Problem 4 - Precalculus

Question

Fill in the blanks. The equation \(r=2 \cos \theta\) represents a __________.

Step-by-Step Solution

Answer

The equation \(r=2\cos\theta\) represents a Circle.

Step 1: Identify the polar equation

Firstly, recognize the given equation \(r=2\cos\theta\) as a polar equation.

Step 2: Identify Conic Section

This type of polar equation, where r depends on the cosine of \(\theta\), represents a specific conic section, particularly when the coefficient of cos(\(\theta\)) is a real number.

Step 3: Name the shape

The general form of the polar equation, \(r=2\cos\theta\), is known to represent a circle when graphed in polar coordinates. Thus, the shape is a Circle

Key Concepts

Polar Equation

A polar equation is an expression used to describe a curve on a plane using polar coordinates. Unlike the Cartesian coordinate system, which uses x and y coordinates, the polar coordinate system uses a radial distance (denoted as \(r\)) from the origin and an angle (denoted as \(\theta\)) from the positive x-axis. This system is especially beneficial when dealing with circular and rotational symmetries.
Recognizing polar equations is crucial as they characterize how distance \(r\) changes with the angle \(\theta\).

• Form: Usually, polar equations take the form \(r = f(\theta)\).
• Use: They are often used to describe conic sections in a rotationally symmetric manner.
• Unique: Allows the depiction of curves that can be both simple or complex in polar coordinates.

When given a polar equation like \(r = 2\cos\theta\), it's a matter of identifying the specific type of curve it describes in terms of polar characteristics.

Conic Sections

Conic sections are a group of curves formed by intersecting a plane with a cone. These sections include ellipses, parabolas, hyperbolas, and circles. Their properties make them significant in geometry and other disciplines, such as physics and engineering.
When considering them in polar coordinates:

• Conics have specific types of equations that define them. For instance, a circle is a special conic section.
• The characteristics of the conics depend on the angle of intersection and the coefficient values in the equations.
• In the polar equation \(r = 2\cos\theta\) they articulate specific properties like eccentricity and directrix in unique ways different from Cartesian formulations.

In the context of this problem, the given equation fits the description of a circle, given its symmetry and form in polar coordinates.

Circle in Polar Coordinates

A circle is perhaps the simplest conic section and is uniquely interesting in polar coordinates. To describe a circle using polar coordinates, an equation like \(r = a\cos\theta\) or \(r = a\sin\theta\) is used, where \(a\) represents the diameter of the circle.
In the context of our example \(r = 2\cos\theta\):

• This denotes a circle with its center positioned at \((a/2, 0)\) on the polar plane.
• Here, \(a = 2\), and the full span from origin to perimeter is 2 units across its diameter, indicating the circle's position along the horizontal axis.
• The use of cosine in the equation signifies its alignment along the x-axis, differentiating it from a sine-based form which would align along the y-axis.

Understanding a circle in polar terms provides clear insights into its rotational symmetry and geometrical properties, which are beneficial both visually and mathematically.