Problem 28 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r=-3 \sec \theta $$

Step-by-Step Solution

Answer

The equivalent Cartesian equation is \(x = -3\), a vertical line.

Step 1: Convert Secant to Cartesian Variables

The given polar equation is given as \( r = -3 \sec \theta \). The secant function \( \sec \theta \) is equal to \( \frac{1}{\cos \theta} \). Therefore, we can rewrite this equation as \( r = \frac{-3}{\cos \theta} \).

Step 2: Substitute Polar to Cartesian Conversion Formulas

Recall from polar-to-Cartesian conversion, \( x = r \cos \theta \) and \( y = r \sin \theta \). We have \( \cos \theta = \frac{x}{r} \). Substitute \( \cos \theta \) in, we have \(r \cos \theta = x\).

Step 3: Rewrite Equation in Cartesian Form

Substitute \( r = \frac{-3}{\cos \theta} \) into \( r \cos \theta = x \) to get \[ \left(\frac{-3}{\cos \theta}\right) \cos \theta = x \]. Simplifying gives us \( x = -3 \).

Step 4: Interpret the Cartesian Equation

The equation \( x = -3 \) represents a vertical line in the Cartesian plane. Every point on this line has an x-coordinate of -3, regardless of the y-coordinate.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance and an angle. Instead of using the familiar x and y coordinates, each point is determined by:

• Its distance from the origin, often called "r".
• An angle "\( \theta \)" from the positive x-axis.

This representation is especially useful in circular or rotational contexts. To remember:

• "r" tells you how far from the origin the point is, akin to a radius of a circle.
• "\( \theta \)" indicates the direction to move from the positive x-axis, measured in radians or degrees.

Example: A point with polar coordinates (3, 45°) is 3 units away from the origin, moving at a 45° angle from the positive x-axis. It provides a different way to look at location data that ties more closely to natural and geometric shapes.

Cartesian Equations

Cartesian equations describe relationships between x and y in a two-dimensional plane. They are the standard way of writing equations we often see:

• Linear, such as \( y = mx + b \)
• Quadratic, such as \( y = ax^2 + bx + c \)

In polar-to-Cartesian conversion, we apply formulas like:

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

These help us translate polar equations into Cartesian coordinates. The conversion emphasizes finding x and y values that correspond to a given r and \( \theta \). This is illustrated effectively by converting the polar equation \( r = -3 \sec \theta \) into the Cartesian form \( x = -3 \). It simplifies the relationship between variables into something we can graph easily.

Graph Interpretation

Interpreting the graph involves understanding what the equations represent visually on the Cartesian plane. Graphs translate abstract numbers and equations into:

• Lines
• Curves
• Shapes

They allow us to "see" the relationships between variables. When the result of converting \( r = -3\sec\theta \) was found to be a straight line \( x = -3 \), it could be visualized as:

• A vertical line crossing the x-axis at -3.

Graph interpretation involves recognizing patterns, symmetry, and intersections, offering a powerful tool to comprehend equations' inherent geometry. Through graphs, complex equations become more intuitive and insightful, as they provide an immediate visual context.