Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ y^{2}-x^{2}=4 $$

Step-by-Step Solution

Answer

The polar equation is \( r^2 \cos(2\theta) = -4 \).

Step 1: Understand the Relationship Between Cartesian and Polar Coordinates

In polar coordinates, a point is given by \((r, \theta)\) where \(r\) is the radius or distance from the origin, and \(\theta\) is the angle from the positive x-axis. The relationships between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\) are \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 2: Substitute Cartesian Formulas into the Original Equation

Begin with the original equation in Cartesian form: \[ y^2 - x^2 = 4 \]Substitute the polar coordinate formulas: \[ (r \sin \theta)^2 - (r \cos \theta)^2 = 4 \]

Step 3: Simplify the Polar Equation

Simplify the expression: \[ r^2 \sin^2 \theta - r^2 \cos^2 \theta = 4 \]Factor out \(r^2\): \[ r^2 (\sin^2 \theta - \cos^2 \theta) = 4 \]

Step 4: Use Trigonometric Identity

Recall the trigonometric identity: \[ \sin^2 \theta - \cos^2 \theta = -\cos(2\theta) \]Substitute this into the equation: \[ r^2 (-\cos(2\theta)) = 4 \]

Step 5: Rearrange for a Clean Polar Equation

Rearrange the equation to get: \[ r^2 \cos(2\theta) = -4 \]This is the required polar equation.

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a common way to describe points on a plane using two values, typically denoted as \(x\) and \(y\). Imagine a flat surface like a sheet of paper. The horizontal line running across it is the x-axis, while the vertical line is the y-axis. Each point on this plane has a unique pair of coordinates \((x, y)\).
Here’s how it works:

• \(x\) is the distance the point is from the y-axis, moving left or right.
• \(y\) is the distance from the x-axis, moving up or down.

If you draw a point to the right 3 units from the y-axis, and 4 units up from the x-axis, that point's coordinates are \(3, 4\). This system is extremely useful for graphing equations and visualizing mathematical concepts.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are universally true. They help in simplifying complex mathematical expressions. When working with polar and Cartesian equations, these identities can be very handy.
A few common trigonometric identities include:

• \( \sin^2 \theta + \cos^2 \theta = 1\), which is the Pythagorean identity.
• \( \tan \theta = \frac{\sin \theta}{\cos \theta}\).
• \( \sin(2\theta) = 2\sin \theta \cos \theta\).

In the context of our problem, we used the identity \( \sin^2 \theta - \cos^2 \theta = -\cos(2\theta)\). This helped us transform the Cartesian equation into a polar form. Understanding these identities allows us to navigate between complex and simplified forms efficiently.

Equation Conversion

Equation conversion involves changing an equation from one form to another. In mathematics, it's often necessary to shift between Cartesian and polar forms to analyze or solve a problem effectively.
This transformation hinges on using known relationships between the two coordinate systems:

• From Cartesian to polar: \(x = r \cos \theta\) and \(y = r \sin \theta\).
• From polar to Cartesian: \(r^2 = x^2 + y^2\) and \( heta = \tan^{-1}(\frac{y}{x})\).

In our exercise, we used the conversion equations to substitute the variables \(x\) and \(y\) in terms of \(r\) and \(\theta\). This allowed us to transition the problem from a familiar Cartesian format into a more complex but insightful polar perspective.

Polar Equations

Polar equations describe the relationship between the radius \(r\) and the angle \(\theta\) of a point from a central origin. Unlike Cartesian equations, where two coordinates determine a point's position, polar equations rely on distance and angle.
Here’s how polar coordinates work:

• The radius \(r\) represents the distance from the origin to the point.
• The angle \(\theta\) represents the direction from the positive x-axis.

A polar equation might look like \(r = 1 + \cos \theta\). Converting from Cartesian to polar equations not only provides a different way to view the relationship between variables, but it can also simplify the problem-solving process in certain cases, such as solving complex integrals or evaluating limits involving circular functions.