Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the given Cartesian equation, and then find the polar equation for it. $$ x=0 $$

Step-by-Step Solution

Answer

A vertical line, \(x=0\), is represented in polar coordinates as \(\theta = \frac{\pi}{2}\) or \(\theta = \frac{3\pi}{2}\).

Step 1: Understand the Cartesian Equation

The Cartesian equation given is \(x=0\). This represents a vertical line on the Cartesian coordinate system, passing through the y-axis where every point on this line has an x-coordinate of 0.

Step 2: Identify Properties of the Line

Since the equation is \(x=0\), the line stretches vertically downwards and upwards through all y-values on the y-axis. Hence, for any arbitrary point \((0,y)\), \(y\) can be any real number.

Step 3: Plot the Cartesian Graph

To plot \(x=0\), draw a vertical line crossing the y-axis at the origin (0,0). This line will continue infinitely in the upward and downward directions along the y-axis.

Step 4: Convert to Polar Coordinates

In polar coordinates, a point is given by \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle with respect to the positive x-axis. A vertical line \(x=0\) does not correspond directly to a polar "equation" line like a constant \(\theta = c\). Instead, it can be described as having infinite possible \(r\) values at \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\).

Step 5: Express the Polar Equation

For the line \(x=0\), in polar terms, it is defined by \(r\cos \theta = 0\). This simplifies to \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}\). Therefore, the polar equation is \(\theta = \frac{\pi}{2}\) or \(\theta = \frac{3\pi}{2}\), representing the angles where the vertical line passes the origin.

Key Concepts

Polar Equations

Polar equations are a fascinating way to represent curves using a polar coordinate system. In this system, each point in the plane is determined by an angle and a distance from a reference point. This is different from Cartesian coordinates, which use x and y axes. In polar coordinates, the position of a point is given by

• Radius (\(r\)): The distance from the origin.
• Angle (\(\theta\)): The counterclockwise direction from the positive x-axis.

This method of plotting is particularly useful for circular and spiral patterns, or any situations where angles and radii are more natural to describe positions than straight-line distances.
For instance, the conversion of the Cartesian equation \(x=0\) into a polar equation simplifies to the condition \(\theta = \frac{\pi}{2}\) or \(\theta = \frac{3\pi}{2}\). This means that at these specific angles, the line extends vertically through the origin, showing the elegance and compactness of polar expressions.

Cartesian Graph

A Cartesian graph is a way to visually represent equations or data by plotting them on a grid formed by two perpendicular axes: the x-axis and the y-axis. Each point on this grid can be represented by a pair of numbers (x, y), which express its horizontal and vertical displacements from the origin.
In our exercise, the equation \(x=0\) corresponds to a vertical line that passes through the origin. Every point on this line has an x-coordinate of 0, and the y-coordinate can be any real number.

• The line is infinite in the upward and downward directions along the y-axis.
• This type of graph is simple yet incredibly useful for visualizing vertical constraints.

Understanding how to move between Cartesian and polar graphs is crucial for visualizing and solving complex problems in both coordinate systems.

Coordinate Systems

Coordinate systems are methods used to determine a point's position on a graph. Two of the most common systems are Cartesian and Polar coordinates. Each serves specific uses and provides unique perspectives on plotting geometric shapes or data points.
The Cartesian coordinate system:

• Uses two perpendicular axes (x and y) to define a point's location.
• Is ideal for functions and equations involving straight lines, parabolas, etc.

Polar coordinate system:

• Uses a radius \(r\) from a central point and an angle \(\theta\) from a reference direction.
• Is perfect for circular and angular plots, like spirals and rose curves.

Each system has its advantages and can be converted from one to the other. For example, converting the line \(x=0\) to a polar format helps us understand its behavior relative to angles, showing how versatile coordinate systems can be for different mathematical problems.