Problem 28 - Algebra and Trigonometry

Question

15–36 Sketch the graph of the polar equation. $$r \theta=1, \quad \theta>0 \quad \text {(reciprocal spiral)}$$

Step-by-Step Solution

Answer

Sketch a reciprocal spiral where \( r = \frac{1}{\theta} \) for \( \theta > 0 \).

Step 1: Understand the Polar Equation

The given polar equation is \( r\theta = 1 \). In polar coordinates, \( r \) represents the radius, and \( \theta \) represents the angle. Our goal is to understand how these variables relate to each other in this equation.

Step 2: Express r in Terms of θ

We can express the equation in the form \( r = \frac{1}{\theta} \). This allows us to see how radius changes as the angle \( \theta \) changes. The equation suggests that as \( \theta \) increases, \( r \) decreases.

Step 3: Identify the Shape of the Graph

The equation \( r = \frac{1}{\theta} \) is known as a reciprocal spiral. As \( \theta \) approaches infinity, \( r \) approaches zero. This means the graph will spiral inward indefinitely as \( \theta \) increases.

Step 4: Plot Points to Visualize the Graph

Choose a set of \( \theta \) values greater than zero, and calculate the corresponding \( r \) values: \( \theta = \frac{\pi}{6} \Rightarrow r = \frac{6}{\pi} \), \( \theta = \frac{\pi}{3} \Rightarrow r = \frac{3}{\pi} \), and \( \theta = \pi \Rightarrow r = \frac{1}{\pi} \). These points can help visualize the graph.

Step 5: Sketch the Reciprocal Spiral

Using the points plotted, sketch a curve that starts at \( (r,\theta) \) with larger \( r \) values and spirals inward towards the origin as \( \theta \) increases. The spiral should get tighter, closing in towards the origin.

Key Concepts

Polar Coordinates

Polar coordinates are a unique way to describe a point in a plane using a distance and an angle. Unlike the usual Cartesian coordinates, which utilize the x and y axes to determine position, polar coordinates rely on:

• Radius (\( r \)): The distance from the origin to the point.
• Angle (\( \theta \)): The angle measured from the positive x-axis to the line connecting the origin to the point.

This system is particularly useful in situations involving circular and spiral patterns. The essential characteristic is that it expresses locations in terms of how far and in what direction a point is from a central point, often making it more intuitive for circular motion and systems. For example, in polar coordinates, a circle of radius 1 centered at the origin is simply described as \( r = 1 \) for all \( \theta \). This makes it easier to handle certain curves like spirals.

Polar Equations

Polar equations are mathematical expressions that relate the radius \( r \) and the angle \( \theta \) to one another. These equations lie at the core of plotting points and graphs in polar coordinates. When dealing with polar equations, it is important to:

• Understand the relationship between \( r \) and \( \theta \).
• Explore how changes in \( \theta \) affect \( r \), or vice versa.
• Identify distinct patterns, such as spirals or circles, through the equation.

For example, consider the polar equation \( r \theta = 1 \). Rearranging this, we express it as \( r = \frac{1}{\theta} \), which shows how \( r \) decreases inversely with increasing \( \theta \). Such equations often guide us in understanding complex patterns common in nature, like the petals of flowers or the coils of shells.

Reciprocal Spiral

The reciprocal spiral is a fascinating curve described by the equation \( r = \frac{1}{\theta} \). This is a specific type of polar equation, and it displays a unique spiraling pattern. As \( \theta \) increases, the radius \( r \) decreases inversely, creating a spiral that moves inward towards the origin. The spiral does not close on itself but continues innately tighter as it approaches the center:

• When \( \theta \) is small, \( r \) is large, and the spiral starts its outward motion.
• As \( \theta \) increases, \( r \) decreases, routing the curve inward.
• Finally, as \( \theta \) tends to infinity, \( r \) approaches zero, indicating the infinite inward spiral.

Understanding reciprocal spirals is essential in fields ranging from physics to art, as they represent naturally occurring structures and aesthetic forms. By plotting specific points and observing the curve, it becomes evident how the spiral coils tighter and demonstrates the elegance and complexity of polar graphing.