Problem 56 - College Algebra

Question

Express each equation in polar form with \(r\) as a function of \(\theta\). $$ x y=2 $$

Step-by-Step Solution

Answer

The polar form of the equation is \( r = \sqrt{\frac{4}{\sin(2\theta)}} \).

Step 1: Recognize the Cartesian Variables

Identify the given equation in Cartesian coordinates. Here, the equation given is \( xy = 2 \). This is in terms of \(x\) and \(y\), which are Cartesian coordinates.

Step 2: Recall the Conversion to Polar Coordinates

In polar coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \). We will use these relationships to convert the equation from Cartesian to polar coordinates.

Step 3: Substitute Polar Equivalents

Substitute \( x \) and \( y \) with their polar coordinate equivalents in the equation: \( xy = 2 \). This gives us \( (r \cos \theta)(r \sin \theta) = 2 \).

Step 4: Simplify the Equation

Simplify the equation from Step 3. We get: \( r^2 \cos \theta \sin \theta = 2 \).

Step 5: Isolate the Term for r^2

We can use the trigonometric identity \( \sin(2\theta) = 2\sin\theta\cos\theta \). Simplify the equation to: \( r^2 \frac{1}{2} \sin(2\theta) = 2 \).

Step 6: Solve for r

Multiply both sides by 2 to isolate \( r^2 \): \( r^2 \sin(2\theta) = 4 \). Divide by \( \sin(2\theta) \) to get \( r^2 = \frac{4}{\sin(2\theta)} \). Then, take the square root: \( r = \sqrt{\frac{4}{\sin(2\theta)}} \).

Key Concepts

Cartesian to Polar Conversion

Cartesian to polar conversion is the process of transforming coordinates from the Cartesian system, which utilizes \(x\) and \(y\) to describe a point, into the polar system that uses \(r\) and \(\theta\). This conversion is helpful in situations where calculations involving angles and distances from a point of origin are more intuitive or simpler.
To convert a Cartesian equation to a polar form, use the following relationships:

• Coordinate \(x\) is represented as \(r \cos \theta\)
• Coordinate \(y\) is represented as \(r \sin \theta\)

These formulas allow us to rewrite Cartesian equations in terms of polar coordinates. For example, substituting these values in the equation \(xy = 2\), \(x\) and \(y\) are replaced with \(r \cos \theta\) and \(r \sin \theta\) respectively. As a result, you'll get \((r \cos \theta)(r \sin \theta) = 2\), which effectively transforms the Cartesian equation into a polar equation.
This process is foundational in translating between these two systems and is particularly useful when solving problems that naturally exhibit symmetry around a central point.

Trigonometric Identities

Trigonometric identities are essential tools in mathematics, providing relationships between the ratios of the sides of triangles.These identities allow us to simplify expressions, especially when dealing with trigonometric functions in polar coordinates.
One of the most useful identities applied in this context is the double angle identity for sine: \(\sin(2\theta) = 2\sin\theta\cos\theta\). This identity helps us rewrite products of sine and cosine as single trigonometric functions,which can significantly simplify polar equations.
In the given exercise, we initially arrived at the equation \(r^2 \cos \theta \sin \theta = 2\). Using the identity mentioned, \(\cos \theta \sin \theta\) was rewritten as \(\frac{1}{2}\sin(2\theta)\). This step is crucial as it allows the combination of trigonometric functions into a more manageable form, resulting in \(r^2 \frac{1}{2} \sin(2\theta) = 2\).
Trigonometric identities bring out a powerful aspect of these conversions, simplifying and making equations more accessible for further manipulation and solution.

Equations in Polar Form

Equations in polar form express mathematical relationships in terms of the radius \(r\) and angle \(\theta\), often leading to simpler representations and solutions for problems involving symmetry. Instead of using Cartesian coordinates \(x\) and \(y\), we describe points based on their distance from the origin and their angular displacement from the positive x-axis.
In the exercise, after converting the Cartesian equation into polar coordinates, we derived \(r^2 \frac{1}{2} \sin(2\theta) = 2\). Simplification led us to is \(r^2 \sin(2\theta) = 4\), and further rearranging gave us the formula \(r^2 = \frac{4}{\sin(2\theta)}\).
The final step is solving for \(r\), resulting in\(r = \sqrt{\frac{4}{\sin(2\theta)}}\). This equation describes \(r\) as a function of \(\theta\) in the polar coordinate system.
Expressing this way is beneficial not only for representing certain curves and patterns in a more straightforward manner but also for solving equations and visualizing them in contexts involving circular motion or wave patterns.