Problem 53 - Algebra and Trigonometry

Question

Convert the polar equation to rectangular coordinates. $$ r \cos \theta=6 $$

Step-by-Step Solution

Answer

The rectangular equation is \( x = 6 \).

Step 1: Understand the Given Equation

The given polar equation is \( r \cos \theta = 6 \). We need to convert this into a rectangular (Cartesian) equation.

Step 2: Recall the Conversion Formulas

In polar coordinates, the relationships between polar and rectangular coordinates are given by: \( x = r \cos \theta \) and \( y = r \sin \theta \). We'll use these to convert the polar equation.

Step 3: Substitute for \( r \cos \theta \)

From the formulas \( x = r \cos \theta \), we can directly substitute \( x \) for \( r \cos \theta \) in the given equation. Thus, our equation becomes \( x = 6 \).

Step 4: Resulting Rectangular Equation

The equation \( x = 6 \) is a rectangular equation representing a vertical line, where for every point on the line, the x-coordinate is 6.

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points in a plane using a distance and an angle. Unlike the usual grid-like layout of rectangular coordinates, polar coordinates focus on the position relative to a starting point—known as the pole, similar to the origin in Cartesian systems—and a direction, typically measured from a fixed line known as the polar axis.

Key components include:

• Radius (\(r\)): The distance from the pole to the point of interest.
• Angle (\(\theta\)): The angle measured from the polar axis to the line connecting the pole with the point.

These coordinates are especially useful in scenarios where relationships are naturally circular or spiral, such as waves or orbits. Understanding how these differ from the rectangular coordinate system involves recognizing their simplicity and effectiveness in dealing with radial patterns.

Rectangular Coordinates

Rectangular coordinates, commonly known as Cartesian coordinates, express points in a plane through two perpendicular axes—traditionally labeled as the \(x\)-axis and \(y\)-axis. In this system, a point is uniquely identified by an \((x, y)\) pair.

This method is perfect for straightforward representations in two-dimensional space. Each coordinate measures the horizontal and vertical distance from the origin, which is where the two axes intersect.

• The \(x\)-coordinate tells you how far to move left or right.
• The \(y\)-coordinate tells you how far to move up or down.

Rectangular coordinates excel in representing linear equations and structures easily. For example, the equation \(x = 6\) signifies a vertical line where the x-value remains constant, demonstrating one of the system's strengths in illustrating straight lines.

Coordinate Conversion

Coordinate conversion is the process of changing the way we describe a point from one coordinate system to another, such as from polar to rectangular coordinates. This skill is invaluable in bridging the gap between various mathematical and scientific analyses and their real-world applications.

To convert from polar to rectangular coordinates, utilize the relationships:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

By understanding these formulas, one can transform complex polar equations into equivalent rectangular forms, which might be more intuitive to work with in certain scenarios.

In our exercise example, we converted the polar equation \(r \cos \theta = 6\) to the rectangular form \(x = 6\), representing a vertical line. Conversion not only allows for different perspectives on data but also simplifies the problem-solving process across various applications.