Problem 64 - A Graphical Approach to Precalculus with Limits

Question

Answer each of the following. Show that the hyperbolic spiral given by \(r \theta=a,\) where \(r^{2}=x^{2}+y^{2},\) is given parametrically by \(x=\frac{a \cos \theta}{\theta}, \quad y=\frac{a \sin \theta}{\theta}, \quad\) for \(\theta\) in \((-\infty, 0) \cup(0, \infty)\)

Step-by-Step Solution

Answer

The hyperbolic spiral is given as \( x = \frac{a \cos \theta}{\theta}, y = \frac{a \sin \theta}{\theta} \).

Step 1: Begin with Polar Coordinates

The hyperbolic spiral is represented in polar coordinates as \( r = \frac{a}{\theta} \). Given that in polar coordinates, \( r \) represents the radial distance, \( \theta \) is the angle.

Step 2: Convert Polar to Parametric Equations

The relations between polar coordinates and Cartesian coordinates are \( x = r \cos\theta \) and \( y = r \sin\theta \). Substitute \( r = \frac{a}{\theta} \) into these equations. This gives:\[x = \left(\frac{a}{\theta}\right) \cos\theta = \frac{a \cos\theta}{\theta},\]\[y = \left(\frac{a}{\theta}\right) \sin\theta = \frac{a \sin\theta}{\theta}.\]

Key Concepts

Polar Coordinates

Polar coordinates are a way of describing the position of a point on a plane using the distance from a reference point, called the pole, and an angle from a reference direction, typically the positive x-axis. This system is especially useful when dealing with problems involving circular or spiral patterns.

In the context of the hyperbolic spiral, the polar equation is given by \( r = \frac{a}{\theta} \). Here, \( r \) is the radial distance from the pole, and \( \theta \) is the angle in radians. This equation describes a curve that spirals towards the pole as \( \theta \) increases or decreases.

The main advantage of using polar coordinates is that they simplify the representation of curves that are circular or spiral in nature. For instance, plotting a hyperbolic spiral is straightforward in polar coordinates but would be more complex in Cartesian coordinates.

Parametric Equations

Parametric equations are used to express coordinates as functions of a parameter. This approach is particularly useful for describing curves, allowing a single variable (often \( t \) or \( \theta \)) to drive the position along the curve.

In our example, the hyperbolic spiral is expressed with parameters: \( x = \frac{a \cos \theta}{\theta} \) and \( y = \frac{a \sin \theta}{\theta} \). Here, \( \theta \) serves as the parameter that defines both horizontal \( x \) and vertical \( y \) positions as it varies. This method provides a clear and complete understanding of how the curve behaves at different angles or positions.

Parametric equations are particularly helpful for:

• Mapping complex curves that can't be easily described with just \( x \) and \( y \).
• Analyzing motion along a path.
• Describing periodic or circular functions.

This flexibility makes them an indispensable tool in both mathematics and physics.

Cartesian Coordinates

Cartesian coordinates offer a way to locate points in a plane by using two perpendicular lines, typically labeled the x-axis and y-axis. Each point is then identified by a pair of values, \( (x, y) \), representing its horizontal and vertical distances from the origin, where the axes intersect.

The Cartesian form of a hyperbolic spiral involves transforming the parametric equations back into x and y terms. By using the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \), and substituting \( r = \frac{a}{\theta} \), we derive the following equations:

• \( x = \frac{a \cos \theta}{\theta} \)
• \( y = \frac{a \sin \theta}{\theta} \)

These coordinates are perhaps the most familiar format for describing shapes and curves in everyday math problems. Because they are based on simple perpendicular measurements, Cartesian coordinates are easy to grasp and widely used in plotting graphs and solving geometric problems.