Problem 36 - Calculus Early Transcendentals

Question

Convert the following equations to Cartesian coordinates. Describe the resulting curve. \(r=\frac{1}{2 \cos \theta+3 \sin \theta}\)

Step-by-Step Solution

Answer

Question: Convert the polar equation \(r = \frac{1}{2 \cos \theta + 3 \sin \theta}\) to Cartesian coordinates and describe the resulting curve. Answer: The Cartesian equation equivalent of the polar equation is \(y = -\frac{1}{3}x\). This curve is a straight line with a slope of \(-\frac{1}{3}\) and a y-intercept of 0. It passes through the origin and slopes downward from left to right.

Step 1: Set up the relationship between polar and Cartesian coordinates

Recall that in polar coordinates, \(x = r\cos\theta\) and \(y = r\sin\theta\). We will use these relationships to substitute for \(r\) and \(\theta\) in the given polar equation.

Step 2: Substitute for r and θ in the equation

We have \(r=\frac{1}{2 \cos \theta+3 \sin \theta}\). Multiply both sides by \((2\cos\theta + 3\sin\theta)\) to get: \(r(2\cos\theta + 3\sin\theta) = 1\)

Step 3: Use the polar-to-Cartesian relationships to substitute

Replace \(r\cos\theta\) with \(x\) and \(r\sin\theta\) with \(y\): \(x(2 + \frac{3y}{x}) = 1\)

Step 4: Clear the fraction and solve for y

Multiply through by \(x\) to eliminate the fraction: \(2x + 3y = x\) Now, solve for \(y\): \(3y = -x\) \(y = -\frac{1}{3}x\)

Step 5: Describe the resulting curve

The resulting Cartesian equation, \(y = -\frac{1}{3}x\), represents a straight line with a slope of \(\displaystyle-\frac{1}{3}\) and a y-intercept of 0. This curve is a linear function that passes through the origin and has a negative slope, meaning it will slope downward from left to right.

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. Unlike Cartesian coordinates, which use x and y axes, polar coordinates use a radial distance from a reference point and an angle from a reference direction.

• The radial coordinate \( r \) indicates how far the point is from the origin.
• The angular coordinate \( \theta \) represents the angle from a fixed direction, typically the positive x-axis.

This system is especially useful for problems involving circular or spiral patterns.
For example, in the exercise, the equation \( r = \frac{1}{2 \cos \theta + 3 \sin \theta} \) is given in polar form. The challenge is to convert this into Cartesian coordinates to find the equivalent representation in terms of x and y.

Coordinate Conversion

Coordinate conversion involves changing the representation of a point or equation from one coordinate system to another. In the context of the given exercise, the task is to convert the equation from polar coordinates to Cartesian coordinates.

• Use the relationships: \( x = r \cos \theta \) and \( y = r \sin \theta \).
• Substitute these relationships into the polar equation.

The main goal is to express everything in terms of \( x \) and \( y \), eliminating \( r \) and \( \theta \).
In the exercise, this involves setting up the equation \( r = \frac{1}{2 \cos \theta + 3 \sin \theta} \), multiplying through by \( (2 \cos \theta + 3 \sin \theta) \) to get \( r(2 \cos \theta + 3 \sin \theta) = 1 \), and substituting \( r \cos \theta \) with \( x \) and \( r \sin \theta \) with \( y \).

Linear Equation

A linear equation is any algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are typically expressed in the form \( ax + by = c \).
They graph as straight lines on a coordinate plane.

• These are the simplest type of relations you can have between two variables.
• Solving a linear equation typically involves finding the value of y in terms of x, or vice versa.

In the original exercise, the polar equation is converted to the linear form \( 3y = -x \), which simplifies to \( y = -\frac{1}{3}x \).
This simple linear equation shows a direct relationship between x and y, with variables appearing only to the first power and no products of different variables.

Slope-Intercept Form

The slope-intercept form gives a clear picture of the line's behavior in a Cartesian plane. It is typically written as \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept.

• "Slope" tells us how steep the line is. A positive slope means the line rises, while a negative slope means it falls, as it moves from left to right.
• "Y-intercept" indicates where the line crosses the y-axis.

In the exercise, the resulting linear equation from the polar equation is \( y = -\frac{1}{3}x \). Here:

• The slope \( m = -\frac{1}{3} \), indicating a downward slope.
• The y-intercept \( b = 0 \), meaning the line passes through the origin (0,0).

Understanding this form helps grasp how the line behaves and where it is positioned on the graph. This form is particularly useful for quick graphing and for understanding the relationship between the variables at a glance.