Problem 64 - Algebra and Trigonometry

Question

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$ r \cos \theta=7 $$

Step-by-Step Solution

Answer

The rectangular form of the polar equation \( r \cos \theta = 7 \) is \( x = 7 \). The graph of the equation is a vertical line passing through the point at x=7 and covers all y values.

Step 1: Convert the polar equation to rectangular form

We can convert a polar equation \( r \cos \theta = 7 \) to rectangular form using the conversion relation \( r \cos \theta = x \). Thus the rectangular form of the equation is \( x = 7 \).

Step 2: Plotting the rectangular equation

The rectangular equation \(x = 7\) represents a vertical line passing through the point (7,0) on the x-axis in the Cartesian coordinate system, i.e., all y values are valid for x = 7. Therefore, plot the vertical line x=7 to complete the graph.

Key Concepts

Polar Equations

Polar equations define relationships based on the polar coordinate system, where each point on a plane is determined by a distance from a reference point, and an angle from a reference direction. In our exercise, the equation \( r \cos \theta = 7 \) is in polar form.
Polar equations are incredibly useful for representing curves and shapes that circularly spread out from a point, such as circles and spirals.

• The variable \( r \) refers to the radius, or the distance from the origin, often called the pole.
• \( \theta \) represents the angle measured from the positive x-axis in a counterclockwise direction.

Converting polar equations to rectangular equations helps in using them within the context of a Cartesian coordinate system, allowing for a different perspective and sometimes a simpler approach to analysis.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way of determining the position of a point on a plane using two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
This system describes each point using an ordered pair \((x, y)\), where \(x\) is the horizontal distance from the origin, and \(y\) is the vertical distance.
In the context of our conversion, the polar coordinate \( r \cos \theta = 7 \) translates to \(x = 7\) in rectangular coordinates.

• Using \(x = r \cos \theta \), we convert the expression directly to find its rectangular counterpart.
• This conversion allows us to work easily with equations in a more familiar plane-coordinate setup, facilitating easier graphing and algebraic manipulation.

Understanding these coordinates is foundational to many fields, such as physics and computer graphics.

Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional system used for graphing and solving equations. Named after the mathematician René Descartes, this system uses two axes: the horizontal x-axis and the vertical y-axis.
The point where these axes intersect is known as the origin, represented as \((0,0)\). From here, any point in the plane can be described by its coordinates \((x, y)\).
When we convert the polar equation \( r \cos \theta = 7 \) to the rectangular form \( x = 7 \), this means we're describing a vertical line on the Cartesian plane.

• The graph of \(x = 7\) indicates a line that crosses the x-axis at the point 7 and extends infinitely in the vertical direction.
• In this system, lines and curves can be described with great precision, facilitating problem-solving and various applications in science and engineering.

Mastering the Cartesian coordinate system is essential for success in many mathematical analyses and for creating models of real-world phenomena.

Graphing Equations

Graphing equations is a powerful tool for visualizing mathematical relationships and understanding their behavior. In this context, graphing can help illustrate the transition from a polar equation to its rectangular equivalent.
For the equation \(x = 7\), graphing this in a Cartesian coordinate system involves making a straight vertical line passing through the point \((7,0)\) on the x-axis.

• No matter the value of \(y\), the value of \(x\) remains constant at 7, representing every possible \((7,y)\) pair.
• This visualization process allows you to see how the abstract equation translates into a geometric feature.

Graphing not only aids in comprehension but also in identifying solutions and features such as intercepts and slopes; it forms a bridge between algebraic expressions and geometric representations.