Problem 26 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r \cos \theta=0 $$

Step-by-Step Solution

Answer

The graph is the y-axis, described by the equation \(x = 0\).

Step 1: Understand the Polar Equation

The given polar equation is \(r \cos \theta = 0\). In polar coordinates, \(r\) represents the radius and \(\theta\) the angle from the positive x-axis. To convert it to Cartesian coordinates, identify how \(r\) and \(\theta\) relate to \(x\) and \(y\).

Step 2: Use Polar to Cartesian Conversion Formulas

Recall that in Polar to Cartesian conversion, \(x = r \cos \theta\) and \(y = r \sin \theta\). Since the given equation is \(r \cos \theta = 0\), we directly substitute \(x\) for \(r \cos \theta\).

Step 3: Convert the Equation

Substituting, we have \(x = 0\). This is the simplified Cartesian equation equivalent to the given polar equation.

Step 4: Identify the Graph

The equation \(x = 0\) represents a vertical line on the Cartesian plane along the y-axis. This means the graph is the y-axis itself.

Key Concepts

polar coordinates

Polar coordinates are a way to describe a point in a plane using two values: a radius and an angle. The radius, denoted by \(r\), tells you how far away the point is from the origin, which is considered the center of the coordinate system. The angle, denoted by \(\theta\), is measured from the positive x-axis and indicates the direction of the radius. Together, \( (r, \theta) \) uniquely defines a point in the polar coordinate system.

Polar coordinates are especially useful in scenarios where circular or rotational symmetries are present. They provide an alternative to the more common Cartesian coordinates, which are usually preferred when dealing with horizontal and vertical translations. Polar coordinates can seem confusing at first but are very powerful for the right situations. They provide a different perspective and can simplify many problems, especially in fields such as physics and engineering.

Cartesian plane

The Cartesian plane is a two-dimensional plane defined by two perpendicular axes: the x-axis and the y-axis. Each point on this plane is described by a pair of values \((x, y)\), with \(x\) indicating the horizontal position and \(y\) indicating the vertical position. In Cartesian coordinates, the axes divide the plane into four quadrants: first, second, third, and fourth.

Unlike polar coordinates that focus on radius and angles, Cartesian coordinates deal directly with distances and directions along straight lines. The geometric interpretation of equations is often more straightforward in the Cartesian plane. It is the most common coordinate system used in mathematics, and many functions and equations are first understood in this framework before considering transformations or conversions.

For instance, any line or curve can be expressed as an equation in Cartesian terms, such as \( y = mx + b \) for a straight line, where \(m\) is the slope and \(b\) is the y-intercept.

equation conversion

Equation conversion involves changing a mathematical equation from one form to another. In this context, we look at converting equations from polar to Cartesian form. This process lets you reinterpret problems using the most convenient framework for solving them.

To convert from polar to Cartesian equations, use the relationships:

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

By substituting these values into a polar equation, you transform it to a Cartesian equation.

For example, given \(r \cos \theta = 0\) in polar coordinates, substituting \(x = r \cos \theta\) gives you \(x = 0\). This translates the equation to a format that could be easily graphed or analyzed in the Cartesian plane.

Equation conversion is a key tool in mathematics, providing flexibility and often simplifying the process of solving complex problems by presenting them in more familiar terms.

vertical line

In the context of the Cartesian plane, a vertical line is a line that runs straight up and down. It is parallel to the y-axis and perpendicular to the x-axis. The defining characteristic of a vertical line is that all points on the line have the same x-coordinate but can have any y-coordinate.

This can be represented by an equation of the form \(x = c\), where \(c\) is a constant. This indicates that no matter the value of \(y\), \(x\) will always equal \(c\).

In this exercise, the polar equation \(r \cos \theta = 0\) converts to the Cartesian equation \(x = 0\), which represents the vertical line corresponding to the y-axis. Vertical lines play an essential role in many geometry and calculus problems, such as analyzing slopes, intercepts, and symmetries in shapes and graphs.