Problem 24 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r \sin \theta=-1 $$

Step-by-Step Solution

Answer

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

Step 1: Understand Polar Coordinates

A polar equation is expressed in terms of the coordinates \(r\) and \(\theta\), where \(r\) is the distance from the origin to the point, and \(\theta\) is the angle from the positive x-axis to the line connecting the origin to the point.

Step 2: Use Conversion Formulas

We use the conversion formulas between polar and Cartesian coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \). The challenge is to rewrite the given polar equation, \( r \sin \theta = -1 \), using these formulas.

Step 3: Substitute and Simplify

Substitute \( y = r \sin \theta \) into the polar equation \( r \sin \theta = -1 \). This gives \( y = -1 \).

Step 4: Describe the Cartesian Equation

The equation \( y = -1 \) is in Cartesian form. This is a horizontal line where the y-coordinate is always -1 for any value of x.

Step 5: Identify the Graph

The graph of the equation \( y = -1 \) is a horizontal line parallel to the x-axis, intersecting the y-axis at -1.

Key Concepts

Polar Coordinates

Polar coordinates are a method of representing points in a two-dimensional plane using a distance and an angle. Instead of using the traditional x and y coordinates to pinpoint a location, polar coordinates rely on:

• r: The radial distance from the origin (center of the plane) to the point of interest. It is always non-negative.
• θ (theta): The angle formed between the positive x-axis and the line connecting the origin to the point. It is measured in degrees or radians.

This system is particularly useful in scenarios involving circular and rotational symmetry, like in navigation and physics. Imagine navigating by telling someone to "go 5 units at a 30-degree angle from due east," rather than "go 3 units north and 4 units east." Polar plots are intuitive when dealing with angles and distances from a central point.

Cartesian Coordinates

Cartesian coordinates are the pairing of two numbers to represent a point in a plane or space. This system is based on the familiar x and y axes, which intersect at the origin:

• x-coordinate: Specifies the horizontal distance from the origin. Positive values are to the right of the origin, whereas negative values go left.
• y-coordinate: Indicates the vertical distance. Positive values are upwards from the origin, and negative values are downwards.

This coordinate system is named after the 17th-century mathematician René Descartes. Cartesian coordinates are widely used in everything from basic geometry to complex computer graphics. They offer a straightforward framework for working with most geometric shapes and are integral to understanding graphs and functions.

Conversion Formulas

The conversion between polar and Cartesian coordinates is based on the relationship between the two systems:

• x = r \(\cos \theta\): This formula converts the radial distance and angle into the x-value of the Cartesian coordinate.
• y = r \(\sin \theta\): Similarly, this formula transforms the radial distance and angle into the y-value.

These formulas are essential because they provide a bridge between polar and Cartesian systems. For instance, to convert the polar equation \(r \sin \theta = -1\) into Cartesian form, we substitute using the formula for y: \( y = r \sin \theta\). Given the original polar equation \(r \sin \theta = -1\), substituting gives us \( y = -1\). This solutions illustrates how the conversion formulas efficiently translate conditions from one system to another, greatly simplifying tasks involving graphing and spatial understanding.