Problem 43 - Precalculus

Question

Graph equation. \(r^{2}=9 \cos (2 \theta)\)

Step-by-Step Solution

Answer

The graph is a polar curve with petals or loops, symmetric about the polar axis.

Step 1: Identify the Type of Equation

The given equation is in polar form: \[ r^{2} = 9 \cos(2\theta) \]. This is a type of polar equation that resembles the general form for limacon or rose curves, particularly those involving cosine.

Step 2: Simplify the Equation

Rewrite the equation by taking the square root of both sides. Given \( r^{2} = 9\cos(2\theta) \), we find \( r = \pm 3\sqrt{\cos(2\theta)} \). Since \( r \) represents a distance and must be non-negative, we take \( r = 3\sqrt{\cos(2\theta)} \). However, this function is undefined for regions where \( \cos(2\theta) < 0 \) due to the square root.

Step 3: Determine The Domain of \(\theta\)

Identify where \( \cos(2\theta) \geq 0 \). Since cosine is positive in the first and fourth quadrants, apply this to \(2\theta\). This implies \(-\frac{\pi}{4} \leq \theta \leq \frac{\pi}{4}\) or \(\frac{3\pi}{4} \leq \theta \leq \frac{5\pi}{4}\).

Step 4: Plot Key Points and Analyze Symmetry

Calculate key points using the range of \(\theta\). For example, at \(\theta = 0\) and \(\theta = \pi\), evaluate \(r\). Also, since \(\cos(2\theta)\) is an even function, the curve is symmetric about the polar axis.

Step 5: Sketch the Polar Curve

Sketch the polar curve based on the computations and analysis. The equation \(r^2 = 9 \cos(2\theta)\) will result in petals or loops depending on the range and symmetry previously determined.

Key Concepts

Polar Equations

Polar equations are a fascinating way of representing curves in the polar coordinate system, which differs greatly from the Cartesian coordinate system that uses \(x\)- and \(y\)-axes. Instead, polar equations rely on two components: \(r\) (the radius or distance from the origin) and \(\theta\) (the angle from the positive \(x\)-axis). This system is especially useful for circles and spirals.

• "\(r\)" is the length from the pole (origin) to the curve.
• "\(\theta\)" is generally measured in radians and represents the direction from the pole.

In the given exercise, the equation \(r^{2} = 9 \cos (2 \theta)\) is a polar equation, specifically dealing with trigonometric functions. Such equations often lead to intriguing shapes like circles, spirals, and petal-like patterns in the polar coordinate system. Polar equations come in various forms, and understanding their classifications like limacons and rose curves helps in predicting the graph's shape.

Limacon

A limacon is a type of polar curve that is generated from equations in the form \(r = a + b \cos(\theta)\) or \(r = a + b \sin(\theta)\). These curves are visually recognized by their distinctive loops or dimple-like structures, depending on the coefficients' values.For example:

• If \(|a| < |b|\), the limacon has an inner loop.
• If \(|a| = |b|\), the limacon touches the pole and forms a cardioid.
• If \(|a| > |b|\), the limacon appears dimpled or without an inner loop.

In the context of the exercise, when dealing with an equation like \(r^2 = 9 \cos(2\theta)\), it's polynomial nature in \(\theta\) can often lead to limacon-like structures when the equation is manipulated. Limacons are important for studying the variability in polar plots due to their changeable symmetry and shapes.

Rose Curves

Rose curves are beautiful plots resulting from equations of the form \(r = a \cos(n\theta)\) or \(r = a \sin(n\theta)\). Depending on \(n\), the curve will sport multiple petal-like structures, which encapsulate the name "rose curves."Key aspects of rose curves include:

• "\(n\)" determines the number of petals.
• If \(n\) is even, the curve will have \(2n\) petals.
• If \(n\) is odd, the curve will boast \(n\) petals.

While the given equation \(r^2 = 9 \cos(2\theta)\) is not typically a simple rose curve, it indicates periodicity similar to rose-like patterns. Recognizing these petal formations helps in graphically portraying the symmetrical beauty of polar plots, showing frequent and even repeats around the pole.

Symmetry in Polar Graphs

Symmetry plays a vital role in polar graphs, making it easier to sketch and understand them. In polar graphs, symmetry commonly occurs across the polar axis, the line \(\theta = \frac{\pi}{2}\), or the pole.Types of symmetry:

• Symmetry about the polar axis (like the \(x\)-axis in Cartesian coordinates).
• Symmetry about the line \(\theta = \frac{\pi}{2}\) (like the \(y\)-axis in Cartesian coordinates).
• Symmetry about the pole, which occurs when the equation holds for opposite points.

The exercise reveals that our equation \(r^2 = 9 \cos(2\theta)\) showcases symmetry about the polar axis due to the even nature of the cosine function. This symmetry is crucial for ensuring the graph appears balanced and helps in predicting uncomputed points on the plot.