Problem 63 - Precalculus Mathematics for Calculus

Question

Convert the polar equation to rectangular coordinates. $$r=\frac{4}{1+2 \sin \theta}$$

Step-by-Step Solution

Answer

The rectangular form is \(x^2 - 3y^2 + 16y - 16 = 0\).

Step 1: Recall the Polar to Rectangular Conversion Formulas

To convert from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), we use: - \(x = r \cos \theta\) - \(y = r \sin \theta\) - \(r^2 = x^2 + y^2\)

Step 2: Substitute \(r\) and Simplify

Given the equation \(r = \frac{4}{1 + 2 \sin \theta}\), multiply both sides by \(1 + 2 \sin \theta\) to clear the fraction:\[ r(1 + 2 \sin \theta) = 4 \]Substitute \(y = r \sin \theta\):\[ r + 2y = 4 \]

Step 3: Substitute \(r = \sqrt{x^2 + y^2}\) and Simplify Further

Substitute \(r = \sqrt{x^2 + y^2}\) into \(r + 2y = 4\):\[ \sqrt{x^2 + y^2} + 2y = 4 \]Rearrange to isolate the square root term:\[ \sqrt{x^2 + y^2} = 4 - 2y \]

Step 4: Square Both Sides to Eliminate the Square Root

Square both sides of \(\sqrt{x^2 + y^2} = 4 - 2y\):\[ x^2 + y^2 = (4 - 2y)^2 \]Expand the right side:\[ x^2 + y^2 = 16 - 16y + 4y^2 \]

Step 5: Rearrange to Form a Rectangular Equation

Bring all terms to one side of the equation to maintain the equation's form:\[ x^2 + y^2 - 4y^2 + 16y - 16 = 0 \]Simplify the equation:\[ x^2 - 3y^2 + 16y - 16 = 0 \]

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to determine a position in a plane using two numbers. These numbers are \(x\) and \(y\), where \(x\) represents the horizontal distance from the origin, and \(y\) represents the vertical distance from the origin.

This coordinate system is based on a grid with perpendicular axes, allowing for easy calculation of distances and angles. Here are the main points to understand about rectangular coordinates:

• The plane is divided into four quadrants by the x-axis (horizontal) and the y-axis (vertical).
• Coordinates are written as \( (x, y) \).
• The origin is the point \( (0, 0) \), where the x-axis and y-axis intersect.

Understanding rectangular coordinates is essential in dealing with various mathematical and scientific problems since it provides a simple but powerful way to express and transform points in space.

Polar Coordinates

Polar coordinates offer a different system for expressing points in a plane, defined by a radius and an angle. Instead of using \(x\) and \(y\) like the rectangular system, polar coordinates use \(r\) and \(\theta\). These represent the distance from the origin and the angle from a reference direction, respectively.

In the polar system:

• The radius \(r\) indicates how far the point is from the origin.
• The angle \(\theta\) is measured in radians or degrees, usually from the positive x-axis.
• Coordinates are described as \( (r, \theta) \).

Polar coordinates are particularly useful in scenarios involving circular motion or periodic functions, as they align naturally with the properties of circles and waves.

Conversion Formulas

The conversion between polar and rectangular coordinates allows us to switch between these two systems of describing locations in a plane. This involves using specific formulas to translate a point from one coordinate system to the other.

To convert from polar coordinates \( (r, \theta) \) to rectangular coordinates \( (x, y) \), use the following formulas:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r^2 = x^2 + y^2 \)

These formulas let you break down any problem involving different equations or coordinate systems into more manageable parts by converting them into a familiar system.

Understanding these conversions is crucial when dealing with problems requiring switches between the coordinate systems, as seen in tasks like converting polar equations to rectangular equations and vice versa.