Problem 47 - 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. $$r=8 \sin \theta$$$

Step-by-Step Solution

Answer

The polar equation \(r = 8 \sin \theta\) converts to \(y = 8\), a horizontal line at \(y = 8\).

Step 1: Understanding Polar and Cartesian Coordinates

Polar coordinates are expressed in terms of \(r\) (distance from the origin) and \(\theta\) (angle from the positive x-axis), whereas Cartesian coordinates are expressed in terms of \(x\) and \(y\). To convert from polar to Cartesian, use the formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 2: Apply the Conversion Formula for \(y\)

Given the polar equation \(r = 8 \sin \theta\), we can use the relationship \(y = r \sin \theta\). Since \(y = r \sin \theta\), we substitute \(r\) to get \(y = 8\) based on \(r \sin \theta = 8 \sin \theta\).

Step 3: Simplify and Rearrange

We already have \(r \sin \theta = y\), and from the equation \(r = 8 \sin \theta\), we substitute to find that \(y = r \sin \theta = 8 \sin \theta = 8\). The Cartesian equation becomes \(y = 8\).

Step 4: Describe the Graph

The Cartesian equation \(y = 8\) represents a horizontal line in the Cartesian plane. This line lies parallel to the x-axis, crossing the y-axis at \(y = 8\).

Key Concepts

Polar Coordinates

Polar coordinates are a system for describing locations on a plane using two parameters: the radial distance and the angular direction from a fixed point, usually the origin.

When using polar coordinates, we typically denote the radial distance from the origin as \( r \), and the angle measured from the positive x-axis, known as the polar angle, as \( \theta \). This is especially useful for problems involving circular and rotational symmetry.

• \( r \) represents the distance from the origin, not negative.
• \( \theta \) is the angle in degrees or radians.

Polar coordinates simplify math with circular patterns or symmetry for graphing and analysis. This system complements Cartesian coordinates by providing an alternative way of interpreting the spatial relationship between points.

Cartesian Coordinates

Cartesian coordinates provide a different way of representing points by using a pair of numerical values: \(x\) and \(y\). These values directly correspond to horizontal and vertical positions in a rectangular grid. The origin

• \( x \): Horizontal distance from the origin.
• \( y \): Vertical distance from the origin.

To switch from polar to Cartesian, we use conversion formulas: - \( x = r \cos \theta \)- \( y = r \sin \theta \)In our example, we start with the polar equation \(r = 8 \sin \theta\). Knowing \( y = r \sin \theta \), and substituting gives us \( y = 8 \). This results in a simple, Cartesian form, allowing us to graph and understand the spatial relationship differently.

Graph Description

Graphs are visually displaying the relationship between mathematical equations and represent data using a coordinate plane. In our exercise, converting the polar equation \( r = 8 \sin \theta \) to Cartesian coordinates simplifies to the equation \( y = 8 \).

This Cartesian equation vividly describes a horizontal line on the graph. The line runs parallel to the x-axis and intersects the y-axis at the point where \( y = 8 \).

• The entire line maintains a constant \( y \) value of 8.
• The line offers a clear representation of the points satisfying the given equation.

This understanding of different graph shapes, like lines or circles, helps in visualizing mathematical concepts and interpreting real-world data through equations.