Problem 30 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r \cos \theta=0$$

Step-by-Step Solution

Answer

The equation \(r \cos \theta = 0\) converts to \(x = 0\), a vertical line on the y-axis.

Step 1: Understanding Polar Coordinates

In polar coordinates, each point is defined by a radius \(r\) from the origin and an angle \(\theta\) from the positive x-axis. The given equation is \(r \cos \theta = 0\).

Step 2: Convert Polar Equation to Cartesian Coordinates

In polar coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\) are the conversions for Cartesian coordinates. Substituting from these formulas, \(r \cos \theta = 0\) converts to \(x = 0\).

Step 3: Describe the Graph in Cartesian Coordinates

The Cartesian equation \(x = 0\) represents a vertical line along the y-axis. This line passes through all points where the x-coordinate is 0.

Key Concepts

Polar Coordinates

When we talk about polar coordinates, we are diving into a system where each point on a plane is defined by a distance from a reference point and an angle from a reference direction.

• The reference point is often known as the "origin" or "pole."
• The reference direction is usually the positive x-axis.

A pair \(r, \theta\) in polar coordinates consists of:

• \(r\): the radial distance from the origin. It represents how far the point is from the center.
• \(\theta\): the angle measured in radians or degrees. It is the direction relative to the positive x-axis.

For example, the polar equation \(r \cos \theta = 0\) can be initially puzzling. This expression implies that the radial component of the x-coordinate should be zero. It emphasizes how polar coordinates are inherently different from the familiar Cartesian system.

Cartesian Coordinates

Cartesian coordinates are usually what we think of first when locating points in a plane. Each point is identified by an \(x, y\) pair. This system is built on two perpendicular lines called the x-axis and y-axis. The intersection is the origin.

• \(x\): the horizontal component. It tells you how far a point is to the right or left of the origin.
• \(y\): the vertical component. It specifies how far a point is up or down from the origin.

When converting from polar to Cartesian coordinates, we use the formulas:

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

For instance, when \(r \cos \theta = 0\) in polar coordinates converts to \(x = 0\) in Cartesian coordinates, it shows every point where the x-value is zero, illustrating the connection between these two systems.

Vertical Line Equation

A vertical line in the Cartesian plane is a line that moves up and down without any sideways movement. The equation for a vertical line is expressed as \(x = c\), where \(c\) is a constant.

• This means that every point on this line has the same x-coordinate, \(c\).
• No matter how much you move up or down, the x-coordinate does not change.

In our exercise, after converting \(r \cos \theta = 0\) to \(x = 0\), we discovered a vertical line along the y-axis.

• This specific line passes through the origin and extends infinitely in both upward and downward directions.
• It's an excellent example of how different equations describe the same visual concept — a line through the vertical axis.