Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Make the required change in the given equation. \(r=2 \sin \theta\) to Cartesian coordinates

Step-by-Step Solution

Answer

The Cartesian equation is \( y = 2 \).

Step 1: Identify Polar Equations

The given equation is in polar coordinates: \( r = 2 \sin \theta \). In polar coordinates, \( r \) represents the radius from the origin, and \( \theta \) is the angle from the positive x-axis. Our goal is to transform this equation into Cartesian coordinates, where equations have variables \( x \) and \( y \).

Step 2: Apply Polar to Cartesian Conversion Formulas

In Cartesian coordinates, we can relate \( x \) and \( y \) to \( r \) and \( \theta \) using these formulas:- \( x = r \cos \theta \)- \( y = r \sin \theta \)- \( r = \sqrt{x^2 + y^2} \)We will primarily use \( y = r \sin \theta \) to help with our conversion.

Step 3: Express \( r \sin \theta \) in Terms of \( y \)

From the formula \( y = r \sin \theta \), we can substitute \( r \sin \theta \) with \( y \). Thus, we can write the equation \( r = 2 \sin \theta \) as:\[ r \sin \theta = y \ \Rightarrow y = 2 \]

Step 4: Understand the Cartesian Equation

The equation \( y = 2 \) is a Cartesian equation that represents a horizontal line two units above the x-axis in the Cartesian plane. This is the conversion of the original polar equation into Cartesian coordinates.

Key Concepts

Polar Coordinates

Polar coordinates system is a method of mapping out points on a plane using a circle-based approach. Instead of using x and y like the Cartesian system, polar coordinates describe a point's position relative to a central point (the origin) using two values:

• Radius \( r \): This is the distance from the origin to the point.
• Angle \( \theta \): This measures the angle from the positive x-axis to the point, counterclockwise.

The strength of polar coordinates lies in its simplicity when dealing with circular and rotational symmetry. For example, an equation might look like \( r = 2 \sin \theta \), indicating that the radius depends on the sine of the angle \( \theta \). This system is especially good for tasks involving angles and distances from a center, such as in designing gears or plotting stars."

Cartesian Coordinates

The Cartesian coordinate system, named after René Descartes, is used to specify each point uniquely on a plane using ordered pairs. It is one of the most commonly used coordinate systems in mathematics and has two main components:

• \( x \): The horizontal position, typically measured along a straight line from the origin.
• \( y \): The vertical position, measured perpendicular to the x-axis.

Cartesian coordinates make it easier to describe linear motions and are ideal for calculations in physics and engineering, such as determining locations on a map or the movement of a particle in a straight line. Their ability to simplify addition and multiplication in problems involving lines and curves is unparalleled when compared to polar coordinates."

Conversion Formulas

To bridge the gap between polar and Cartesian coordinates, conversion formulas are essential. These formulas allow us to express a point defined in one system into the other one.

• \( x = r \cos \theta \): Converts the radius and angle into the horizontal Cartesian coordinate.
• \( y = r \sin \theta \): Transforms polar coordinates to the vertical Cartesian coordinate.
• \( r = \sqrt{x^2 + y^2} \): Used to find the radius from Cartesian coordinates.
• \( \theta = \tan^{-1}(\frac{y}{x}) \): Assists in finding the angle from the x-axis.

These formulas let us move between systems seamlessly, taking the complexity out of circular paths in Cartesian grids or straight paths in polar circles. By using these transformations, problems like converting \( r = 2 \sin \theta \) into Cartesian form become straightforward, leading us to discover simple linear equations such as \( y = 2 \) in the Cartesian plane."

Horizontal Line in Cartesian Plane

In the Cartesian coordinate system, a horizontal line holds a special position. It is characterized by having the same y-coordinate for all points on the line, which means it runs parallel to the x-axis.

• The equation \( y = c \) describes a horizontal line, where \( c \) is a constant value.
• For example, \( y = 2 \) indicates a line that crosses the y-axis at 2 and extends in both horizontal directions.

This is crucial when interpreting conversion results from polar to Cartesian coordinates. From the polar equation \( r = 2 \sin \theta \), we derived the Cartesian equation \( y = 2 \), representing such a line. Horizontal lines in Cartesian planes are simple yet potent tools for visualizing and solving mathematical problems in various fields like architecture and computer graphics."