Problem 34 - A Graphical Approach to Precalculus with Limits

Question

For each rectangular equation, give its equivalent polar equation and sketch its graph. $$3 x-2 y=6$$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{6}{3\cos(\theta) - 2\sin(\theta)} \), representing a line.

Step 1: Understand the Relationship between Cartesian and Polar Coordinates

In Cartesian coordinates, a point is represented as \((x, y)\). In polar coordinates, the same point is represented using the distance from the origin, \(r\), and the angle \(\theta\) from the positive x-axis. These coordinate systems are related by the equations \[x = r\cos(\theta) \y = r\sin(\theta)\]

Step 2: Substitute Polar Coordinate Expressions into the Original Equation

The given rectangular equation is \(3x - 2y = 6\). Substitute \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\) into this equation:\[3(r\cos(\theta)) - 2(r\sin(\theta)) = 6\] This simplifies to:\[r(3\cos(\theta) - 2\sin(\theta)) = 6\]

Step 3: Solve for the Polar Equation

To express \(r\) as a function of \(\theta\), divide both sides of the equation by \(3\cos(\theta) - 2\sin(\theta)\):\[r = \frac{6}{3\cos(\theta) - 2\sin(\theta)}\]This is the polar form of the given equation.

Step 4: Sketch the Graph

To sketch the graph, understand that the original equation \(3x - 2y = 6\) is a line. In Cartesian coordinates, this line has a slope of \(\frac{3}{2}\) and intersects the x-axis at \((2, 0)\). In polar form, the line will be represented as above, reflecting the linear relationship in polar coordinates.

Key Concepts

Cartesian Coordinates

Cartesian Coordinates are a way to pinpoint a location on a 2D plane using two values, x and y. Think of it like a city grid where you use street names and numbers to find your destination. In this coordinate system:

• The horizontal line is called the x-axis.
• The vertical line is called the y-axis.
• Together, they create a grid where each point can be defined with an ordered pair (x, y).

Each point on this plane is found by counting how far along the x-axis first, and then moving parallel to the y-axis. This system is very useful for forming equations of shapes and curves, and for this reason, it's often called a "rectangular coordinate system." Cartesian coordinates are usually the starting point when working with graphs before translating the information into polar coordinates.

Rectangular Equation

A Rectangular Equation is an equation that defines a relationship between x and y in the Cartesian plane. In our example, the rectangular equation is given as \(3x - 2y = 6\), which describes a straight line.To understand it, note:

• The equation involves only the x and y variables.
• It can be transformed into the slope-intercept form \(y = mx + b\), which reveals its slope and y-intercept.

Identifying the form and the operations involved helps in understanding the nature of the geometric figure it represents. Every rectangular equation can potentially be converted into a polar equation through a change of coordinates, which turns the line into a radial form based on distance and angle.

Coordinate Conversion

Coordinate Conversion is the process of changing one kind of coordinate (like Cartesian) into another format (like Polar). This step involves using trigonometric identities and relationships.For our line \(3x - 2y = 6\):

• Replace \(x\) with \(r\cos(\theta)\).
• Replace \(y\) with \(r\sin(\theta)\).

By substituting, we get the polar form: \(r(3\cos(\theta) - 2\sin(\theta)) = 6\). Solving for \(r\), you arrive at \(r = \frac{6}{3\cos(\theta) - 2\sin(\theta)}\).This conversion facilitates the understanding of how shapes and lines in the plane relate to their corresponding angles and radial distances from a central point, enriching the contextual comprehension of geometric forms.

Graph Sketching

Graph Sketching involves drawing the visual representation of an equation on the coordinate plane. For our exercise, we focus on both the Cartesian and Polar forms of the line.Steps to sketch:

• Start with the rectangular equation, \(3x - 2y = 6\), and convert it to the form \(y = mx + b\) for easy sketching.
• Identify key points such as where the line crosses the x and y axes.
• Translate this information into polar coordinates.

In Polar coordinates, instead of x and y, use distance and angle. The result is a line considering its angle and distance from the origin, providing an alternative viewpoint that is particularly useful for problems involving circular or radial symmetry.Graph sketching helps visualize the relationship and improve understanding of the geometry involved, whether you're using Cartesian or Polar Coordinates.