Problem 16 - The Calculus with Analytic Geometry

Question

The graph of the given equation intersects itself. Find the points at which this occurs.\(r=1+2 \cos 2 \theta\)

Step-by-Step Solution

Answer

The points of self-intersection are solved by \[ \theta_1 = n \pi/2 + (-1)^n \theta_2 \]

Step 1 - Understand the given polar equation

The given polar equation is: \(r = 1 + 2 \cos 2 \theta\). This equation needs to be solved to find the points where the graph intersects itself.

Step 2 - Set up the conditions for self-intersection

For self-intersection, two pairs \(r, \theta\) and \(\tilde{r}, \tilde{\theta}\) must satisfy the same polar equation. At the same time, they are different points on the graph. This implies: \( r_1 = 1 + 2 \cos 2 \theta_1\) and \( r_2 = 1 + 2 \cos 2 \theta_2\)

Step 3 - Equate the two radii to find condition for intersection

Set \(r_1 = r_2\), to find the condition for the radius: \[1 + 2 \cos 2 \theta_1 = 1 + 2 \cos 2 \theta_2 \Rightarrow \cos 2 \theta_1 = \cos 2 \theta_2\]

Step 4 - Solve for \theta_1 and \theta_2

The equation \ \cos 2 \theta_1 = \cos 2 \theta_2\ \Rightarrow \theta_1 = n\pi + (-1)^n \theta_2 \(for \ n \in Z\), can be converted back to angles to find the possible self-intersection values. Choosing the principal value: \2 \theta_1 = 2\pi n \pm 2 \theta_2\. Thus, \theta_1 = n\pi/2 + (-1)^n \theta_2\.

Key Concepts

Self-Intersection Points

In polar coordinates, self-intersection points occur where the graph of the polar equation meets itself at two or more different values of the angle, \theta\, and radius, \r\. For the given polar equation \( r = 1 + 2 \cos 2\theta \), self-intersections are found when different pairs \((r_1, \theta_1\)) and \((r_2, \theta_2\)) both satisfy the equation. This means setting the conditions so that the radius remains the same, but the angles are different. The goal is to find various \theta-values where this condition holds true to identify those unique points on the graph where it intersects itself.

Trigonometric Identities

In solving polar equations like \( r = 1 + 2 \cos 2\theta \), we often leverage trigonometric identities to simplify the solutions. A very common trigonometric identity used here is: \( \cos 2\theta = \cos 2\theta \). For self-intersections, this can be transformed to show that \( \theta_1 = n\pi + (-1)^n \theta_2 \), where \( n \) is an integer. This reveals the symmetrical properties and periodic nature of the trigonometric functions, helping identify the potential angles where self-intersections could occur.

Solving Polar Equations

Solving polar equations involves finding solutions for both \r\ and \theta\. The given equation, \( r = 1 + 2 \cos 2\theta \), can be solved by equating two radius expressions set to be equal:\(1 + 2 \cos 2\theta_1 = 1 + 2 \cos 2\theta_2 \). From this, it simplifies to \( \cos 2\theta_1 = \cos 2\theta_2 \), which means \( \theta_1 = n\pi + (-1)^n \theta_2 \). By examining different values of \( n \) (such as \(n = 0, ±1, ±2\), etc.), we discover multiple angle pairs where the equation holds true, indicating points of intersection.

Graphical Analysis of Functions

Graphical analysis helps in visualizing functions and their behavior, especially for complex polar equations. For the equation \( r = 1 + 2 \cos 2\theta \), plotting this reveals a graph with petals or loops that intersect themselves. By scrutinizing these intersections via angles \( \theta_1 \) and \( \theta_2 \), we can determine exactly where the self-intersections appear. This visual approach not only supports the algebraic solution but also provides an intuitive understanding of the self-intersecting nature of the curve. Using graphing tools or software can greatly aid in this analysis, making it easier for students to grasp the concepts.