Problem 52 - Precalculus with Calculus Previews

Question

Find a rectangular equation that has the same graph as the given polar equation. $$ r \cos \theta=-4 $$

Step-by-Step Solution

Answer

The rectangular equation is \( x = -4 \).

Step 1: Understand Polar Coordinates

In polar coordinates, any point in the plane is defined by its distance from the origin, denoted as \( r \), and the angle \( \theta \) formed with the positive x-axis.

Step 2: Use Polar to Rectangular Conversion Formulas

Recall the conversion formulas from polar to rectangular coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \). In this step, identify the connection to convert using these formulas.

Step 3: Identify Equation Component in Rectangular Form

The given equation is \( r \cos \theta = -4 \). Notice that \( x = r \cos \theta \), so substitute \( x \) for \( r \cos \theta \) in the equation, resulting in \( x = -4 \).

Step 4: Write the Rectangular Equation

Since \( x \) and \( y \) are the rectangular coordinates, and we determined that \( x = -4 \), the rectangular form of the given equation is simply \( x = -4 \). This represents a vertical line 4 units to the left of the y-axis in the rectangular coordinate system.

Key Concepts

Polar Coordinates

Polar coordinates provide a method to locate points in a plane using two measurements: the radial distance from the origin (denoted as \( r \)) and the angular measurement from the positive x-axis (denoted as \( \theta \)).
Unlike the rectangular coordinate system, which uses Cartesian coordinates \( (x, y) \), polar coordinates give a unique advantage by streamlining equations that involve circles or rotations.
Here's why polar coordinates are significant:

• They simplify the representation of curves and shapes like spirals and circles.
• They are particularly useful in fields that require rotational symmetry, such as physics and engineering.
• Using \( r \) and \( \theta \), you can easily describe positions in circular layouts or systems.

In our exercise, we begin with the polar equation \( r \cos \theta = -4 \), which describes a line based on its distance and angle from the origin.
Understanding that \( r \) reflects distance aids in interpreting and transforming such equations into different forms.

Polar to Rectangular Conversion

To convert polar equations to rectangular form, we utilize the relation of polar components to Cartesian components.
The foundational conversion formulas are:

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

These relationships allow translation from one coordinate system to the other.
In the given exercise where the polar equation is \( r \cos \theta = -4 \), converting to rectangular form is straightforward.
By observing that \( x = r \cos \theta \), you can substitute \( x \) directly in place of \( r \cos \theta \) in the equation.
This gives the rectangular equation as \( x = -4 \).
Using these polar to rectangular conversions is essential for shifting between different coordinate systems,

• Allowing simplification or further manipulation of equations.
• Helping compare geometric interpretations of different equations.

In this context, conversion aids in understanding the positioning and forms of graphical representations.

Equation Transformation

Transforming equations between polar and rectangular formats reveals various properties of lines and curves in a coordinate system.
For our exercise, we have transformed the polar equation \( r \cos \theta = -4 \) into its rectangular form: \( x = -4 \).
Some key aspects of equation transformation:

• Simplification of Understanding: By transforming into a familiar form (like rectangular), one can interpret the equation's real-world representation, as this becomes a vertical line to the left of the y-axis.
• Geometric Insights: Rectangular transformations help show lines as collections of coordinates \( (x, y) \) directly, assisting in graphic visualization.
• System Compatibility: Switching forms ensures compatibility with systems primarily understood or represented in Cartesian coordinates.

In practice, equation transformation is crucial for:

• Facilitating problem-solving across different disciplines.
• Ensuring that equations can be worked out and solved using preferred or required coordinate systems.

Understanding the final line of \( x = -4 \) in this exercise confirms the efficacy of utilizing transformations to present data in diverse forms.