equations in a Cartesian plane provides a clear representation of the path without constraints of polar plots.

To graph the parametric equations \( x(\theta) = \frac{4 \cos \theta}{2 - \cos \theta} \) and \( y(\theta) = \frac{4 \sin \theta}{2 - \cos \theta} \), set your graphing device to interpret \( \theta \) as the range parameter, often going from 0 to \( 2\pi \). This range covers a full revolution in polar terms, ensuring the completeness of the graph in Cartesian coordinates.

By analyzing the resulting graph, we can visually interpret complex mathematical relationships and appreciate the beauty of mathematical curves. Graphing technology brings clarity and enables interactive exploration of the parameterized paths originally defined by polar equations.