Problem 39 - Algebra and Trigonometry

Question

Sketch a graph of the polar equation. $$ r=\theta, \quad \theta \geq 0 \quad(\text { spiral }) $$

Step-by-Step Solution

Answer

The graph is an Archimedean spiral starting at the origin and spiraling outwards.

Step 1: Understand the Polar Equation

The polar equation given is \( r = \theta \), where \( \theta \geq 0 \). This means that the radius \( r \) is equal to the angle \( \theta \). The point \( (r, \theta) \) in polar coordinates represents the position on the graph.

Step 2: Plot Basic Points

Start by substituting small, incremental values of \( \theta \) to find corresponding values of \( r \). For example: \( \theta = 0 \) gives \( r = 0 \), \( \theta = \frac{\pi}{4} \) gives \( r = \frac{\pi}{4} \), and so on. Mark these points on a polar grid.

Step 3: Identify the Pattern

Observe the pattern as \( \theta \) increases and \( r \) also increases. The points plotted will start to spiral outwards from the origin because the distance \( r \) becomes larger as the angle \( \theta \) increases.

Step 4: Sketch the Graph

Using the points and understanding of the pattern, sketch a smooth curve that spirals outwards from the origin. The curve becomes more spaced out as \( \theta \) increases because \( r = \theta \) means \( r \) must increase linearly as \( \theta \) increases.

Step 5: Label the Graph

Ensure the graph is labeled properly, indicating that it is a spiral starting from the origin. You can indicate some example points to show the relation \( r = \theta \).

Key Concepts

Polar Equation

Polar coordinates provide a unique way of describing a point's location on a plane using a radius and an angle. In a polar equation, such as \( r = \theta \), \( r \) denotes the radial distance from the origin, while \( \theta \) signifies the angle formed with the positive x-axis. Unlike Cartesian coordinates that use x and y, polar coordinates emphasize radial distance and direction. This makes them particularly useful for equations describing curves like spirals and circles.

To work with a polar equation, you treat \( \theta \) as your variable and calculate \( r \) for varying values of \( \theta \). For example, if \( \theta = 0 \), then \( r = 0 \). As \( \theta \) increases, you'll generate a series of points \((r, \theta)\), defining the curve's outline on a polar grid. This equation simplifies computations for certain types of curves, allowing the demonstration of spirals in a neat and comprehensible way.

Spiral Graph

When plotting a spiral graph, you encounter an enchanting pattern that naturally expands outward. The given polar equation \( r = \theta \) perfectly exemplifies this, forming an Archimedean spiral. The key characteristic of such a spiral is its constant increase in radial distance at a linear rate as the angle increases.

In practice, as \( \theta \) moves from 0 and grows, \( r = \theta \) ensures that each point \((r, \theta)\) is further from the origin than the last, leading to the iconic spiral shape.

• The spiral begins at the origin when both \( r \) and \( \theta \) equal zero.
• Moving outward, \( r \) is directly proportional to \( \theta \), creating evenly spaced loops.
• The result is a beautifully expanding curve that elegantly demonstrates the interrelation between radius and angle in polar graphs.

This simple yet fascinating relationship is frequently observed in natural phenomena and is a staple in the study of dynamical systems.

Graph Sketching

Sketching a spiral from the polar equation \( r = \theta \) involves a step-by-step approach relying on plotting key points and intuitively connecting them. Begin by selecting specific values of \( \theta \), such as \( 0, \frac{\pi}{4}, \frac{\pi}{2}, \pi \), and calculating the corresponding \( r \) for each. By marking these points on a polar grid, you form the skeleton of your graph.

Next, identify trends and patterns: here, observe how each new point lies further from the origin than the previous one, confirming the spiral nature. Place more points to get an accurate idea of the curve's path. Using these guideposts, draw a smooth, continuous curve that gracefully extends outward from the origin.

• Avoid sharp angles; the curve should spiral out smoothly.
• Consistently increment \( \theta \) in small steps to ensure accuracy and a clear pattern.
• Label the graph to clarify it represents a spiral, specifying that \( r \) equates \( \theta \).

Successfully sketching this graph aids in visualizing and understanding complex functions depicted through polar coordinates.