Problem 67 - Precalculus with Limits

Question

In Exercises 65-84, convert the rectangular equation to polar form. Assume \(a>0\). \(y=4\)

Step-by-Step Solution

Answer

The polar form of the rectangular equation \( y = 4 \) is \( r = \frac{4}{\sin(\theta)} \).

Step 1: Understanding the given Rectangular Equation

We know the rectangular equation is given as \( y = 4 \). This is a straight line parallel to the x-axis at y=4.

Step 2: Transform into Polar Form

From the equations defining the relationship between rectangular and polar coordinates, we have \( y = r \sin(\theta) \). Thus, the polar form of the rectangular equation \( y = 4 \) is \( r \sin(\theta) = 4 \).

Step 3: Expressing r in terms of theta

We rearrange the equation to solve for 'r'. Hence, our final resulting polar equation from \( y = 4 \) is \( r = \frac{4}{\sin(\theta)} \), which implies 'r' depends on \( \theta \) thus, 'r' will vary as \( \theta \) goes to various angles, maintaining y value constant at 4.

Key Concepts

Rectangular to Polar Conversion

Converting rectangular equations to polar form is a key skill in mathematics that allows us to explore how equations behave on different coordinate planes. The rectangular system, often referred to as the Cartesian coordinate system, uses x and y coordinates to describe locations. In contrast, the polar coordinate system uses a radius (r) and an angle (θ).

To transform a rectangular equation like \(y = 4\) into its polar form, we utilize the relationship between these two systems:

• From rectangular: \(x = r \cos(θ)\)
• \(y = r \sin(θ)\)

For \(y = 4\), substitute \(y\) with \(r \sin(θ)\):
\[ r \sin(θ) = 4 \]
This polar equation represents the same line as \(y = 4\), but described in terms of how far out \(r\) extends at each angle \(θ\). Converting equations helps in contexts like physics, where problems might be simplified in polar coordinates.

Trigonometric Identities

Trigonometric identities are fundamental tools that allow us to relate angles and sides of triangles. In the conversion process between rectangular and polar coordinates, these identities play a crucial role.

Common trigonometric identities include:

• \(\sin^2(θ) + \cos^2(θ) = 1\)
• \(\tan(θ) = \frac{\sin(θ)}{\cos(θ)}\)
• \(1 + \tan^2(θ) = \sec^2(θ)\)

In the rectangular to polar conversion, we make use of the identities:

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

These help translate the clarity of a rectangular line equation into an adaptable polar description. With the equation \(r \sin(θ) = 4\), it depicts how \(r\) changes based on \(θ\) to maintain the constant y-coordinate.

Coordinate Systems

Coordinate systems serve as different ways to describe points in space. The rectangular system uses a grid formed by two perpendicular lines, usually labeled x and y, creating an easy reference for plotting linear and quadratic equations.

Switching to a polar coordinate system can simplify situations involving circles or spirals. The polar system is made up of:

• Radius (r): Distance from the origin
• Angle (θ): Measured from the positive x-axis

Understanding the strengths of each system allows us to choose the best approach for solving mathematical problems. Converting the equation \(y = 4\) into polar form demonstrates this flexibility. While it remains a horizontal line in rectangular coordinates, its polar form \(r = \frac{4}{\sin(θ)}\) illustrates how the radius adapts to different angles while preserving its definition.