Problem 18 - The Calculus with Analytic Geometry

Question

Find a polar equation of the graph having the given cartesian equation.\(x^{2}-y^{2}=16\)

Step-by-Step Solution

Answer

The polar equation is \[ r = \pm \sqrt{\frac{16}{\cos(2\theta)}} \].

Step 1: Identify Cartesian Equation

The given Cartesian equation is \[ x^{2}-y^{2}=16 \].

Step 2: Convert Cartesian Coordinates to Polar Coordinates

Use the polar coordinates transformations: \[ x = r \cos\theta \] and \[ y = r \sin\theta \].

Step 3: Substitute Polar Equations Into the Cartesian Equation

Substitute \[ x = r \cos\theta \] and \[ y = r \sin\theta \] into the given equation: \[ (r \cos\theta)^{2} - (r \sin\theta)^{2} = 16 \].

Step 4: Simplify the Equation

Simplify the equation: \[ r^{2} \cos^{2}\theta - r^{2} \sin^{2}\theta = 16 \].

Step 5: Factor Out Common Terms

Factor out \( r^{2} \): \[ r^{2} (\cos^{2}\theta - \sin^{2}\theta) = 16 \].

Step 6: Use Trigonometric Identity

Recall the trigonometric identity: \[ \cos^{2}\theta - \sin^{2}\theta = \cos(2\theta) \]. Substituting this into the equation gives: \[ r^{2} \cos(2\theta) = 16 \].

Step 7: Solve for r

Solve for \( r^{2} \): \[ r^{2} = \frac{16}{\cos(2\theta)} \]. Taking the square root of both sides, we get: \[ r = \pm \sqrt{\frac{16}{\cos(2\theta)}} \].

Key Concepts

Cartesian to Polar Conversion

Converting Cartesian coordinates to polar coordinates is crucial for solving and visualizing equations. Cartesian coordinates are represented by \(x,y\), while polar coordinates are represented by \(r,\theta\). The formulas to convert between these two systems are:

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

Here, \(r\) is the distance from the origin (0,0) to the point, and \(\theta\) is the angle between the positive x-axis and the line connecting the origin to the point. For the given problem, you'd use these conversions to change the Cartesian equation \(x^2 - y^2 = 16\) into a polar form.

Trigonometric Identities

Trigonometric identities simplify equations involving trigonometric functions like sine and cosine. One key identity used in this problem is:

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

This identity helps to reduce the complexity of the equation. Instead of dealing with squares of sine and cosine individually, you combine them using the 2\theta angle. This was vital in transforming \(r^2 (\cos^2\theta - \sin^2\theta) = 16\) into the simpler form \(r^2 \cos(2\theta) = 16\).

Equation Simplification

Simplifying equations makes it easier to solve them. In this exercise, simplification involved multiple steps.
Firstly, you substituted \(x = r \cos\theta\) and \(y = r \sin\theta\) into \(x^2 - y^2 = 16\): \(r^2 \cos^2\theta - r^2 \sin^2\theta = 16\).
Breaking down this step:

• Combine \cos and \sin terms using trigonometric identities.
• Simplify grouped terms to find common factors.

Here, extracting \(r^2\) as a common factor streamlined our equation.

Factorization

Factorization is a mathematical technique used to rewrite an equation as a product of simpler expressions. For instance, in our problem once we substitute the polar coordinates, our equation looks like \(r^2 \cos^2\theta - r^2 \sin^2\theta = 16\).
We factor \(r^2\) out of the entire equation:

• \r^2 (\cos^2\theta - \sin^2\theta) = 16\
• Use trigonometric identity to replace \cos^2\theta - \sin^2\theta with \cos(2\theta)\

Factorization makes it manageable to solve: \(r^2 = \frac{16}{\cos(2\theta)}\) and \(r = \pm \sqrt{\frac{16}{\cos(2\theta)}}\), enabling us to find the desired output easily.