Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

In each of Problems 11-16, sketch the graph of the given Cartesian equation, and then find the polar equation for it. \(x-3 y+2=0\)

Step-by-Step Solution

Answer

Polar equation: \(r = \frac{-2}{\cos(\theta) - 3\sin(\theta)}\).

Step 1: Understanding the Cartesian Equation

The given Cartesian equation is a linear equation of the form \(x - 3y + 2 = 0\). This equation represents a straight line in the Cartesian plane.

Step 2: Rearrange the Cartesian Equation

Rearrange the given equation to slope-intercept form \(y = mx + b\). Start by subtracting \(x\) and \(2\) from both sides to isolate terms of \(y\): \(-3y = -x - 2\). Now divide every term by \(-3\): \(y = \frac{x}{3} + \frac{2}{3}\). This line has a slope \(m = \frac{1}{3}\) and a y-intercept \(b = \frac{2}{3}\).

Step 3: Sketch the Graph of the Line

Construct the graph by identifying the y-intercept and using the slope. Start at \((0, \frac{2}{3})\) on the y-axis, then use the slope \(\frac{1}{3}\) by going up 1 unit and right 3 units to plot another point, \((3, 1\frac{2}{3})\). Draw the line through these points.

Step 4: Convert the Cartesian Equation to Polar Form

To convert to polar form, use the relationships \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Substitute these into the original Cartesian equation: \(r\cos(\theta) - 3r\sin(\theta) + 2 = 0\). Factor out \(r\): \(r(\cos(\theta) - 3\sin(\theta)) + 2 = 0\). Rearrange for \(r\): \(r = \frac{-2}{\cos(\theta) - 3\sin(\theta)}\). This equation is the polar form of the given Cartesian line.

Key Concepts

Cartesian Equation

A Cartesian equation represents points on a plane using rectangular coordinates, usually denoted by the variables \(x\) and \(y\). These equations describe the relationship between the horizontal and vertical coordinates of a point. For instance, the equation \(x - 3y + 2 = 0\) is a linear Cartesian equation that describes a straight line in the plane.

A Cartesian equation can have different forms, such as the slope-intercept form \(y = mx + b\) or the standard form \(ax + by = c\). Each form is useful for different purposes, such as finding the slope of a line or determining intercepts with axes. Understanding how to manipulate and interpret these forms helps in graphing and analyzing linear equations on a Cartesian plane.

Linear Equation

Linear equations are algebraic equations that form straight lines when graphed on a Cartesian plane. The general form of a linear equation in two dimensions is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.

• Slope-Intercept Form: The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) represents the slope of the line, indicating steepness and direction, and \(b\) is the y-intercept, the point where the line crosses the y-axis.
• Standard Form: Expressed as \(ax + by = c\), this form is useful for analyzing intercepts easily.

Linear equations can help describe real-world phenomena where there is a constant rate of change. Understanding linear equations is crucial for modeling and predicting such relationships.

Graphing

Graphing is the visual representation of equations on a coordinate plane, which helps in understanding the behavior of a function or relation. To graph the line from our Cartesian equation \(x - 3y + 2 = 0\), we first convert it into the slope-intercept form: \(y = \frac{1}{3}x + \frac{2}{3}\).

Here's how to graph the line based on the slope-intercept form:

• Identify the y-intercept: The line crosses the y-axis at \(b = \frac{2}{3}\).
• Use the slope \(m = \frac{1}{3}\): From the y-intercept, move up one unit and right three units to find another point on the line.
• Draw the line through these points, extending it in both directions.

Graphing helps in visualizing how lines and curves behave, revealing intersections, parallels, and other relationships.

Coordinate Conversion

Coordinate conversion involves changing from one coordinate system to another. In this exercise, we convert from Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\).

Polar coordinates express a point's position using a distance \(r\) from the origin and an angle \(\theta\) from the positive x-axis. The formulas for conversion are:

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

For our line \(x - 3y + 2 = 0\), substituting these into the equation gives \(r(\cos(\theta) - 3\sin(\theta)) + 2 = 0\). Solving for \(r\) results in the polar equation: \[r = \frac{-2}{\cos(\theta) - 3\sin(\theta)}\]

Conversion aids in problem-solving, as some problems are simpler in Cartesian form while others benefit from the properties of polar coordinates.