Problem 6 - Thomas Calculus

Question

Identify the symmetries of the curves. Then sketch the curves in the \(x y\) -plane. \(r=1+2 \sin \theta\)

Step-by-Step Solution

Answer

The curve \(r = 1 + 2 \sin \theta\) is symmetric about the line \(\theta = \frac{\pi}{2}\) and is a limaçon with an inner loop.

Step 1: Identify Symmetries

To determine the symmetries of the polar curve, we look for symmetries with respect to the polar axis (x-axis), the line \(\theta=\frac{\pi}{2}\) (y-axis), or the pole (origin). For the given equation \( r = 1 + 2 \sin \theta \), we check:1. **Symmetry with respect to the polar axis**: Replace \(\theta\) with \(-\theta\). If the equation remains the same, the graph is symmetric. \[ r = 1 + 2 \sin(-\theta) = 1 - 2 \sin(\theta) \] The equation changes, so it is not symmetric with respect to the polar axis.2. **Symmetry with respect to the line \(\theta=\frac{\pi}{2}\)**: Replace \(\theta\) with \(\pi - \theta\). If the equation remains the same, the graph is symmetric. \[ r = 1 + 2 \sin(\pi - \theta) = 1 + 2 \sin \theta \] The equation remains the same, indicating symmetry with respect to \(\theta = \frac{\pi}{2}\).3. **Symmetry with respect to the origin**: Replace \(r\) with \(-r\) and \(\theta\) with \(\theta + \pi\). \[ -r = 1 + 2 \sin(\theta + \pi) = 1 - 2 \sin \theta \] The equation is not the same, so it is not symmetric with respect to the origin.

Step 2: Sketch the Curve using Polar Equations

To sketch the polar curve \(r = 1 + 2 \sin \theta\), it is useful to understand the general shape of similar equations. The form \(r = a + b \sin \theta\) often corresponds to a limaçon.1. **Plot Key Angles**: - At \(\theta = 0\), \(r = 1 + 2 \cdot 0 = 1\) - At \(\theta = \frac{\pi}{2}\), \(r = 1 + 2 \cdot 1 = 3\) - At \(\theta = \pi\), \(r = 1 + 2 \cdot 0 = 1\) - At \(\theta = \frac{3\pi}{2}\), \(r = 1 + 2 \cdot (-1) = -1\) (negative \(r\) indicates that the point is in the opposite direction)2. **Connection Points**: - The highest value of \(r\) occurs at \(\theta = \frac{\pi}{2}\), making the graph extend up to \(r=3\) at the top. - The lowest value at \(\theta = \frac{3\pi}{2}\) suggests the graph loops inward past the origin. - The shape is typically a "limaçon with an inner loop" due to \(b > a\).3. **Draw the Curve**: - Start from the smallest angle (\(\theta = 0\)) and reach the highest point at \(\theta = \frac{\pi}{2}\), loop back to start as \(\theta o\pi\), and continue to \(\frac{3\pi}{2}\) with the negative radius indicating an inward loop.

Key Concepts

Symmetry in Polar Graphs

In polar coordinate systems, symmetry plays a crucial role in understanding and sketching the behavior of polar graphs. Recognizing symmetry can simplify graphing tremendously, especially when certain symmetries mean that you only need to compute values for a portion of the angle range.
For the polar equation \( r = 1 + 2 \sin \theta \), determining symmetry is essential. Here are some checks for symmetry in polar graphs:

• **Polar axis symmetry (x-axis)**: Replace \( \theta \) with \( -\theta \). If the equation remains unchanged, the graph is symmetric. For our equation, this test failed, as \( 1 + 2 \sin(-\theta) eq 1 + 2 \sin(\theta) \).
• **Y-axis symmetry (line \( \theta=\frac{\pi}{2} \))**: Substitute \( \theta \) with \( \pi-\theta \). Our equation stays the same (\( 1 + 2 \sin(\pi-\theta) = 1 + 2 \sin(\theta) \)), revealing symmetry about this line.
• **Origin symmetry**: Test by replacing \( r \) with \( -r \) and \( \theta \) with \( \theta + \pi \). This does not apply here as the equation does not remain valid when these substitutions are made.

Recognizing when a graph has symmetry makes it easier to predict its overall shape and to simplify the plotting process. This is because symmetries replicate the graph in defined portions, reducing the workload for sketching the complete graph.

Limaçon Curves

Limaçon curves are a fascinating family of curves that are defined by equations of the form \( r = a + b \sin \theta \) or \( r = a + b \cos \theta \). These curves, which can have a variety of shapes, are particularly known for their "snail-like" appearance.
The nature of the limaçon curve largely depends on the relationship between \( a \) and \( b \). Depending on the values:

• If \( |b| > |a| \), the limaçon has an "inner loop." This applies to our equation \( r = 1 + 2 \sin \theta \) where \( b = 2 \) and \( a = 1 \), resulting in \( b > a \), so expect a loop inside the curve.
• If \( |b| < |a| \), the curve still resembles a limaçon but without a loop.
• If \( |b| = |a| \), the result is a cardioid, which is a special form of the limaçon curve.

Understanding the type of limaçon you're dealing with helps in predicting its shape and in confirming the features observed in the graph once plotted. The inner loop feature in our case means that part of the curve will double back and cross through the origin, adding complexity and beauty to its form.

Polar Sketching Techniques

Drawing polar curves can be an artistic and analytical activity, enhanced by recognizing patterns and using specific techniques. Getting started requires attention to characteristic points and an understanding of the curve type. Here's how you can approach sketching a polar curve like \( r = 1 + 2 \sin \theta \):
**1. Identify Key Angles and Radii**
Begin with a few strategic angles:

• At \( \theta = 0 \), calculate \( r = 1 \).
• At \( \theta = \frac{\pi}{2} \), find \( r = 3 \), this will be the maximum radius as \( \sin \theta \) achieves its peak value of 1.
• At \( \theta = \pi \), \( r \) returns to 1.
• At \( \theta = \frac{3\pi}{2} \), \( r = -1 \), indicating the point is reflected across the origin.

**2. Connect the Points**
Link these calculated points smoothly, mindful of both the gradual change in \( r \) and directional changes.
**3. Analyze The Shape**
Recognize features like loops: our polar equation forms a limaçon with an inner loop. Such shapes comprise repeated sections or curves leading back to the core.
Visualizing and sketching polar curves evolve not just from computation but also from visual interpretation and understanding curve symmetries.