Problem 9 - Thomas Calculus

Question

Identify the symmetries of the curves in Exercises \(1-12 .\) Then sketch the curves. $$ r^{2}=\cos \theta $$

Step-by-Step Solution

Answer

The curve is symmetric with respect to the polar axis and the pole.

Step 1: Check for Symmetry with Respect to the Polar Axis

For symmetry with respect to the polar axis (the horizontal axis), replace \(\theta\) with \(-\theta\) in the equation and simplify. The equation becomes \(r^2 = \cos(-\theta)\). Since \(\cos(-\theta) = \cos \theta\), the equation remains unchanged. Hence, the curve has symmetry with respect to the polar axis.

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

For this symmetry, replace \(r\) with \(-r\) and \(\theta\) with \(\pi - \theta\) in the equation: \((-r)^2 = \cos(\pi - \theta)\). Since \(\cos(\pi - \theta) = -\cos \theta\), the equation \(r^2 = -\cos \theta\) does not match the original. Therefore, there is no symmetry with respect to the line \(\theta = \frac{\pi}{2}\).

Step 3: Check for Symmetry with Respect to the Pole

For symmetry with respect to the pole (origin), replace \(r\) with \(-r\). The equation becomes \((-r)^2 = \cos \theta\), which simplifies to \(r^2 = \cos \theta\). The equation is unchanged, indicating symmetry with respect to the pole.

Step 4: Sketch the Curve

Given the symmetries identified, sketch the curve by considering points that satisfy the equation \(r^2 = \cos \theta\). Convert typical angles like \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\) to Cartesian coordinates for plotting. The behavior of \(r\) at these angles helps to outline a lemniscate-shaped curve, symmetric about the pole and the polar axis.

Key Concepts

Symmetry in Polar Graphs

Polar graphs have unique symmetries that help to simplify sketching and analyzing their curves. Identifying symmetries in polar graphs involves checking reflections across specific lines or points.

• Polar Axis Symmetry: This is the symmetry with respect to the polar axis, analogous to the x-axis in Cartesian coordinates. For a curve to have this symmetry, replacing \( \theta \) with \(-\theta\) in its equation should yield an equation identical to the original.
• Symmetry about the Line \( \theta = \frac{\pi}{2} \): Also known as y-axis symmetry, this requires substituting \( r \) with \(-r\) and \( \theta \) with \( \pi - \theta \).
• Symmetry about the Pole: This form of symmetry needs \( r \) to be swapped with \(-r \), and if the equation remains constant, the curve is symmetrical around the pole.

Understanding the symmetry helps in predicting how the graph will look, ensuring that fewer calculations are needed for accurate sketching.

Lemniscate Curves

Lemniscates are intriguing and visually striking curves that often appear in polar coordinates. They are defined by equations that typically involve squares of sine or cosine functions. In the equation \( r^2 = \cos \theta \), we see an example of a lemniscate curve.

• Double-Loop Shape: Lemniscates often take on a figure-eight or infinity-like loop shape. This occurs because the square relation between \( r \) and the trigonometric function allows for negative and positive values of \( \theta \) to satisfy both loops.
• Symmetry Characteristics: Many lemniscates exhibit complete symmetry about both the pole and one of the axes, making them straightforward to analyze despite their complex appearance.
• Applications: These curves are not just pretty; they feature in various fields, including physics problems dealing with electric fields or orbits.

Lemniscates are a perfect illustration of the beauty and complexity of polar geometry, providing a challenge and delight for students and mathematicians alike.

Polar Axis Symmetry

The polar axis in polar coordinates is similar to the x-axis in Cartesian coordinates. Checking for symmetry around this axis can simplify graphing polar equations like \( r^2 = \cos \theta \).
To verify polar axis symmetry, the process involves replacing \( \theta \) with \(-\theta \) in the equation:

• For the given curve \( r^2 = \cos \theta \), substituting \(-\theta \) results in the equation \( r^2 = \cos(-\theta) \).Since \( \cos(-\theta) = \cos \theta \), the equation remains unchanged, confirming symmetry with respect to the polar axis.
• This means the curve mirrors itself across the polar axis, simplifying the sketching process significantly.

By recognizing this symmetry early, we can predict and draw the graph's expansive loops with more precision and confidence.

Graphing Techniques in Polar Coordinates

Graphing in polar coordinates can initially be challenging, but some helpful techniques can make it easier. Polar graphs use coordinates based on a fixed point, the pole, and an angle \( \theta \).

• Converting Angles: Use standard angles such as \( 0, \frac{\pi}{2}, \pi, \frac{3\\pi}{2} \) to find key positions on the polar graph. Calculating the corresponding radius \( r \) at these angles gives critical points for plotting.
• Symmetrical Properties: Utilize any symmetry the curve may have to reduce the number of points you need to calculate. For example, knowing that the lemniscate is symmetric about specific axes can halve your workload in determining the curve's shape.
• Incremental Plotting: Create a table of values for \( \theta \) and \( r \) to map out specific points. This helps in drawing a more accurate curve.

By mastering these techniques, students can skillfully create accurate polar graphs, turning challenging plots into manageable tasks.