Problem 53 - Algebra and Trigonometry

Question

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $$ \left(x^{2}+y^{2}\right)^{3}=4 x^{2} y^{2} $$

Step-by-Step Solution

Answer

Convert to polar form: \( r = |\sin(2\theta)| \). Sketch a four-leaf rose.

Step 1: Express as Polar Coordinates

In polar coordinates, we use the relationships \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). Therefore, \( x^2 + y^2 = r^2 \). Substituting these into the equation, we get \( (r^2)^3 = 4 (r \cos(\theta))^2 (r \sin(\theta))^2 \). Simplify this to obtain \( r^6 = 4r^4 \cos^2(\theta) \sin^2(\theta) \).

Step 2: Simplify the Polar Equation

Divide both sides by \( r^4 \) (assuming \( r eq 0 \)) to simplify. The equation becomes \( r^2 = 4\cos^2(\theta)\sin^2(\theta) \). Using the double angle identity for sine, \( \, \sin(2\theta) = 2\sin(\theta)\cos(\theta) \), rewrite \( \cos^2(\theta)\sin^2(\theta) \) as \( \left(\frac{\sin(2\theta)}{2}\right)^2 \). Thus, \( r^2 = \left(\frac{\sin(2\theta)}{2}\right)^2 \times 4 \).

Step 3: Further Simplify and Solve for "r"

So, \( r^2 = \sin^2(2\theta) \). Taking the square root, we get \( r = \left|\sin(2\theta)\right| \). This tells us that the radius \( r \) is the absolute value of \( \sin(2\theta) \), describing a limacon type shape without an inner loop.

Step 4: Sketch the Graph

To sketch the graph: note that when \( \sin(2\theta) = 0 \), \( r = 0 \), which occurs at every \( \pi/2 \) increment starting from \( 0 \). The maximum value of \( r \) will be \( 1 \). Plot points for values of \( \theta \) from \( 0 \) to \( \pi \) multiples to see how \( r \) behaves, which results in a four-leaf rose shape.

Key Concepts

Rectangular Equation

Rectangular equations are typically expressed using Cartesian coordinates, represented by \( x \) and \( y \). In this context, turning a rectangular equation into a polar one involves substituting these variables using polar relationships. For example, the equation \((x^2+y^2)^3=4x^2y^2\) can be converted by substituting \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). This transformation is crucial for simplifying complex equations by leveraging the uniformity of polar coordinates, particularly when dealing with curves or graphs that naturally complement circular symmetry.

Graph Sketching

Graph sketching involves understanding the behavior and shape of a mathematical function visually. It's a powerful tool in recognizing patterns and getting insights into mathematical relationships. When exploring polar equations, the behavior of radius \( r \) as a function of angle \( \theta \) gives important clues.

• Look for key values where the function resolves to zero, as these typically indicate intercepts on the physical graph.
• Consider the maximum values to determine the outermost reach of the curve.
• Symmetry checks can reduce workload; many polar graphs are symmetric about certain lines.

This specific case's graph, expressed by \( r = |\sin(2\theta)| \), informs us that a four-leaf rose shape will appear, repeating the pattern at intervals of \( \pi/2 \).

Limacon

A limacon is a type of polar graph characterized by its unique heart shape. However, limacons can exhibit various structures including inner loops or dimples, which are influenced by the parameters in their equations.
Our equation simplifies to \( r = |\sin(2\theta)| \), showing a limacon without inner loops, similar to a four-leaf rose. The absence of an inner loop results from how the sine function affects the radius. The limacon's shape often varies, allowing us to see mathematical beauty through diverse geometrical forms. By knowing it’s a limacon, we predict symmetric patterns which simplify the plotting process. It's important to note how closely related they are to cardioids, which are special types of limacons.

Double Angle Identity

The double angle identities provide essential tools in trigonometry. For this problem, the identity \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \) was pivotal. Applying it, we transformed the product of squares \((\cos^2(\theta)\sin^2(\theta))\) into something more manageable.

• These identities aid in simplifying trigonometric equations, allowing for easier graphing and analysis.
• They resolve complex functions into recognizable segments, thus providing clarity on otherwise complicated expressions.

In this problem, using the double angle identity allowed further simplification to the core equation \( r^2 = \sin^2(2\theta) \), making the final shape and behavior far more apparent for graphing purposes.