Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Answer each of the following. The spiral of Archimedes has polar equation \(r=a \theta\) where \(r^{2}=x^{2}+y^{2}\). Show that a parametric representation of the spiral of Archimedes is \(x=a \theta \cos \theta, \quad y=a \theta \sin \theta, \quad\) for \(\theta\) in \((-\infty, \infty)\)

Step-by-Step Solution

Answer

Use polar-to-rectangular conversion: \(x = a \theta \cos \theta, y = a \theta \sin \theta\).

Step 1: Understand Polar Coordinates

In a polar coordinate system, a point in the plane is determined by its distance from the origin \( r \), and the angle \( \theta \) it makes with the positive x-axis. \( r = a \theta \) is the polar equation of the Archimedean spiral.

Step 2: Convert to Rectangular Coordinates

In rectangular coordinates, the relationships are \( x = r \cos \theta \) and \( y = r \sin \theta \). Since \( r = a \theta \), substitute this into the equations: - \( x = a \theta \cos \theta \) - \( y = a \theta \sin \theta \).

Step 3: Parametric Representation

The parametric equations for the Archimedean spiral are given as a set of equations where \( \theta \) varies over a certain range. Using the results from Step 2, these are already in the form: - \( x(\theta) = a \theta \cos \theta \) - \( y(\theta) = a \theta \sin \theta \). These represent the parametric form of the spiral as \( \theta \) varies from \(-\infty\) to \(\infty\).

Key Concepts

Parametric Equations

Parametric equations are a way to express a relationship between coordinates using a third variable, known as a parameter. In many cases, the parameter is time, but it doesn’t have to be. It can be any variable that allows you to describe the curve or geometric object you are interested in.
For the Archimedean spiral, the parameter is the angle \(\theta\). By using \(\theta\), we describe both the \(x\) and \(y\) coordinates in terms of a single changing value. This parameter helps in illustrating the path traced out by the spiral.

• The equations look like this: \( x(\theta) = a \theta \cos \theta \)
• \( y(\theta) = a \theta \sin \theta \)

By changing \(\theta\), we move along the spiral. This provides a clean and simple way to understand how the spiral is structured.

Archimedean Spiral

The Archimedean spiral is an interesting and straightforward type of curve which is easy to understand and visualize. Its polar form is given by \( r = a \theta \).
The value \( r \) represents the distance from the origin, effectively establishing how far out the point is on the spiral as a function of \(\theta\). This distance increases linearly with \(\theta\), unlike some other spirals where the growth might be exponential or logarithmic.

• This linear relationship is key to the spiral's structure.
• It shows a uniform spacing between the loops of the spiral.
• The bigger the \(a\), the more spread out the spirals.

When you walk along the spiral by increasing \(\theta\), it's much like walking away from the center in a very systematic and predictable way.

Rectangular Coordinates

Rectangular coordinates (or Cartesian coordinates) are the most familiar way to describe points in a plane. We use two values: \(x\) and \(y\), which tell us how far the point is to the right/left and up/down from the origin respectively.

To connect these two-coordinate systems—polar and rectangular—we apply transformations using trigonometry:

• The relation \( x = r \cos \theta \) tells us how much of the radius is projected on the x-axis.
• The relation \( y = r \sin \theta \) tells us how much of the radius projects on the y-axis.

For the Archimedean spiral, substituting \( r = a \theta \) into these equations gives:
- \( x = a \theta \cos \theta \)
- \( y = a \theta \sin \theta \)
This successfully translates the spiral from polar to rectangular coordinates, helping to visualize and analyze the spiral through a more familiar lens.