Problem 48 - Algebra and Trigonometry

Question

47–50 Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $$\left(x^{2}+y^{2}\right)^{3}=\left(x^{2}-y^{2}\right)^{2}$$

Step-by-Step Solution

Answer

The graph is a rose curve with four petals centered on the axes.

Step 1: Convert to Polar Coordinates

We start by converting the given rectangular equation to polar coordinates. Recall the relationships: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). Therefore, \( x^2 + y^2 = r^2 \) and \( x^2 - y^2 = r^2 \cos(2\theta) \). Substituting these into the equation \( (x^2 + y^2)^3 = (x^2 - y^2)^2 \), we get \( (r^2)^3 = (r^2 \cos(2\theta))^2 \).

Step 2: Simplify the Polar Equation

We simplify the equation \( (r^2)^3 = (r^2 \cos(2\theta))^2 \) to \( r^6 = r^4 \cos^2(2\theta) \). Dividing both sides by \( r^4 \) (assuming \( r eq 0 \)), we find \( r^2 = \cos^2(2\theta) \). The equation in polar coordinates is now \( r = \pm \cos(2\theta) \).

Step 3: Analyze and Sketch the Graph

The polar equation \( r = \pm \cos(2\theta) \) suggests a rose curve with four petals, since the \( \cos(2\theta) \) term has a frequency that suggests repetition every 90 degrees (or \( \frac{\pi}{2} \) radians). Each petal corresponds to the angles where \( \cos(2\theta) \) is \( 1, 0, -1, 0 \) at respective \( \theta \) intervals. Thus, sketch these petals symmetrically centered along the axes, covering the full rotation.

Key Concepts

Rectangular Equation

A rectangular equation is one where we use standard Cartesian coordinates, which are based on the x-axis and y-axis in a plane. In this exercise, the equation is given as \((x^2 + y^2)^3 = (x^2 - y^2)^2\). These types of equations are common in algebra and calculus and allow us to describe geometric shapes and curves.
Rectangular equations are straightforward for plotting points on a standard graph where each point is identified by two numbers: \(x\) and \(y\). The goal is to understand how to transform these equations to different forms for easier analysis or graphing, which will involve converting them into polar coordinates.

Polar Conversion

Polar conversion involves transforming a rectangular equation into polar coordinates. This step is crucial for better understanding certain types of curves that are symmetrical or have repeating patterns.
In a polar coordinate system, any point in the plane is represented by the distance from the origin, \(r\), and the angle from the positive x-axis, \(\theta\).
To convert, use the equations:

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

By substituting these into the given rectangular equation, the equation simplifies to \(r^2 = \cos^2(2\theta)\), showing how polar equations can often reveal the symmetry of a curve more easily than rectangular equations.

Rose Curve

The polar equation \(r = \pm \cos(2\theta)\) describes a rose curve, which is a type of mathematical curve that looks like a flower. This curve is called a "rose" because of its petal-like appearance.
Typically, the number of petals in a rose curve equation like \(r = \cos(n\theta)\) depends on the value of \(n\). If \(n\) is even, the curve will have \(2n\) petals; if \(n\) is odd, it will have \(n\) petals. In our case, since \(n\) is 2, the curve has 4 petals.
This symmetrical pattern arises because the cosine function repeats its values over specific intervals, creating the alternating petal formation at consistent angles.

Graph Sketching

Graph sketching is the process of drawing the graph of an equation based on its mathematical formulation. For this exercise, we have converted the given equation into the polar form and identified it as a rose curve.
We choose key values for \(\theta\) to see how \(r\) changes:

• When \(\theta = 0\), \(\pi/2\), \(\pi\), \(3\pi/2\), the value of \(r\) will be maximum or minimum, providing the tips of the petals.
• These points repeat every \(\frac{\pi}{2}\) radians due to the \(\cos(2\theta)\) term.

By plotting these points and their associated radii, you can draw the curve, ensuring a symmetrical flower-like pattern around the origin. This technique helps visualize complex geometric structures using simpler, intuitive steps.