Problem 21 - Calculus Early Transcendentals

Question

\(21-26\) Find a polar equation for the curve represented by the given Cartesian equation. $$x=3$$

Step-by-Step Solution

Answer

The polar equation is \(r \cos \theta = 3\).

Step 1: Understand the Cartesian Equation

The Cartesian equation provided is \(x = 3\). This implies that the graph is a vertical line in the Cartesian coordinate system.

Step 2: Recall the Polar Coordinate Principles

In polar coordinates, each point is described by a distance \(r\) from the origin and an angle \(\theta\) from the positive x-axis. The transformations between these two coordinate systems are: \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 3: Transform the Cartesian Equation

Replace \(x\) in the Cartesian equation with its polar representation: \(x = r \cos \theta\). Therefore, it becomes \(r \cos \theta = 3\).

Step 4: Verify the Polar Equation

Ensure that the derived polar equation \(r \cos \theta = 3\) matches the original Cartesian form, which represents all points where the distance from the origin projected onto the x-axis equals 3. Since \(r \cos \theta\) denotes this component in polar terms, the translation is accurate.

Key Concepts

Cartesian equation

A Cartesian equation describes a curve by linking the x and y coordinates of each point on the curve within the Cartesian coordinate system. In this system, the x-axis and y-axis meet at a right angle to form a 2D plane. Equations in Cartesian coordinates have two variables: x and y.
For example, in the equation \(x = 3\), every point on this vertical line shares the same x-coordinate value of 3.

• This implies that no matter the y value, x remains constant.
• This is why it's considered a 'vertical line'.

Understanding Cartesian equations is crucial as they are used extensively to describe geometric shapes and functions on a two-dimensional plane.

coordinate transformation

Coordinate transformation involves converting information from one coordinate system to another. Here, the conversion is from Cartesian to polar coordinates. This transformation is often necessary because different problems are simplified or better represented by using different coordinate systems.
When converting from Cartesian to polar coordinates:

• The formula \(x = r \cos \theta\) is used to express x in terms of polar coordinates.
• Similarly, \(y = r \sin \theta\) expresses y.

Transformation relies on the relationships between trigonometric functions and circle properties.This transformation provides a way to see the geometric nature of equations more clearly in many cases.

polar equation

A polar equation describes a relationship between the distance from the origin, \(r\), and the angle, \(\theta\), in a polar coordinate system. Unlike Cartesian coordinates, polar coordinates revolve around the circle's center, making them perfect for circular and periodic phenomena.
When dealing with the equation \(r \cos \theta = 3\), it shows that the distance an object's shadow would cast onto the x-axis is always 3 units away from the center. This setup allows for a different perspective on the same line that was originally described by \(x = 3\) in Cartesian form.

• This perspective can be beneficial when working with circles or angles more naturally expressed in polar terms.

By understanding polar equations, you gain an alternative approach to geometry and trigonometry.

vertical line

A vertical line in geometry is a line that runs from top to bottom and does not change in its horizontal position, represented in Cartesian coordinates as \(x = k\), where \(k\) is a constant. For such a line, all points have the same x-coordinate, reflecting vertical symmetry.
Visualizing a vertical line helps in understanding it as a simple linear structure.

• In polar coordinates, this line becomes more nuanced with \(r \cos \theta = c\), marking where the horizontal projection remains constant on the x-axis at c.
• It signifies infinite possible y-values since y is not fixed.

This conversion exemplifies how geometry remains consistent across different mathematical systems, enhancing conceptual comprehension across coordinate systems.