Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r^{2} \sin 2 \theta=4 $$

Step-by-Step Solution

Answer

The equation is \( xy = 2 \).

Step 1: Recall Polar to Cartesian Transformations

To convert a polar equation to a Cartesian equation, recall the transformations: \( r^2 = x^2 + y^2 \) and \( \sin 2\theta = 2 \sin \theta \cos \theta \). Use the identities \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 2: Express \( \sin 2\theta \) in terms of \( x \) and \( y \)

Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \), express this in terms of \( x \) and \( y \). Note \( \sin \theta = \frac{y}{r} \) and \( \cos \theta = \frac{x}{r} \). Thus, \( \sin 2\theta = 2 \left(\frac{y}{r}\right)\left(\frac{x}{r}\right) = \frac{2xy}{r^2} \).

Step 3: Substitute into the Polar Equation

Insert the expression for \( \sin 2\theta \) into the equation \( r^2 \sin 2\theta = 4 \). This gives \( r^2 \cdot \frac{2xy}{r^2} = 4 \), simplifying it to \( 2xy = 4 \).

Step 4: Solve for \( xy \) in Cartesian Coordinates

The equation \( 2xy = 4 \) can be simplified further by dividing both sides by 2, resulting in \( xy = 2 \). This is the Cartesian equation corresponding to the original polar equation.

Step 5: Sketch the Graph

The equation \( xy = 2 \) represents a hyperbola in the Cartesian plane. Therefore, in the \( r\theta \)-plane, the graph will reflect transformation into Cartesian coordinates but sketched according to its polar characteristics.

Key Concepts

Polar Equations: Understanding the Basics

Polar equations play a crucial role in mathematics, especially when linking geometric shapes with algebraic equations. Unlike Cartesian coordinates, which define a point using horizontal and vertical distances (x, y), polar equations use the radius and angle (r, θ). This means

• the radius \( r \) denotes how far the point is from the origin,
• and the angle \( \theta \) indicates the direction of that point from the polar axis.

These equations can describe complex curves more naturally than their Cartesian counterparts in many scenarios. For instance, circles and spirals are often expressed more straightforwardly using polar equations. In the given exercise, we have a polar equation \( r^2 \sin 2\theta = 4 \), where understanding its transformation involves converting it into a familiar form within Cartesian coordinates.

Cartesian Coordinates: Translating from Polar Form

Cartesian coordinates are a standard method to describe any location in a 2D plane using two values: x and y. These coordinates provide a convenient way for graphing and solving equations because they align perfectly with the straight lines of graph paper.
To translate polar expressions into Cartesian forms, we use some helpful identities:

• The whole distance formula \( r^2 = x^2 + y^2 \), capturing both the horizontal and vertical components as squared sums,
• \( x = r \cos \theta \) to get the x-component from polar form,
• \( y = r \sin \theta \) to derive the y-component.

Our task using these relationships is to express the polar equation \( r^2 \sin 2\theta = 4 \) in terms of x and y. As we did this, it brought us to the more familiar Cartesian equation \( xy = 2 \). This conversion is essential for graphing and understanding the behavior of the graph without switching between coordinate systems.

Trigonometric Identities: Bridging Polar and Cartesian Systems

Trigonometric identities are vital in transforming equations between polar and Cartesian systems. These identities allow us to relate angles and side lengths to various trigonometric functions, such as sine and cosine.

• One key identity is \( \sin 2\theta = 2\sin\theta\cos\theta \), used to simplify expression transformations.
• Using identities, we can express \( \sin \theta \) as \( \frac{y}{r} \) and \( \cos \theta \) as \( \frac{x}{r} \), bridging the gap between r, \(\theta\), x, and y.

In our problem, we rewrote \( \sin 2\theta \) as \( \frac{2xy}{r^2} \), allowing us to transform the polar equation into a format that includes x and y. This enables us to connect seemingly complex geometric representations with standard algebraic expressions, facilitating easier understanding and manipulation of equations.

Graph Sketching: Visualizing Polar and Cartesian Equations

Graph sketching is a fundamental skill for visualizing mathematical relationships. In the context of polar to Cartesian conversions, sketching helps us see the geometric representation of the corresponding equation.
When dealing with the converted Cartesian equation \( xy = 2 \), it's helpful to recognize the graph as a hyperbola. In this scenario:

• The equation implies that for any point on the graph, the product of its x and y coordinates is constant (2).
• The Cartesian graph provides a straightforward depiction of this equation and its geometric properties.
• This hyperbola opens along the coordinate axes, reflective of the polar equation's power in representing curves differently.

In the \( r\theta \)-plane, understanding the graph associated with its transformation into Cartesian coordinates allows us to comprehend how these coordinate systems interrelate and exhibit identical geometric structures. This duality is crucial for deeper insights into the mathematics of different coordinate systems.