Problem 107 - Precalculus with Limits

Question

In Exercises 85-108, convert the polar equation to rectangular form. \(r=\dfrac{6}{2-3\sin\ \theta}\)

Step-by-Step Solution

Answer

The rectangular form of the polar equation \(r=\dfrac{6}{2-3\sin\ \theta}\) is \(4x^4 + 4xy^2 - 3x^2y^2 - 3y^4 = 36x^2 + 36y^2\)

Step 1: Implement x and y relationships

Start by substituting \(x = r\cos \theta\) and \(y = r\sin \theta\) into the given polar equation \(r = \dfrac{6}{2-3\sin\ \theta}\). Rewrite the equation by multiplying both sides by \(r\), that result in \(r^2 = r \times \dfrac{6}{2-3\sin\ \theta}\). Therefore \(r^2 = \dfrac{6r}{2-3\sin\ \theta}\). Now use the relationships \(x = r\cos \theta\) and \(y = r\sin \theta\) where \(r^2 = x^2 + y^2\) and multiply both sides by \(2-3\sin \theta\).

Step 2: Substitute and Simplify It

Substitute \(r^2\) for \(x^2 + y^2\) and simplify the equation. \((2-3\sin \theta)(x^2 + y^2) = 6r. \)here we replace \(r\sin \theta\) with \(y\) and \(r\) with \(\sqrt{x^2+y^2}\). So the equation becomes, \((2-3y)(x^2 + y^2) = 6\sqrt{x^2+y^2}\)

Step 3: Square Both Sides to Eliminate the Square Root

When you square both sides of equation, you get: \(4x^4 + 4xy^2 - 3x^2y^2 - 3y^4 = 36x^2 + 36y^2\). This equation is the rectangular form of the given polar equation.

Key Concepts

Polar Equations

Polar equations are a way of expressing mathematical relationships using polar coordinates, which are characterized by the variables \(r\) and \(\theta\). Here, \(r\) represents the radial distance from the origin, and \(\theta\) signifies the angular displacement from the positive x-axis.

Polar equations are especially useful when dealing with problems that have symmetrical properties or circular paths. They provide a unique perspective, differing from the more commonly used rectangular form which relies on the x-y coordinate system.

• The given equation \(r = \dfrac{6}{2-3\sin\ \theta}\) is expressed in polar form.
• This form is advantageous for describing curves and shapes that are more naturally radial, such as spirals or circles.

To convert a polar equation into a rectangular one, we utilize relationships that connect polar and rectangular coordinates. This transition is essential for solving different kinds of problems, particularly in fields like physics and engineering.

Rectangular Form

The rectangular form of an equation is based on the Cartesian coordinate system, using \(x\) and \(y\) coordinates. This form is more familiar to most students because it forms the foundation of most algebra and calculus operations.

To convert a polar equation like \(r = \dfrac{6}{2-3\sin\ \theta}\) into rectangular form, we utilize the relationships between the two systems. Here’s how:

• First, we note that \(x = r\cos \theta\) and \(y = r\sin \theta\).
• We also know \(r^2 = x^2 + y^2\).

With these relationships, we substitute in: \(r\sin\theta\) becomes \(y\) and \(r\) becomes \(\sqrt{x^2+y^2}\).

By manipulating and simplifying these expressions, we achieve a rectangular equation. In this problem, simplifying and substituting gives us the rectangular form \(4x^4 + 4xy^2 - 3x^2y^2 - 3y^4 = 36x^2 + 36y^2\). This form is useful for plotting equations on a standard x-y graph and analyzing their geometric properties.

Coordinate Systems

Coordinate systems are frameworks for locating points within a given space. Two primary types are the polar and rectangular coordinate systems.

The **rectangular coordinate system** is the most familiar; it uses a grid of perpendicular lines (x and y axes) to describe any point in two-dimensional space with an ordered pair \((x, y)\).

The **polar coordinate system**, however, uses a point's distance from a fixed origin \(r\), and an angle \(\theta\) from a reference direction (usually the positive x-axis). Points are denoted as \((r, \theta)\).

• These systems each have their advantages. Rectangular coordinates are preferred for straightforward mathematical calculations and describing linear relationships.
• Polar coordinates are often used in situations involving curves, rotations, or radial symmetry.

Understanding and converting between these systems is key as you work with different types of mathematical problems. This skill is especially vital in advanced mathematics, physics, and engineering, where each system provides insights suited for particular types of problems.