Problem 41 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r=\csc \theta e^{r \cos \theta}$$

Step-by-Step Solution

Answer

The Cartesian equation is \(y = 1\), a horizontal line.

Step 1: Recall Polar to Cartesian Conversion

To convert from polar to Cartesian coordinates, use the equations: - \(x = r \cos \theta\) - \(y = r \sin \theta\) - \(r^2 = x^2 + y^2\) - \(\tan \theta = \frac{y}{x}\)

Step 2: Substitute for \(\csc \theta\)

Since \(\csc \theta = \frac{1}{\sin \theta}\), rewrite the polar equation as: \[r = \frac{1}{\sin \theta} e^{r \cos \theta}\]. This simplifies to: \[r \sin \theta = 1\] and \[r e^{r \cos \theta} = 1\].

Step 3: Convert \(r \sin \theta\) to Cartesian

Since \(r \sin \theta = y\), use this to rewrite the left side of the equation: \[y = 1\].

Step 4: Analyze \(e^{r \cos \theta}\)

Since \(r \cos \theta = x\), substitute into the equation \(r e^{r \cos \theta} = 1\) to get: \[r e^x = 1\].

Step 5: Solve the Equations

From Step 3, we have \(y = 1\). Since \(r^2 = x^2 + y^2\), substitute \(y = 1\) into this equation: \[r^2 = x^2 + 1\]. Combine this with \(r e^x = 1\) from Step 4 to solve for the other variables as needed.

Step 6: Identify the Graph

From Step 5, the equation \(y = 1\) represents a horizontal line on the Cartesian plane. The transformation in exponential terms doesn't change that basic form drastically without further constraints on \(x\), but it confirms \(y = 1\) is core to the solution.

Key Concepts

Polar Coordinates

Polar coordinates offer a way to represent points on a plane using a combination of a distance and an angle. This system is different from the rectangular Cartesian coordinate system most people are familiar with. Here, every point on a plane is determined by:

• Distance from a reference point, usually known as the pole (akin to the origin in Cartesian coordinates).
• An angle from a reference direction, typically the positive x-axis, expressed in degrees or radians.

The two components are typically denoted as \(r\) for the radius (distance) and \(\theta\) for the angle. A polar coordinate pair \((r, \theta)\) might describe a position in terms of how far the point is from the pole and at what angle from the horizontal axis.
Conversions between polar and Cartesian coordinates use trigonometric functions, where \(x = r \cos \theta\) and \(y = r \sin \theta\). This interrelation is central to understanding how to shift between these systems for applications in physics, engineering, and navigation.

Cartesian Coordinates

Cartesian coordinates describe a point's specific location on a two-dimensional plane using two values, typically represented as \((x, y)\). This system relies on a grid of horizontal and vertical lines. Here's how it functions:

• \(x\) represents the horizontal distance from the origin.
• \(y\) represents the vertical distance from the origin.

In contrast to polar coordinates, Cartesian coordinates use perpendicular axes intersecting at the origin to specify coordinates. Transitioning from polar to Cartesian coordinates involves using equations like \(x = r \cos \theta\) and \(y = r \sin \theta\), allowing a seamless conversion between the two systems.
Understanding Cartesian coordinates is critical for graphing on a coordinate plane, as it forms the basis for drawing shapes, interpreting functions, and analyzing movement within that space. Many geometric transformations and plotting tasks start with these principles.

Polar Equations

Polar equations express relationships between the coordinates \((r, \theta)\) in polar form. Converting these equations into the Cartesian form can reveal familiar graph shapes and structures. Polar equations often have intricate patterns, forming spirals, circles, or other loops which might not appear straightforward at first glance.
To convert polar equations into Cartesian, substitutions are essential:

• Since \(\csc \theta = \frac{1}{\sin \theta}\), polar equations involving \(\csc \theta\) can transform by replacing it with its Cartesian equivalent using \(y = r \sin \theta\).
• For exponential functions involving \(r \cos \theta\), use \(x = r \cos \theta\) to simplify the conversion process.

As seen in the equation \(r \csc \theta e^{r \cos \theta} = 1\), transforming into Cartesian form involves recognizing these trigonometric and exponential relationships. The ultimate goal is to rewrite the polar equation into a familiar Cartesian format, such as the linear equation \(y = 1\), aiding easier graphing and function analysis.