Problem 54 - Precalculus with Calculus Previews

Question

Find a rectangular equation that has the same graph as the given polar equation. $$ 2 r=\tan \theta $$

Step-by-Step Solution

Answer

The rectangular equation is \(4x^4 + 4x^2y^2 - y^2 = 0\).

Step 1: Understand the Given Polar Equation

The given polar equation is \(2r = \tan\theta\). In polar coordinates, \(r\) is the radius or distance from the origin, and \(\theta\) is the angle.

Step 2: Convert Polar to Rectangular Coordinates

To convert from polar to rectangular coordinates, use the formulas: - \( x = r \cos\theta\) - \( y = r \sin\theta \) - \(\tan\theta = \frac{y}{x}\)

Step 3: Substitute the Polar Formulas

Substitute \(\tan\theta = \frac{y}{x}\) into the equation \(2r = \tan\theta\) to get \(2r = \frac{y}{x}\).

Step 4: Solve for \(r\) in Terms of \(x\) and \(y\)

Multiply both sides by \(x\): \[ 2rx = y \]

Step 5: Use the Pythagorean Identity for \(r\)

In polar coordinates, \(r = \sqrt{x^2 + y^2}\). Substitute this into the equation and simplify: \[ 2x\sqrt{x^2 + y^2} = y \] Now, you have a rectangular equation in terms of \(x\) and \(y\).

Step 6: Simplify the Rectangular Equation

Isolate \(\sqrt{x^2 + y^2}\) by rewriting the equation: \[ 2x \sqrt{x^2 + y^2} = y \] becomes \[ 4x^2(x^2 + y^2) = y^2 \] after squaring both sides to eliminate the square root. Expand and simplify as needed to express in a more standard form if possible: \[ 4x^4 + 4x^2y^2 = y^2 \].

Step 7: Rearrange to Form Final Rectangular Equation

Rearrange the equation: \[ 4x^4 + 4x^2y^2 - y^2 = 0 \]. This is the rectangular equation equivalent to the given polar equation.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two axes (usually labeled as the x-axis and y-axis) to define the position of a point in a plane. These coordinates are incredibly useful in many areas of mathematics and science due to their straightforwardness and the ease with which they can describe geometric figures.

In rectangular coordinates:

• The x-coordinate represents the horizontal distance from the origin (the point where the axes intersect).
• The y-coordinate represents the vertical distance from the origin.
• A point is expressed as \( (x, y) \).

The intersection of the x and y axes forms the origin, denoted as (0,0), serving as a reference point. This system uniquely describes the position of any point in the 2-dimensional space. When converting from polar to rectangular coordinates, it is essential to understand how these axes function because they relate to different components of polar coordinates.

Polar Equations

Polar equations relate to a coordinate system defined by a radius and an angle rather than x and y distances. In the polar coordinate system, a point is represented as \( (r, \theta) \) where:

• \( r \) is the distance from the origin, also known as the radial coordinate.
• \( \theta \) is the angle measured from the positive x-axis, known as the angular coordinate.

Polar equations are particularly useful in scenarios involving circular and rotational symmetry, such as those found in physics or in plotting spirals and roses.

An equation in polar form expresses a relationship between \( r \) and \( \theta \). For example, the given polar equation \( 2r = \tan\theta \) implies a specific relationship between the radial distance of points on a curve and their angles. Understanding polar equations allows mathematicians to analyze and convert them into more familiar rectangular forms using specific transformations.

Coordinate Transformation

Coordinate transformation involves moving from one coordinate system to another while maintaining the geometric representation of the system. It is a key mathematical operation that enhances flexibility in analyzing and interpreting different types of problems. For polar to rectangular conversion, we apply specific transformations.

To convert polar coordinates \( (r, \theta) \) to rectangular coordinates \( (x, y) \):

• The x-coordinate is found using \( x = r \cos\theta \).
• The y-coordinate is found using \( y = r \sin\theta \).
• Additionally, \( \tan\theta = \frac{y}{x} \) provides a connection between the angle and the rectangular coordinates.

This set of transformations helps in converting equations as well. For example, when you have the polar equation \(2r = \tan\theta\), you first express it in terms of \(x\) and \(y\) using the known relationships. Solving and manipulating these expressions gives you the equivalent rectangular equation. Understanding coordinate transformation allows you to easily switch between different perspectives of the same problem, offering deeper insights and new methods for solving mathematical challenges.