Problem 39 - Precalculus with Calculus Previews

Question

Find a polar equation that has the same graph as the given rectangular equation. $$ y=5 $$

Step-by-Step Solution

Answer

The polar equation with the same graph is \( r\sin\theta = 5 \).

Step 1: Understand Rectangular Equation

The given rectangular equation is \( y = 5 \), which represents a horizontal line parallel to the x-axis at a distance of 5 units above it.

Step 2: Convert y-coordinate into Polar Form

In polar coordinates, \( y \) can be expressed as \( r\sin\theta \). Therefore, the equation \( y = 5 \) becomes \( r\sin\theta = 5 \).

Step 3: Simplify and Rearrange Polar Equation

Now, rearrange the polar equation: \( r\sin\theta = 5 \). This is already in a simple polar form, where \( r \) is expressed as a function of \( \theta \).

Step 4: Review Polar Equation

Ensure the polar equation \( r\sin\theta = 5 \) represents the same geometrical graph as \( y=5 \). They both describe a horizontal line at \( y = 5 \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way of describing the location of points on a plane using two numbers. These numbers represent the distances from two perpendicular lines known as the x-axis and y-axis.
Typically, these coordinates are written as pairs \( (x, y) \), where \( x \) is the horizontal distance and \( y \) is the vertical distance.

Key Characteristics of Rectangular Coordinates:

• They are based on a grid layout formed by two intersecting lines.
• Each point on this grid is described by an \( (x, y) \) pair.
• A horizontal line like \( y = 5 \) indicates that all points on this line have the same \( y \)-coordinate but varying \( x \)-coordinates.

Understanding this system is crucial in fields such as physics, engineering, and computer graphics, where precise position measurements are important. In our example, the line \( y = 5 \) is parallel to the x-axis and situated 5 units above it.

Equation Conversion

Equation conversion involves translating equations from one form of coordinates to another. This task sometimes requires replacing variables and adjusting mathematical expressions.
In this context, we focus on converting rectangular equations into polar equations.

Here is how this works:

• Identify the rectangular variables \( x \) and \( y \). In polar coordinates, these are represented by \( r \) (radius) and \( \theta \) (angle).
• Use the relationships \( x = r\cos\theta \) and \( y = r\sin\theta \) to convert the equation.
• For the example of \( y = 5 \), convert it into the polar form as \( r\sin\theta = 5 \).

Through these steps, you derive a polar equation that graphically matches the given rectangular equation. This specific conversion allows easier analysis and understanding of certain mathematical patterns, particularly those involving curves and circular shapes.

Graphing Polar Equations

Graphing polar equations can feel different than plotting rectangular ones since the focus shifts to angles and distances from a central point known as the pole.
Instead of moving outwards from an origin using grids, polar graphs use a radial layout and angles.

Steps to Graph Polar Equations:

• Determine points using values of \( \theta \) and calculate the corresponding \( r \).
• Plot these points on a polar grid where each point is some distance \( r \) from the pole at an angle \( \theta \).
• Draw smooth curves through the plotted points to represent the figure.

For our example, \( r\sin\theta = 5 \), each point lies on a horizontal line at the height where \( y \) equals 5. This shows how a line, originally described in rectangular terms, maintains its structure when expressed in polar terms. Understanding such conversions and graphing in polar coordinates allows better insights into the relationships between different parts of geometry and their mathematical representations.