Problem 19 - Calculus Early Transcendentals

Question

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$r=\csc \theta$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Polar Equation

The given polar equation is \( r = \csc \theta \). The function \( \csc \theta \) is the cosecant of \( \theta \), which is the reciprocal of \( \sin \theta \). So, we have the polar equation \( r = \frac{1}{\sin \theta} \).

Step 2: Express in Terms of Sine

Since \( \csc \theta = \frac{1}{\sin \theta} \), we rewrite the equation as: \[ r = \frac{1}{\sin \theta} \] Multiplying both sides by \( \sin \theta \) gives: \[ r \sin \theta = 1 \]

Step 3: Convert to Cartesian Coordinates

In polar coordinates, \( r \sin \theta \) can be rewritten using the Cartesian coordinate \( y \) since \( r \sin \theta = y \). Thus, the equation becomes: \[ y = 1 \]

Step 4: Identify the Curve

The equation \( y = 1 \) represents a horizontal line in the Cartesian coordinate plane, parallel to the x-axis, located one unit above it.

Key Concepts

Cartesian Coordinates

In mathematics, Cartesian coordinates are used to specify the location of points on a plane. Named after the French philosopher and mathematician René Descartes, this system uses two perpendicular axes: generally called the x-axis (horizontal) and the y-axis (vertical). Each point in the plane is defined by a pair of numerical coordinates \(x, y\), which indicate its distance from these axes.

• The origin, where the axes intersect, is denoted as \(0,0\).
• Moving right along the x-axis increases the x-value, while moving left decreases it.
• Similarly, moving up along the y-axis increases the y-value, while moving down decreases it.

This coordinate system is incredibly useful for plotting and visualizing mathematical equations and relationships, as it provides a straightforward way to represent linear and curved graph shapes.

Polar Coordinates

Polar coordinates provide an alternative to the Cartesian system for representing points on a plane, especially when dealing with circular and angular relationships. Unlike Cartesian coordinates, which use a grid of right angles, polar coordinates are expressed as \(r, \theta\).

• Here, \(r\) represents the radius or distance from the origin to the point.
• \(\theta \), usually measured in radians or degrees, represents the angle formed with the positive x-axis.

This system is particularly useful to describe phenomena that have a central point or revolve around a central axis. For example, by using the polar coordinates, you can represent the position of a point on a circle with ease. When converting polar to Cartesian coordinates, remember these key relationships:

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

Horizontal Line Equation

A horizontal line in the Cartesian plane is straightforward to understand and work with. The equation for a horizontal line is given by \(y = c\), where \(c\) is a constant indicating the vertical position of the line.

• The line is parallel to the x-axis.
• As all points on this line have the same y-coordinate, the x-coordinate can vary freely.

In the context of the solution, converting the polar equation \(r \sin \theta = 1\) into Cartesian coordinates reveals the equation \(y = 1\), which is a horizontal line one unit above the x-axis. This simple linear representation is a fundamental component in understanding how polar and Cartesian systems interconnect, particularly by showcasing how angular relationships are translated into straight lines.