Problem 27 - Calculus Early Transcendentals

Question

Convert the following equations to Cartesian coordinates. Describe the resulting curve. \(r \cos \theta=-4\)

Step-by-Step Solution

Answer

Question: Convert the polar equation \(r \cos \theta = -4\) to Cartesian coordinates and describe the resulting curve. Answer: The polar equation is equivalent to the Cartesian equation \(x = -4\), which represents a vertical line passing through the point \((-4,0)\).

Step 1: Identify the conversion formulas

To convert a polar equation to Cartesian coordinates, remember that: \(x = r \cos \theta\) \(y = r \sin \theta\)

Step 2: Convert the given equation to Cartesian coordinates

We are given the polar equation \(r \cos \theta = -4\). We can see that the left side of the equation is the expression for the \(x\)-coordinate in Cartesian coordinates. Therefore, we can directly convert the polar equation into the Cartesian equation: \(x = -4\)

Step 3: Describe the resulting curve

The Cartesian equation that we got is \(x = -4\). This equation represents a vertical line that passes through the point \((-4, 0)\).

Key Concepts

Polar to Cartesian Conversion

When converting from polar to Cartesian coordinates, we utilize two fundamental equations:

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

Polar coordinates express a point's location using a radius \(r\), which is the distance from the origin, and an angle \(\theta\), which is measured from the positive x-axis. To find the Cartesian coordinates (\(x, y\)), we replace \(r\cos\theta\) with \(x\) and \(r\sin\theta\) with \(y\).
For instance, if you're given an equation like \(r \cos \theta = -4\), you can identify that the expression is the \(x\)-coordinate in Cartesian form. Thus, converting it gives the equation \(x = -4\) in Cartesian coordinates. Converting successfully between these coordinate systems is essential for graphing and understanding the relationships of equations in different realms.

Vertical Line Equation

In Cartesian coordinates, a vertical line can be described by a simple equation, \(x = a\), where \(a\) is a constant. This indicates that every point on the line has the same \(x\)-coordinate of \(a\), but the \(y\)-coordinate can be any real number.

For example, the equation \(x = -4\) represents a vertical line passing through \((-4, 0)\), capturing all points where the horizontal distance from the origin is consistently \(-4\). It's important to note that vertical lines do not have a defined slope, as their rise is undefined compared to their run, and thus their slope is considered infinite.
When interpreting and drawing these lines on a graph, remember that the vertical line will cross the x-axis at the point \((a, 0)\). Practicing with different constants will help reinforce how these lines appear visually.

Graph of a Line

To understand the graphing of lines, it's beneficial to recognize the basic types: vertical, horizontal, and sloped lines. Each type has distinct characteristics and equations.

• Vertical Lines: As mentioned, are represented by \(x = a\) and are parallel to the y-axis.
• Horizontal Lines: These are denoted by \(y = b\) where \(b\) is a constant, indicating that all points have the same \(y\)-coordinate. These lines run parallel to the x-axis.
• Sloped Lines: Characterized by the equation \(y = mx + c\), where \(m\) represents the slope of the line and \(c\) is the y-intercept.

When converting from polar to Cartesian and plotting these lines, understanding the derivatives of these equations helps define the geometry of the line. Whether it's identifying intercepts or determining slope, mastering these principles allows for easier graph interpretation and construction.
Graphing different types of lines reveals patterns and relationships in equations, aiding in mathematical visualization and problem-solving.