Problem 12 - College Trigonometry

Question

Graph the following equations. $$ r=\frac{2}{1+\sin (\theta)} $$

Step-by-Step Solution

Answer

The graph of the given equation is a limaçon with a loop, symmetric about the vertical axis.

Step 1: Identify the Polar Equation

The given equation is in the form of a polar equation: \( r = \frac{2}{1+\sin(\theta)} \). This resembles the equation of a conic section, specifically a limaçon or a conic with a directrix, depending on the terms present.

Step 2: Recognize the Conic Section Type

The equation looks like the general form of a conic in polar coordinates \( r = \frac{ed}{1 + e\sin(\theta)} \). Here, \( e = 1 \) and \( d = 2 \), indicating that this is a conic section with \( e = 1 \), interpreting it as a circle or parabola depending on the setup and context.

Step 3: Analyze Symmetry and Features

Since the equation has \( \sin(\theta) \), it is symmetric around the vertical (\( \theta = \frac{\pi}{2} \)) line. For \( \theta = \frac{\pi}{2} \), \( r = 2 \). As \( \theta \) changes from \( 0 \) to \( 2\pi \), \( r \) will change accordingly.

Step 4: Plot Key Points

Starting at \( \theta = 0 \), \( \sin(0) = 0 \) and \( r = 2 \). At \( \theta = \frac{\pi}{2} \), \( \sin\left(\frac{\pi}{2}\right) = 1 \), making \( r = 1 \). At \( \theta = \pi \), \( \sin(\pi) = 0 \), so \( r = 2 \). This pattern repeats, sketching out a closed curve.

Step 5: Sketch the Polar Graph

Using the points from the previous step, plot them in polar coordinates. At each \( \theta \), plot the corresponding \( r \). Connect these points to form a complete curve, which resembles a limaçon with a loop since the entire range confirms an ellipse-like shape due to consistent symmetry.

Key Concepts

Limaçon

A limaçon is a fascinating type of curve that can often be discovered while working with polar coordinates. This curve is usually grouped amongst other conic sections like ellipses and parabolas. However, it has its distinctive charm and peculiarities.

Limaçons come in different shapes, depending on the exact parameters of the equation. They can have an inner loop, appear heart-shaped, or even morph into circles. When you hear **limaçon**, think of the snail-like shape it sometimes exhibits; in fact, limaçon means snail in French!

To identify a limaçon in a polar equation, look for expressions such as:

• \( r = a  ± b \sin(\theta) \)
• \( r = a  ± b \cos(\theta) \)

If the coefficient of \( \sin \) or \( \cos \) decides how the curve wraps around in the plane, varying the values of parameters \(a\) and \(b\) changes the limaçon’s shape dramatically.

A limaçon with an inner loop occurs when \( |b| > |a| \). It becomes dimpled when \( |b|\) is nearly equal to \( |a| \), and simply a circle if \( |a| = |b| \). This example equation, \( r = \frac{2}{1 + \sin(\theta)} \), forms a limaçon, and because the terms aim to mimic a conic definition, it bears similarities to these fascinating curves.

Conic Section

Conic sections are crucial geometric shapes resulting from the intersection of a plane with a cone. The fundamental conic sections include:

• Circle
• Ellipse
• Parabola
• Hyperbola

Each form of conic section is determined by the angle and position of the intersecting plane relative to the cone. In polar coordinates, conics are expressed elegantly. They typically follow the general form:\[ r = \frac{ed}{1 + e\sin(\theta)} \text{ or } r = \frac{ed}{1 + e\cos(\theta)} \]

In these equations:

• \(e\) represents the eccentricity
• \(d\) is a constant indicating the directrix distance factor

The value of \(e\) determines the type of conic:

• \(e = 0\) yields a circle
• \(0 < e < 1\) yields an ellipse
• \(e = 1\) makes a parabola
• \(e > 1\) gives us a hyperbola

In the provided equation, \( r = \frac{2}{1+\sin(\theta)} \), we ascertain that \( e = 1 \), translating into a parabola, signifying that this equation takes the form of a specific conic section. Though in practical graphing, it manifests as a specific kind of limaçon due to the harmonic change of \( \sin \).

Graphing Polar Equations

Graphing polar equations can be an enjoyable venture into a world of curves like spirals, roses, and limaçons. The polar coordinate system uses angles and radii to estimate points in a plane, distinguishing itself from the traditional Cartesian approach.

Here’s how you would graph an equation like \( r = \frac{2}{1+\sin(\theta)} \):

• **Calculate Key Points:** Start with specific angle values of \( \theta \), such as \(0, \frac{\pi}{2}, \pi,\) and \(2\pi\).
• **Evaluate \(r\):** Find the corresponding radius for each angle.
• **Plot on Polar Grid:** Plot these points on a polar grid where each point is given by \((r, \theta)\).

Let’s delve into the given example further. For \( \theta = 0 \), \( r \) evaluates to 2. As \( \theta \) moves to \( \frac{\pi}{2} \), \( r \) decreases to 1. At \( \pi \), \( r \) returns to 2, demonstrating a symmetric pattern about the vertical line \( \theta = \frac{\pi}{2} \).

Connecting these plotted points results in a curve distinct to its parameters. This particular route, due to the sinusoidal element, forms a limaçon. Its shape can be appreciated even more when adjusting parameter values and observing the mesmerizing transformations these polar equations offer.