Problem 51 - College Trigonometry

Question

$$ r=\theta, 0 \leq \theta \leq 12 \pi $$

Step-by-Step Solution

Answer

The curve is an Archimedean spiral that makes 6 full turns.

Step 1: Understand the Problem

This problem involves polar coordinates, where you're given a polar equation \( r = \theta \) with \( \theta \) ranging from \( 0 \) to \( 12\pi \). The task is to understand how the curve behaves over this interval.

Step 2: Analyze the Polar Equation

In the equation \( r = \theta \), both the radius \( r \) and the angle \( \theta \) are the same. As \( \theta \) increases, \( r \) increases proportionally. This means that as you rotate around the origin, the distance from the origin also increases.

Step 3: Recognize the Curve Type

The equation \( r = \theta \) is known as an Archimedean spiral. In this type of curve, as \( \theta \) increases, the radius increases linearly, creating a spiral that moves away from the origin.

Step 4: Consider the Interval

The interval \( 0 \leq \theta \leq 12\pi \) indicates that the curve wraps around the origin several times. Since \( 12\pi \) is equivalent to 6 full rotations (because \(2\pi\) is one full circle), the spiral will make 6 complete turns.

Step 5: Identify Characteristics of the Spiral

For each complete rotation of \( 2\pi \), the radius increases by \( 2\pi \) units. In this case, after 6 rotations, the radius will reach \( 12\pi \). Thus, the spiral starts at the origin and spirals outwards, reaching the radius of \( 12\pi \) at \( \theta = 12\pi \).

Key Concepts

Archimedean Spiral

An Archimedean spiral is a fascinating mathematical curve that expands outward as it revolves around a central point. The defining characteristic of this spiral is its linear growth: the distance from the center increases at a constant rate as you move around the spiral. This means that if you imagine drawing the spiral from the center outwards, it looks like a growing coil that continuously gets further away from the origin.

In the polar coordinate system, an Archimedean spiral is expressed by the equation \( r = a + b\theta \), where \( a \) and \( b \) are constants. In the case of \( r = \theta \), it is a special form of the Archimedean spiral where \( a = 0 \) and \( b = 1 \). This makes it particularly simple, as the radius \( r \) directly matches the angle \( \theta \) in radians. As \( \theta \) increases, the radial distance increases in such a way that a smooth spiral is formed.

• The spiral is equi-angular, meaning the angle between successive turns of the spiral is constant.
• For each full rotation of \(2\pi\), the spiral's radius increases by the same amount.
• This characteristic makes it a tool for understanding wave patterns and other phenomena that revolve around a point.

Polar Equation

The concept of a polar equation moves away from the traditional x-y Cartesian coordinates, instead using a combination of an angle and a distance from a central point (often called the pole or origin).

A polar equation, such as \( r = \theta \), defines a curve by the relationship between the radius \( r \) and the angle \( \theta \). The task is to understand how this equation dictates the shape or path of the curve.

• In our specific equation \( r = \theta \), the radius is equal to the angle, linking the two dimensions directly.
• This dynamic allows one to imagine that as the angle \( \theta \) increases by small increments, the radius increases as well, creating a spiraled pattern outward.
• Each increment in \( \theta \) translates to an equal increment in the distance from the origin, resulting in a consistent and predictable path, notably producing an Archimedean spiral.

r = θ

The shorthand expression \( r = \theta \) encapsulates a simple yet powerful relationship in polar coordinates. Here, each increase in \( \theta \) directly corresponds to an increase in the radius \( r \).

This straight-forward equation sets the stage for constructing spirals systematically.

• The given interval \( 0 \leq \theta \leq 12\pi \) signifies that as \( \theta \) rotates from 0 to \( 12\pi \), the curve will complete multiple loops around the central point.
• Because \(2\pi\) is one complete revolution in radians, \( 12\pi \) constitutes six full rotations.
• During these rotations, the spiral lengthens with every circular path, capturing both the beauty and mathematical elegance of how a simple linear function can model growth patterns through curvilinear motion.