Problem 49 - Precalculus with Calculus Previews

Question

Use \((4)\) to parameterize the curve whose polar equation is given. Use a graphing utility to obtain the graph of the resulting set of parametric equations. $$ r=2 \cos \frac{\theta}{5}, \quad 0 \leq \theta \leq 5 \pi $$

Step-by-Step Solution

Answer

Parametric equations are \( x(t) = 2 \cos \frac{t}{5} \cos t \) and \( y(t) = 2 \cos \frac{t}{5} \sin t \) for \( 0 \leq t \leq 5\pi \).

Step 1: Understand Polar Equations

In a polar coordinate system, the variable \( r \) represents the radius or distance from the origin, while \( \theta \) represents the angle in radians from the positive x-axis. The given polar equation is \( r = 2 \cos \frac{\theta}{5} \), which defines a curve in polar coordinates.

Step 2: Convert Polar to Parametric Equations

To convert the polar equation to parametric equations, we use the relationships: \( x = r \cos \theta \) and \( y = r \sin \theta \). Substituting \( r = 2 \cos \frac{\theta}{5} \), we get:\[ x = 2 \cos \frac{\theta}{5} \cos \theta \]\[ y = 2 \cos \frac{\theta}{5} \sin \theta \]

Step 3: Define the Parameter and Limits

In parametric equations, \( \theta \) typically becomes the parameter \( t \). We set \( t = \theta \), with \( 0 \leq t \leq 5\pi \).Thus, the parametric equations can be written as:\[ x(t) = 2 \cos \frac{t}{5} \cos t \]\[ y(t) = 2 \cos \frac{t}{5} \sin t \]

Step 4: Use Graphing Utility

To visualize the parametric equations, input them into a graphing utility (like Desmos or GeoGebra). Set \( t \) from 0 to \( 5\pi \) and plot the curve. Observe the shape of the graph, which demonstrates how \( r = 2 \cos \frac{\theta}{5} \) is represented in the Cartesian plane.

Key Concepts

Polar Coordinates

Polar coordinates offer a different way to represent points on a plane, in contrast to the standard Cartesian (or rectangular) coordinate system. In polar coordinates, a point's location is determined by two values:

• Radius (r): Represents the distance from a fixed point known as the origin.
• Angle (\( \theta \)): Denotes the direction of the radius relative to a reference direction, usually the positive x-axis in radians.

For example, in the polar equation \( r = 2 \cos \frac{\theta}{5} \), the variable \( r \) is defined by the trigonometric function of \( \theta \). This particular equation creates a pattern as \( \theta \) varies from \( 0 \) to \( 5\pi \). As this angle increases, the curve is traced out by the distance from the origin changing following the cosine value.

Parametric Equations

Transforming polar equations into parametric equations allows the curve to be expressed in a way that is useful for graphing and analysis. Parametric equations are given with separate functions for \( x \) and \( y \), both as functions of a third variable, often called the parameter t.To convert a polar equation like \( r = 2 \cos \frac{\theta}{5} \) into parametric form:

• Use the relationships \( x = r \cos \theta \)
• and \( y = r \sin \theta \)

Thus, replacing \( r \) results in:

• \( x(t) = 2 \cos \frac{t}{5} \cos t \)
• \( y(t) = 2 \cos \frac{t}{5} \sin t \)

Here, t takes the place of \( \theta \), and the range of \( 0 \leq t \leq 5\pi \) covers the same values.

Graphing Utility

Graphing utilities, like Desmos or GeoGebra, are powerful tools to visualize equations in both Cartesian and polar forms. They handle computations that would be otherwise tedious and quickly generate graphs based on inputted equations.When plotting parametric equations from \( t = 0 \) to \( 5\pi \) in a graphing utility:

• Set up the equations \( x(t) = 2 \cos \frac{t}{5} \cos t \) and \( y(t) = 2 \cos \frac{t}{5} \sin t \)
• Observe how the graph unfolds as t changes.

The graph reveals the geometric interpretation of the polar equation, showcasing how every t value maps to a point in the plane, providing a comprehensive view of the curve's behavior.

Trigonometric Functions

Trigonometric functions like cosine and sine are essential in transforming polar equations. Their periodic nature is ideal for representing the continuous rotation and oscillation involved in polar curves.In the polar equation \( r = 2 \cos \frac{\theta}{5} \):

• Cosine: Determines the radial distance r continuously as \( \theta \) varies, affecting the radius.

In parametric conversion:

• >The expressions \( x = r \cos \theta \) and \( y = r \sin \theta \) rely directly on these trigonometric relationships.
• Cosine and sine functions are fundamental in changing between angular movement (\( \theta \)) and linear coordinates (x, y).

Their cyclic nature not only helps create beautiful and intricate curves but also makes trigonometric functions the backbone of both polar and parametric representations.