Problem 30 - Precalculus: Functions and Graphs

Question

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x^{2}-y^{2}=9$$

Step-by-Step Solution

Answer

The polar equation is \( r^2 = \frac{9}{\cos 2\theta} \).

Step 1: Convert to Polar Coordinates

In polar coordinates, the relationships between \(x, y\) and \(r, \theta\) are given by: \(x = r \cos \theta\) and \(y = r \sin \theta\). Replace \(x\) and \(y\) in the given equation with their polar expressions: \(x^2 = (r \cos \theta)^2\) and \(y^2 = (r \sin \theta)^2\). This transforms the equation into \( (r \cos \theta)^2 - (r \sin \theta)^2 = 9\).

Step 2: Simplify the Expression

Factor \(r^2\) out of the expression: \( r^2 (\cos^2 \theta - \sin^2 \theta) = 9\). Recognize the trigonometric identity for cosine of double angle: \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\), then substitute it in: \( r^2 \cos 2\theta = 9\).

Step 3: Solve for the Polar Equation

Solve for \(r^2\) by dividing each side of the equation by \ (\cos 2\theta) \: \( r^2 = \frac{9}{\cos 2\theta}\). The polar equation equivalent to the original Cartesian equation is thus \( r^2 = \frac{9}{\cos 2\theta}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable where the functions are defined. These identities are invaluable tools in mathematics, particularly when solving complex equations. In our exercise, the identity \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \) plays a pivotal role. This specific identity, known as the double-angle identity for cosine, simplifies expressions involving trigonometric functions by reducing the power from a sum of squares to a simpler form. It's crucial to master these identities as they allow us to transform and solve equations efficiently. Here, it enables the conversion from a Cartesian form to a polar form by simplifying \( r^2 (\cos^2 \theta - \sin^2 \theta) \) to \( r^2 \cos 2\theta \). Understanding these transformations and the relationships between different trigonometric functions simplifies solving a variety of mathematical problems. Remember to always look for opportunities to apply these identities, as they often transform difficult problems into more manageable ones.

Coordinate Conversion

Coordinate conversion between different systems, such as Cartesian (rectangular) and polar, is a foundation of understanding and visualizing different types of equations. In our exercise, we start with the Cartesian equation \( x^2 - y^2 = 9 \) and aim to convert it into a polar expression. The key relationships are \( x = r \cos \theta \) and \( y = r \sin \theta \), which express rectangular coordinates in terms of polar coordinates \( r \) and \( \theta \), where \( r \) is the radius (distance from the origin) and \( \theta \) is the angle from the positive x-axis.

• Begin by substituting \( x \) and \( y \) with their polar forms in the original equation.
• This gives us an expression involving \( r \) and \( \theta \), thus beginning the process of translating into polar format.

This conversion is integral to understanding systems where the geometry isn't easily expressed in rectangular terms, such as circular or spiral patterns.

Polar Equation

A polar equation represents a relationship involving the distance from the origin \( r \) and the angle \( \theta \). The advantage of polar coordinates is evident when graphing curves centered on the origin or exhibiting symmetry about the origin. The final polar equation we derived is \( r^2 = \frac{9}{\cos 2\theta} \). This format leverages polar functions to express what can be a complex Cartesian relation in simpler terms.

• Recognize that equations expressed this way are often more versatile for analyzing symmetry.
• The expression \( r^2 = \frac{9}{\cos 2\theta} \) indicates that \( r \) varies with \( \theta \) to maintain the graph's structure.

Understanding polar equations is crucial for both solving equations like this and applying them to real-world scenarios, such as calculating paths in navigation and patterns in radiation fields.

Cartesian Equations

Cartesian equations express relationships between x and y coordinates in a standard, rectangular format. These equations are well-suited to straightforward linear and plane geometry problems but can be more challenging when describing curves or complex spatial patterns. In the exercise, \( x^2 - y^2 = 9 \) is a simple form but can represent more complex dynamics, like hyperbolic distances in theoretical physics.

• The goal of converting these equations is often to simplify interaction with geometry, which polar coordinates can achieve more gracefully.
• Converting to polar form, as shown, provides insight into how these same relationships manifest differently.

The core principle behind Cartesian equations remains central to geometry, but converting them demonstrates the adaptability and utility of other forms such as polar, especially when approaching modeling challenges involving rotations or radial distributions.