Problem 26 - Calculus Early Transcendentals

Question

\(21-26\) Find a polar equation for the curve represented by the given Cartesian equation. $$x y=4$$

Step-by-Step Solution

Answer

The polar equation is \( r^2 \sin(2\theta) = 8 \).

Step 1: Identify Cartesian Coordinates in Terms of Polar Coordinates

In polar coordinates, any point on the plane is described by \[ x = r \cos(\theta) \]\[ y = r \sin(\theta) \]where \(r\) is the radius and \(\theta\) is the angle with the positive x-axis.

Step 2: Substitute Polar Coordinates into Cartesian Equation

Substitute the polar expressions for \(x\) and \(y\) into the given Cartesian equation \( xy = 4 \):\[ (r \cos(\theta))(r \sin(\theta)) = 4 \]

Step 3: Simplify the Equation

Simplify the equation \[ r^2 \cos(\theta) \sin(\theta) = 4 \]We can use the trigonometric identity \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\) to further simplify it.

Step 4: Apply Trigonometric Identity

Apply the trigonometric identity:\[ \cos(\theta) \sin(\theta) = \frac{1}{2} \sin(2\theta) \] So, the equation becomes \[ r^2 \frac{1}{2} \sin(2\theta) = 4 \]

Step 5: Solve for the Polar Equation

To isolate \(r^2\), multiply both sides by 2:\[ r^2 \sin(2\theta) = 8 \]This is the polar equation for the given curve.

Key Concepts

Cartesian coordinates

To begin understanding the problem, we first delve into Cartesian coordinates. This coordinate system is the familiar grid of x and y axes we've used in basic geometry. Every point in this system is defined using a pair of values:

• x-coordinate: the distance from the y-axis, measured horizontally.
• y-coordinate: the distance from the x-axis, measured vertically.

In the exercise given, the Cartesian equation is expressed as \( xy = 4 \). This represents a curve, specifically a hyperbola, on the Cartesian plane. Here, each pair \((x, y)\) on the curve satisfies this equation.

trigonometric identity

The trigonometric identity used in this solution is a key concept that simplifies terms when working with angles and their sine and cosine. One essential identity is:

• \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)

This identity is immensely helpful when you encounter products of sine and cosine terms, like in the exercise where \( \cos(\theta) \sin(\theta) \) appears. By recognizing and using the identity, we can transform products into a more manageable form, simplifying complex equations. This makes solving for variables, like \( r \) in polar equations, much more straightforward.

coordinate conversion

Converting coordinates from Cartesian to polar is a fundamental process in analyzing different mathematical perspectives. To convert a point from Cartesian \((x, y)\) to polar coordinates, you use these relationships:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)

This transformation is critical when the problem requires understanding curves in terms of radius and angle rather than grid-based x and y values. It allows for expressing the curve in the plane uniquely and often more conveniently, particularly for circular or angled aspects. Understanding these formulas helps in making the bridge between the two representations seamless.

polar equation

The conversion from Cartesian to polar culminates in the formation of a polar equation. Polar equations represent curves using parameters \( r \) (radius) and \( \theta \) (angle), providing a different yet valuable perspective. In this exercise, we started with \( xy = 4 \) and needed to convert it to a polar form. After substituting and simplifying with trigonometric identities, we arrive at the polar equation:

• \( r^2 \sin(2\theta) = 8 \)

This new representation gives insight into the curve's structure, orientation, and distance from the origin, which can be less apparent in Cartesian form. Converting this way often reveals symmetries and properties crucial for deeper mathematical analysis.