Problem 1 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The curve is symmetric about the polar axis and is a limacon with an inner loop.

Step 1: Identify Symmetry with Respect to the Polar Axis

To check for symmetry about the polar axis (the horizontal axis when converted to Cartesian coordinates), we replace \( \theta \) with \( -\theta \) and see if the equation remains unchanged:\[r = 1 + \cos(-\theta) = 1 + \cos \theta\]The equation remains the same, indicating the curve is symmetric with respect to the polar axis.

Step 2: Convert to Cartesian Coordinates

To better understand the curve, we can convert the polar equation into Cartesian coordinates using the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \). Given \( r = 1 + \cos \theta \), we find:\[x = (1 + \cos \theta) \cos \theta\]\[y = (1 + \cos \theta) \sin \theta\]

Step 3: Identify Symmetry with Respect to the Pole

For symmetry about the pole (origin), replace \( r \) with \( -r \):\[-r = 1 + \cos \theta\]This does not simplify to the original equation, thus there is no symmetry with respect to the origin.

Step 4: Identify Symmetry with Respect to the Line \(\theta = \frac{\pi}{2}\)

For symmetry about the vertical line \(\theta = \frac{\pi}{2}\), replace \( \theta \) with \( \pi - \theta \):\[r = 1 + \cos(\pi - \theta) = 1 - \cos \theta\]This transformation does not result in the original expression, hence the curve lacks symmetry with respect to \(\theta = \frac{\pi}{2}\).

Step 5: Analyze and Sketch the Curve

The identified symmetry is only with respect to the polar axis. Known as a 'limacon', this graph can be sketched as follows: it has a small loop inside, touching the origin (since when \(\theta = \pi\), \(r = 0\)). For \(\theta = 0\), \(r = 2\), meaning it reaches maximum distance along the positive x-axis.

Key Concepts

Curve Symmetry

Identifying symmetry in polar curves helps in sketching and understanding the shape better. Symmetry can simplify graphing by recognizing patterns in the curve. In polar coordinates, curves can have symmetry about various lines or points:

• Polar Axis (Horizontal Axis): Check symmetry by replacing \(\theta\) with \(-\theta\). If the equation remains unchanged, the curve is symmetric about the horizontal axis. For example, the limacon \(r = 1 + \cos \theta\) retains its form, indicating it has polar axis symmetry.
• Pole (Origin): To check for symmetry about the origin, replace \(r\) with \(-r\). If the original curve results, it is symmetric about the pole. In our example, this check doesn’t hold, so there's no symmetry about the origin.
• Vertical Line (\(\theta = \frac{\pi}{2}\)): Replace \(\theta\) with \(\pi - \theta\). Symmetry is confirmed if the curve is unchanged. For the limacon \(r = 1 + \cos \theta\), this does not confirm symmetry either.

Recognizing these symmetries assists not only in drawing these curves but also in solving related mathematical problems more easily.

Limacon

A limacon is a type of polar curve characterized by its unique shape. It has the formula \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\). Depending upon the values of \(a\) and \(b\), different shapes of limacons emerge:

• Inner Loop: If \(|b| > |a|\), the curve will have an inner loop.
• Cardioid Shape: When \(a = b\), the curve looks like a heart, known as a cardioid.
• Dimpled Limacon: If \(|a| > |b|\) but not too much larger, the curve will have a dimple.
• Convex Limacon: When \(|a| \geq |b|\), it appears as simple convex.

For the equation \(r = 1 + \cos \theta\), it creates a limacon with a small loop. At \(\theta = 0\), it reaches 2 (since \(r = 2\)), and at \(\theta = \pi\), the inner loop touches the origin. This structural understanding helps visualize and sketch polar curves effectively on the plane.

Conversion to Cartesian Coordinates

Converting polar coordinates to Cartesian coordinates is crucial for graphing polar equations in the familiar \(xy\)-plane. Polar coordinates describe a point by its distance from the origin \(r\) and angle \(\theta\), while Cartesian coordinates express a location via \(x\) and \(y\) values.Here's how to perform the conversion:

• \(x\)-Coordinate: Calculate using \(x = r \cos \theta\).
• \(y\)-Coordinate: Find through \(y = r \sin \theta\).

For the given problem, \(r = 1 + \cos \theta\), we convert to:

• \(x = (1 + \cos \theta) \cos \theta\)
• \(y = (1 + \cos \theta) \sin \theta\)

These equations allow us to plot the points in the Cartesian plane, bridging the gap between polar and rectilinear systems. By adapting polar curves to Cartesian coordinates, you can enjoy a more intuitive understanding of their graphs and symmetries.