Problem 73 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-78: Sketch the graph of the polar equation. $$ r=2 \theta, \theta \geq 0 $$

Step-by-Step Solution

Answer

The graph is a spiral starting at the origin, expanding outward indefinitely.

Step 1: Understand the Polar Equation

The given polar equation is \( r = 2\theta \), where \( \theta \geq 0 \). This means that the radius \( r \) varies linearly with \( \theta \). As \( \theta \) increases, so does \( r \), and the graph will spiral outward.

Step 2: Calculate Key Points

To sketch the graph, calculate key points at common angles for \( \theta \). For example, for \( \theta = 0 \), \( r = 0 \); for \( \theta = \frac{\pi}{4} \), \( r = \frac{\pi}{2} \); and for \( \theta = \frac{\pi}{2} \), \( r = \pi \).

Step 3: Sketch the Points on Polar Plane

Plot these calculated points on the polar coordinate plane. Start with \( (0,0) \), then \( \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \), and finally \( \left( \frac{\pi}{2}, \pi \right) \).

Step 4: Connect Points to Form the Spiral

Connect the plotted points smoothly. Since \( r \) increases with \( \theta \), the graph will form an Archimedean spiral that starts at the origin and spirals outward, covering every angle.

Step 5: Review the Spiral Nature

Since the equation is linear and constantly increasing, the spiral is uniform and keeps expanding outward indefinitely as \( \theta \) keeps increasing.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way of representing points on a plane using the distance from a central point and the angle from a reference direction. Unlike Cartesian coordinates that use the x and y axes, polar coordinates are defined as \( (r, \theta) \), where:

• \( r \) represents the radial distance from the origin (also known as the pole).
• \( \theta \) is the angle measured from the positive x-axis (counterclockwise).

This system is particularly useful in scenarios where problems exhibit radial symmetry or where properties depend on the angle and distance from a single point. For example, polar coordinates are ideal for graphing circles, spirals, and other curves that naturally follow a circular path.
If you think about how a clock's hand moves, polar coordinates capture a similar idea -- the hand rotates by an angle \( \theta \) and extends out to a length \( r \). Understanding these coordinates will make graphing equations like \( r = 2\theta \) more intuitive. Here, the concept of varying \( r \) with \( \theta \) leads to intriguing spiral patterns typical of many natural phenomena.

Archimedean Spiral

The Archimedean spiral is a famous and simple type of spiral characterized by linear growth in distance from the center as the angle increases. The basic form of an Archimedean spiral is captured by the equation \( r = a + b\theta \). In this exercise, the equation \( r = 2\theta \) is a specific case where \( a = 0 \) and \( b = 2 \), indicating that it starts from the origin and spirals outward consistently.

• This spiral is a uniform spiral, meaning the distance between turns remains constant as it winds outward.
• Unlike logarithmic spirals (such as those observed in shell formations), the Archimedean spiral maintains the same spacing between each successive turn, making it easier to predict its path.

In practical applications, Archimedean spirals appear in various fields, including physics and engineering. They're seen in the grooves of vinyl records and, interestingly, when plotting the motion paths of certain celestial objects. Their predictable, evenly spaced nature makes them a straightforward yet fascinating mathematical object.

Graph Sketching

Graph sketching in polar coordinates involves plotting points by using angles and radial distances, then connecting these points to visualize the curve defined by a given polar equation. For the equation \( r = 2\theta \), a systematic approach helps create an accurate representation:

• Identify easy-to-calculate points by selecting specific \( \theta \) values, such as multiples of \( \frac{\pi}{4} \) -- known angles make calculations straightforward.
• For example, calculate \( r \) for each chosen \( \theta \):
  - When \( \theta = 0 \), \( r = 0 \), indicating the graph starts from the origin. - For \( \theta = \frac{\pi}{4} \), \( r = \frac{\pi}{2} \), and \( \theta = \frac{\pi}{2} \), \( r = \pi \).
• Plotting these points on a polar grid allows them to be connected to visualize the spiral pattern.

By connecting points smoothly, the unique shape of the Archimedean spiral emerges, showcasing how \( r \) grows with \( \theta \). This visual approach highlights the practical and artistic beauty of mathematics, making graph sketching an engaging and enlightening task.