Problem 1 - Calculus Early Transcendentals

Question

Express the polar equation \(r=f(\theta)\) in parametric form in Cartesian coordinates, where \(\theta\) is the parameter.

Step-by-Step Solution

Answer

Answer: The parametric equations in Cartesian coordinates are given by \(x = f(\theta)\cos(\theta)\) and \(y = f(\theta)\sin(\theta)\) with \(\theta\) as the parameter.

Step 1: Identify the polar equation

Given, the polar equation is \(r=f(\theta)\).

Step 2: Convert from polar to Cartesian coordinates

To convert from polar to Cartesian coordinates, we use the following formulas: \(x = r\cos(\theta)\), \(y = r\sin(\theta)\). Since \(r=f(\theta)\), we can substitute the given function \(f(\theta)\) into these formulas to obtain the parametric equations in Cartesian coordinates.

Step 3: Substitute the function f(θ) into the formulas

We replace \(r\) with \(f(\theta)\) in our conversion formulas: \(x = f(\theta)\cos(\theta)\), \(y = f(\theta)\sin(\theta)\). These are the parametric equations in Cartesian coordinates with \(\theta\) as the parameter.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to represent points in a plane using two values: the distance from the origin (known as the radius, or "r") and the angle ("\(\theta\)") formed with the positive x-axis. This system is particularly useful for dealing with circular or spiral patterns where Cartesian coordinates might be cumbersome.

In the polar coordinate system:

• \(r\) represents the radial distance from the origin to the point.
• \(\theta\) is the angle measured in a counter-clockwise direction from the positive x-axis.

Unlike the rectangular grid of Cartesian coordinates, the grid formed by polar coordinates is circular, making it particularly useful in fields involving rotations, such as engineering and physics.

Understanding polar coordinates allows us to switch perspectives between these systems, leveraging the strengths of each.

Cartesian Coordinates

Cartesian coordinates describe a point in a two-dimensional plane using an "x" and "y" value. These coordinates form the backbone of most geometry studied in school. They define a straightforward way to describe any location on a flat surface through a horizontal and vertical axis.

The key features are:

• The x-coordinate measures horizontal distance from the origin.
• The y-coordinate measures vertical distance from the origin.

The simplicity and familiarity of Cartesian coordinates make them ideal for linear calculations and designing graphs.

When converting between polar and Cartesian, these coordinates provide the rectangular lens through which we can reinterpret circular paths and relationships into straight lines.

Conversion Formulas

Converting polar coordinates to Cartesian coordinates involves simple trigonometric relationships that help bring curved paths into a linear perspective. This conversion is essential for analyzing complex shapes and motion using straightforward algebraic techniques.

Here are the key conversion formulas:

• \(x = r\cos(\theta)\) translates the radius and angle into the x-axis coordinate.
• \(y = r\sin(\theta)\) translates the radius and angle into the y-axis coordinate.

These conversions allow us to express any polar equation like \(r=f(\theta)\) in Cartesian terms by substituting the function \(f(\theta)\) for \(r\).

By understanding and applying these formulas, we can easily switch between polar and Cartesian systems, depending on which best suits our current problem. This skill helps in solving mathematical problems involving curves, circles, and complex shapes with ease.