Problem 6 - Thomas Calculus

Question

Identify the symmetries of the curves in Exercises \(1-12 .\) Then sketch the curves. $$ r=1+2 \sin \theta $$

Step-by-Step Solution

Answer

The curve is symmetric about the line \(\theta = \frac{\pi}{2}\).

Step 1: Understanding the Equation

The given equation is in polar coordinates and is expressed as \( r = 1 + 2 \sin \theta \). Our goal is to identify any symmetries present in this polar curve. A common approach is to test for symmetry about the polar axis (horizontal), the line \( \theta = \frac{\pi}{2} \) (vertical), and the pole (origin).

Step 2: Test for Polar Axis Symmetry

To test for symmetry about the polar axis, replace \( \theta \) with \( -\theta \) in the equation. If the equation remains unchanged, it has polar axis symmetry.Substituting, we get:\[ r = 1 + 2 \sin(-\theta) \]Since \( \sin(-\theta) = -\sin(\theta) \), the equation becomes:\[ r = 1 - 2 \sin \theta \]This is different from the original equation, so there is no polar axis symmetry.

Step 3: Test for Line Symmetry (\(\theta = \frac{\pi}{2}\))

To test symmetry about the line \( \theta = \frac{\pi}{2} \), replace \( \theta \) with \( \pi - \theta \). If the equation remains unchanged, it has this symmetry.Substituting, we get:\[ r = 1 + 2 \sin(\pi - \theta) \]Since \( \sin(\pi - \theta) = \sin(\theta) \), the equation simplifies back to the original:\[ r = 1 + 2 \sin \theta \]The equation remains unchanged, so it is symmetric about the line \( \theta = \frac{\pi}{2} \).

Step 4: Test for Pole Symmetry

To test for symmetry about the pole (origin), replace \( r \) with \(-r\). This will generate the equation:\[ -r = 1 + 2 \sin \theta \]This simplifies to:\[ r = -1 - 2 \sin \theta \]This is different from the original equation, indicating there is no symmetry about the pole.

Step 5: Conclusion on Symmetry

The curve \( r = 1 + 2 \sin \theta \) is symmetric about the line \( \theta = \frac{\pi}{2} \). There is no symmetry about the polar axis or the pole.

Step 6: Sketching the Curve

Now that we know the symmetry, we sketch the curve. The equation \( r = 1 + 2 \sin \theta \) represents a limaçon with an inner loop. The presence of \( 2\sin\theta \) indicates a peculiar shape influenced by \( \theta \). Use the identified symmetry to ensure the sketch is accurate across the specified line \( \theta = \frac{\pi}{2} \).

Key Concepts

Symmetry in Polar Curves

Symmetry in polar curves helps us understand the shape and behavior of these curves in the polar coordinate system. There are three main types of symmetry you may encounter:

• Polar Axis Symmetry: If replacing \(\theta\) with \(-\theta\) leaves the equation unchanged, the curve has symmetry about the polar axis (often visualized as the x-axis in Cartesian coordinates).
• Symmetry About the Line \(\theta = \frac{\pi}{2}\): Replace \(\theta\) with \(\pi - \theta\). If the equation remains the same, the curve is symmetric about the vertical line \(\theta = \frac{\pi}{2}\).
• Symmetry About the Pole: Replace \(r\) with \(-r\). If you still get the original equation, the curve is symmetric about the origin.

These symmetries allow for easier graphing and deeper understanding of the curve's behavior. In our exercise, the curve \( r = 1 + 2 \sin \theta \) exhibits symmetry about the line \( \theta = \frac{\pi}{2} \). This means you only need to sketch the curve for half the range of \( \theta \), and mirror it to complete your sketch.

Polar Curve Sketching

Sketching polar curves involves plotting points and understanding the impact of symmetries and shapes given by the polar equation. Here's a straightforward approach:

• Identify Key Points: Start by plugging in angles that make calculations simple, such as \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \( 2\pi \).
• Use Symmetries: Determine any symmetries and use them to reduce the amount of sketching needed, as it allows you to reflect parts of the curve over axes or lines.
• Identify Curve Types: Recognize patterns, such as circles, roses, or limaçons, to anticipate general shapes.
• Plot Points and Connect: Plot calculated points and join them smoothly, paying attention to changes in direction or magnitude.

For the limaçon \( r = 1 + 2 \sin \theta \), it’s helpful to note the points and leverage the symmetry about \( \theta = \frac{\pi}{2} \) to accurately sketch the curve, resulting in an inner loop characteristic of some limaçons.

Limaçon Curves

Limaçon curves are a fascinating class of polar curves, characterized by their heart-shaped appearance, often with loops. They come from equations of the form \( r = a + b \sin(\theta) \) or \( r = a + b \cos(\theta) \), where \(a\) and \(b\) are constants.

• Key Characteristics: Depending on the values of \(a\) and \(b\), limaçons can have no loop, one inner loop, or appear similar to a cardioid (a special type of limaçon).
• Relationship to Symmetry: These curves may show symmetry about the line \(\theta = \frac{\pi}{2} \) or the polar axis, which helps in sketching.
• Understanding Shape Variations: When \( |b| > |a| \), a limaçon will exhibit an inner loop, as in our equation \( r = 1 + 2 \sin \theta \). This happens because the varying \( \theta \) causes \( r \) to switch between positive and negative rather elegantly.

In practice, understanding limaçons lets you recognize unique properties of these polar equations, making them easier to work with and sketch accurately.