Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 17-22, find the Cartesian equations of the graphs of the given polar equations. \(r \cos \theta+3=0\)

Step-by-Step Solution

Answer

The Cartesian equation is \( x = -3 \).

Step 1: Convert Polar Equation to Cartesian Form

The given polar equation is \( r \cos \theta + 3 = 0 \). To convert this into Cartesian coordinates, we recall the relation between polar and Cartesian systems: \( x = r \cos \theta \) and \( y = r \sin \theta \). Substitute \( x \) for \( r \cos \theta \), resulting in the equation \( x + 3 = 0 \).

Step 2: Rearrange the Equation

The equation from the previous step is \( x + 3 = 0 \). Isolate \( x \) by subtracting 3 from both sides to obtain \( x = -3 \).

Step 3: Interpret the Solution

The Cartesian equation \( x = -3 \) represents a vertical line in the Cartesian plane that crosses the x-axis at -3. This line is parallel to the y-axis and every point on this line has the same x-coordinate, which is -3.

Key Concepts

Polar Coordinates

Polar coordinates provide a unique way to express points on a plane. Instead of using the typical xy-coordinate system, polar coordinates use a different approach. Here, each point on the plane is defined by:

• A distance from a specific point called the origin (denoted as \( r \)).
• An angle \( \theta \), measured in radians or degrees, from a fixed direction.

The point \((r, \theta)\) represents the location of a point based on these two parameters. This system is especially useful when dealing with circular and rotational symmetries. Remember:

• If \( r \) is positive, it indicates the distance out from the origin in the direction given by \( \theta \).
• If \( r \) is negative, it represents a point in the opposite direction along the same line.

Conversion Between Systems

When working with problems involving different systems of coordinates, conversion becomes important. Switching from polar to Cartesian coordinates is a common task, involving these simple relationships:

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

This transformation allows any point given in polar form \((r, \theta)\) to be translated into the familiar Cartesian form \((x, y)\). Conversely, to convert from Cartesian to polar:

• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \tan^{-1}(\frac{y}{x}) \)

Understanding these equations is key to moving seamlessly between coordinate systems. Having these conversions at your fingertips helps in solving complex mathematical problems with various representations.

Polar to Cartesian Conversion

The process of converting equations from polar to Cartesian form involves using the right substitutions. As demonstrated in the example, if you start with a polar equation like:
\( r \cos \theta + 3 = 0 \)
You can use the conversion formula \( x = r \cos \theta \). By substituting \( x \) for \( r \cos \theta \), the equation simplifies to:

\( x + 3 = 0 \)

This is a typical step in converting polar equations into a system that is often easier to work within specific contexts, especially when sketching graphs on a Cartesian plane. It's about identifying these common substitutions and systematically applying them to simplify and interpret the given mathematical relationship.

Equation of a Line

In Cartesian coordinates, lines have several standard forms, and each tells us something different about the line's characteristics. The equation derived from the conversion, \( x = -3 \), is a straightforward example of a vertical line. Vertical lines in the Cartesian plane:

• Run parallel to the y-axis.
• Have an undefined slope.
• Can be expressed using equations of the form \( x = a \), where "a" is a constant.

This particular line, \( x = -3 \), tells us that every point on this line has an x-coordinate of -3, regardless of the y-coordinate. It's crucial to visualize such equations as lines that divide the plane and guide us on which side of the line particular points might fall.