Question

Graph the following spirals. Indicate the direction in which the spiral is generated as \(\theta\) increases, where \(\theta>0 .\) Let \(a=1\) and \(a=-1\). Hyperbolic spiral: \(r=a / \theta\)

Short answer

Question: Describe the two types of hyperbolic spirals, their graphical representation, and the direction they move when \(\theta\) increases. Answer: The two types of hyperbolic spirals are represented by the equation \(r = a/\theta\), with cases \(a = 1\) and \(a = -1\). In rectangular coordinates, the case \(a = 1\) has parametric equations \(x(\theta) = \frac{\cos\theta}{\theta}\) and \(y(\theta) = \frac{\sin\theta}{\theta}\), while the case \(a = -1\) has parametric equations \(x(\theta) = \frac{-\cos\theta}{\theta}\) and \(y(\theta) = \frac{-\sin\theta}{\theta}\). As \(\theta\) increases, the hyperbolic spiral with \(a = 1\) moves outward in the clockwise direction, while the hyperbolic spiral with \(a = -1\) moves outward in the counterclockwise direction.

Step-by-step solution

Recognize the equation

The given equation is \(r = a/\theta\) which represents a hyperbolic spiral in polar coordinates. We have two cases: \(a = 1\) and \(a = -1\).

Convert polar equation to rectangular coordinates

To convert the polar equation \(r = a/\theta\) to rectangular coordinates, we can use the conversion formulas: \(x = r\cos\theta\) and \(y = r\sin\theta\). For \(a=1\): \(x = \frac{\cos\theta}{\theta}\) \(y = \frac{\sin\theta}{\theta}\) For \(a=-1\): \(x = \frac{-\cos\theta}{\theta}\) \(y = \frac{-\sin\theta}{\theta}\)

Graph the equations and indicate the direction of \(\theta\)

We will graph the equations for the two cases and indicate the direction when \(\theta\) increases. For \(a=1\): Graph the curve with parametric equations \(x(\theta) = \frac{\cos\theta}{\theta}\) and \(y(\theta) = \frac{\sin\theta}{\theta}\) for \(\theta > 0\). Observe that the curve spirals outward in the clockwise direction as \(\theta\) increases. For \(a=-1\): Graph the curve with parametric equations \(x(\theta) = \frac{-\cos\theta}{\theta}\) and \(y(\theta) = \frac{-\sin\theta}{\theta}\) for \(\theta > 0\). Observe that the curve spirals outward in the counterclockwise direction as \(\theta\) increases.

Final observations

The hyperbolic spiral \(r = a/\theta\) has two cases we analyzed: 1. For \(a = 1\), the spiral moves outward in the clockwise direction. 2. For \(a = -1\), the spiral moves outward in the counterclockwise direction.

Key concepts

Polar Coordinates

To understand the hyperbolic spiral in the context of the exercise, we need to start with the basics of polar coordinates. Polar coordinates provide a way of expressing a point in a plane using a distance from a reference point (usually the origin) and an angle from a reference direction (usually the positive x-axis).

The two elements in polar coordinates are the radial distance, denoted as r, and the angle, denoted as \(\theta\). A point in polar coordinates is written as (r, \theta). Unlike rectangular coordinates, which use (x, y) to show horizontal and vertical positions, polar coordinates are especially useful in scenarios where symmetry or central forces are involved, such as in the graphing of spirals.

Parametric Equations

Moving on, parametric equations play a significant role in representing the hyperbolic spiral seen in our exercise. Parametric equations express sets of related quantities as explicit functions of an independent variable, known as the parameter. Here, \(\theta\) is the parameter.

In the context of the exercise, the hyperbolic spiral's parametric equations allow us to express each coordinate, x and y, in terms of \(\theta\). For a=1 and a=-1, our equations are:
x(\theta) = \frac{a \cdot \cos(\theta)}{\theta} and y(\theta) = \frac{a \cdot \sin(\theta)}{\theta}.

These parametric equations help us understand how the values of x and y evolve as \(\theta\) changes, which is crucial for graphing spirals. By manipulating the parameter \(\theta\), we can observe the behavior and shape of the spiral directly on a Cartesian coordinate system.

Graphing Spirals

Finally, let's discuss graphing spirals, which is the ultimate goal of the exercise. Graphing a spiral involves plotting points that follow the spiral's formula, like the hyperbolic spiral \(r = a/\theta\). Spirals can be generated by plotting points using a range of \(\theta\) values and linking these points smoothly to display the characteristic shape of the spiral.

For the given exercise, as \(\theta\) increases from zero, the hyperbolic spiral behaves differently based on the value of a. When a=1, the spiral winds clockwise and outward, while for a=-1, it winds counterclockwise and outward. These observations are vital for visualizing the spiral's directionality.

To graph the hyperbolic spiral accurately, you would start with a large value of \(\theta\) near the pole and gradually decrease \(\theta\) to see the spiral arms expand outward. Illustrating the behavior of the hyperbolic spiral requires understanding its asymptotic nature, as it will never cross the pole (the origin), despite winding ever closer to it.