Problem 37 - Algebra and Trigonometry

Question

Sketch a graph of the polar equation. $$ r^{2}=\cos 2 \theta $$

Step-by-Step Solution

Answer

It's a 4-petal rose graph, symmetric to both axes.

Step 1: Understand Polar Coordinates and Equation

The given equation is in polar form, expressed as \( r^2 = \cos(2\theta) \). In polar coordinates, each point on the plane is determined by a distance \( r \) from the origin and an angle \( \theta \) from the positive x-axis.

Step 2: Identify Symmetry

Since the equation has \( \cos(2\theta) \), it is symmetric about the polar axis (x-axis), the line \( \theta = \pi/2 \), and the origin. This is due to the properties of cosine and the fact that the equation involves \(2\theta\).

Step 3: Analyze Zero and Negative Regions of Cosine

Cosine values range from -1 to 1. When \( \cos(2\theta) \leq 0 \), \( r^2 \leq 0 \) implies \( r \) is non-real (since \( r^2 \) must be non-negative), thus those \( \theta \) values contribute no points on the graph. These zero points primarily occur when \( 2\theta = \pi/2 + n\pi \) for odd integers \( n \).

Step 4: Calculate Significant Points

Determine points by evaluating \( r^2 \) for select \( \theta \) values. At \( \theta = 0 \), \( r^2 = \cos(0) = 1 \), so \( r = \pm1 \). At \( \theta = \pi/4 \), \( r^2 = \cos(\pi/2) = 0 \), so \( r = 0 \). This pattern repeats every \( \pi/2 \).

Step 5: Construct Petals for 0 to \( \pi \) Range

The pattern of \( r \) expanding and then shrinking forms 'petals'. From 0 to \( \pi/4 \), \( r \) moves from maximum to zero forming one half-petal, and from \( 3\pi/4 \) to \( \pi \), the pattern repeats in opposite direction. Repeat this analysis for negative values of \( \theta \).

Step 6: Sketch the Complete Polar Graph

By combining all information, we find the graph has petals centred at the origin with points at maximum \( r = 1 \) along the x-axis, zero magnitude at angles such as \( \pi/4 \). There are four petals total oriented symmetrically to axes due to the symmetry of \( \cos(2\theta) \).

Key Concepts

Polar Equation Graphing

Graphing polar equations can seem daunting at first, especially if you're more familiar with Cartesian coordinates. However, with a bit of understanding, it can become rather intuitive. The polar equation given here is \( r^2 = \cos(2\theta) \). In polar coordinates, each point is defined by the radial distance \( r \) from the origin and an angle \( \theta \) from the positive x-axis.
For this equation, the graph is formed based on the behavior of \( r \) as \( \theta \) varies. The expression for \( r^2 \) indicates points' distance from the origin:

• When \( \cos(2\theta) > 0 \), \( r^2 > 0 \), meaning \( r \) is real, and points are plotted accordingly on the graph.
• If \( \cos(2\theta) = 0 \), \( r^2 = 0 \), so \( r = 0 \) indicating critical points where the graph touches the origin.
• For \( \cos(2\theta) < 0 \), \( r^2 \) would be negative, suggesting points do not exist because \( r^2 \) must be non-negative in real scenarios.

By sketching this polar equation, it reveals a distinctive symmetrical pattern, known as a rose curve, due to the trigonometric influence of the cosine function.

Symmetry in Polar Graphs

Symmetry can greatly simplify the task of plotting polar graphs. In our polar equation \( r^2 = \cos(2\theta) \), understanding its symmetry can save time and improve accuracy. This particular graph exhibits multiple symmetry properties:

• Symmetry about the polar axis (x-axis): The cosine function itself is symmetric about the x-axis, so the entire graph will reflect this quality.
• Symmetry about the line \( \theta = \pi/2 \): The property of \( \cos(2\theta) \) allows the graph to be symmetric to the vertical line, dividing the graph into mirror images.
• Symmetry about the origin: For polar equations, general symmetry about the origin means if \((r, \theta)\) is on the graph, \((-r, \theta)\) is also on the graph.

Recognizing these symmetries not only aids in efficient graphing but also helps in identifying repetitive patterns, leading to a better grasp of how the graph develops as \( \theta \) changes.

Cosine Function in Polar Coordinates

The cosine function plays an integral role in shaping polar equations. In the context of \( r^2 = \cos(2\theta) \), it dictates how the graph behaves across different values of \( \theta \).
Here's how the cosine function contributes:

• Amplitude and frequency: The number \( 2 \) in \( \cos(2\theta) \) affects the frequency of the graph's pattern. In polar graphs like this, the doubling of the angle \( \theta \) results in four petals within one complete revolution (from 0 to \( 2\pi \)).
• Significance of cosine values: Cosine provides a wave-like pattern that repeats every \( 2\pi \). This periodicity is crucial for determining when \( r^2 \) becomes zero, thereby forming identifiable points like petals and their alignment.

When graphing such equations, understanding the behavior and influence of the cosine function helps predict and verify the structure of the graph visually.

Analyzing Zero Regions

Understanding where the polar graph reaches zero points is essential for plotting equations accurately. For \( r^2 = \cos(2\theta) \), zero regions occur at angles where the cosine term becomes zero or negative.
These points can be directly analyzed by determining angle values fulfilling \( \cos(2\theta) = 0 \):

• These zeros primarily occur at \( 2\theta = \frac{(2n + 1)\pi}{2} \) where \( n \) is any integer. For \( r \), it means no real solutions emerge, contributing no points to the graph when mapped at such angles.
• Since \( r^2 \) must be non-negative, negative cosine values contribute to the graph's invisible segments, seen as gaps or non-existent sections.

Mastering this analysis ensures a coherent drawing of the polar graph, as it correctly displays where the graph dips to zero, illustrating points of intersection and complete petal formations due to these zero regions.