Problem 90 - Precalculus: Functions and Graphs

Question

Sketch the graph of the polar equation. \(r \theta=1, \theta>0\) (spiral)

Step-by-Step Solution

Answer

The equation describes a spiral that coils toward the origin as \(\theta\) increases.

Step 1: Understand the Polar Equation

The given equation is in polar form: \( r \theta = 1 \). In polar coordinates, \( r \) represents the radius, and \( \theta \) represents the angle. The product of these two quantities must remain constant, which indicates the equation is describing a spiral.

Step 2: Solve for r in Terms of θ

Rearrange the equation \( r \theta = 1 \) to express \( r \) in terms of \( \theta \). The equation becomes\[ r = \frac{1}{\theta} \]for \( \theta > 0 \). This implies that as \( \theta \) increases, \( r \) decreases, forming a spiral.

Step 3: Identify Key Points

To sketch the graph, it helps to identify key points. Calculate \( r \) for several values of \( \theta \):\[ \begin{align*} \text{When } \theta = \frac{\pi}{4}, &\ r = \frac{1}{\pi/4} = \frac{4}{\pi} \ \text{When } \theta = \frac{\pi}{2}, &\ r = \frac{1}{\pi/2} = \frac{2}{\pi} \ \text{When } \theta = \pi, &\ r = \frac{1}{\pi} \ \text{When } \theta = 2\pi, &\ r = \frac{1}{2\pi} \ \end{align*} \]These calculations show the spiral's nature with decreasing radius as \( \theta \) increases.

Step 4: Sketch the Graph

Using the calculated points, begin sketching. Start at the point closest to the origin where \( r \) is largest and draw the spiral inwards. As \( \theta \) increases, continue drawing the curve in a spiral shape, ensuring \( r \) decreases.

Step 5: Describe the Spiral

The graph is a spiral starting at a large radius when \( \theta \) is small, then curving towards the origin as \( \theta \) becomes larger. The spiral continuously gets closer to the origin but never actually reaches it, as \( \theta \) approaches infinity.

Key Concepts

Polar Equations

Polar equations are a fascinating mathematical concept used to describe curves in a two-dimensional plane. Unlike Cartesian coordinates, where points are determined using an x and y axis, polar coordinates rely on a center point (or pole) and an angle, as well as a distance, called the radius or r. In a polar equation, the relationship between the radius and the angle is crucial. Consider our equation, \( r \theta = 1 \). Here, the product of \( r \) (the radius) and \( \theta \) (the angle) is always equal to one. This was derived by rearranging the terms to \( r = \frac{1}{\theta} \), illustrating a distinct relationship where as the angle increases, the radius decreases. Polar equations are particularly useful for plotting curves such as circles, roses, and spirals, which are complex to describe using Cartesian coordinates. With a polar equation like \( r \theta = 1 \), we can effectively represent a spiral by focusing on the relationship between radius and angle, showcasing the versatility and beauty of polar graphs.

Graphing Spirals

Graphing spirals in polar coordinates requires us to understand how changes in the angle \(\theta\) impact the radius \(r\). In our example \( r \theta = 1 \), solving for \( r \) gives us \( r = \frac{1}{\theta} \). This tells us that with each increment in \( \theta \), the radius starts to shrink, naturally forming a spiral.To effectively plot a spiral, it can be beneficial to calculate specific points. For instance, by computing \( r \) for particular values of \(\theta\), such as \(\frac{\pi}{4}, \frac{\pi}{2}, \pi, \text{and } 2\pi\), we gain insight into how the spiral behaves. As exemplified:

• When \( \theta = \frac{\pi}{4} \), \( r = \frac{4}{\pi} \)
• When \( \theta = \frac{\pi}{2} \), \( r = \frac{2}{\pi} \)
• When \( \theta = \pi \), \( r = \frac{1}{\pi} \)
• When \( \theta = 2\pi \), \( r = \frac{1}{2\pi} \)

These points help shape the spiral, showing a smooth transition towards the center as \( \theta \) increases. Understanding these relationships helps in manually sketching the curve or using software for a more precise rendering. The spiral will continue indefinitely towards the origin without crossing it.

Radius and Angle Relationship

The relationship between radius and angle in polar coordinates is essential for understanding and visualizing graphs, especially spirals. In our given equation \( r \theta = 1 \), the interaction between \( r \) and \( \theta \) is particularly intriguing.As \( \theta \) increases, \( r \) must decrease to maintain their constant product of 1, leading to the creation of a spiral that moves inward. This inverse relationship is fundamental to understanding why the graph of this particular equation forms a spiral. Such dynamics illustrate that with every increase in \( \theta \), you are effectively moving 'tighter' along the spiral.Practically, this means that if you were to plot every possible pair of \( (r, \theta) \) values, the plotted points would trace out a pattern that curves continuously towards the center (origin). The spiral will always approach the origin closer and closer but will never reach it as long as \( \theta \) grows without bounds. This pattern reflects the smooth yet never-ending nature of spirals, which is both mathematically intriguing and visually beautiful.