Problem 27 - Precalculus: Functions and Graphs

Question

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: Identify the Cartesian Equation

The given equation is in Cartesian coordinates: \[ y^2 - x^2 = 4 \] This is the equation of a hyperbola centered at the origin.

Step 2: Convert Cartesian to Polar Coordinates

In polar coordinates, we use the relationships \( x = r \cos\theta \) and \( y = r \sin\theta \). Substitute these into the Cartesian equation: \[ (r \sin\theta)^2 - (r \cos\theta)^2 = 4 \] Simplify this to find the polar form.

Step 3: Simplify the Polar Equation

Expanding the squared terms gives:\[ r^2 \sin^2\theta - r^2 \cos^2\theta = 4 \]Factor out \( r^2 \):\[ r^2 (\sin^2\theta - \cos^2\theta) = 4 \] Using the trigonometric identity \( \sin^2\theta - \cos^2\theta = -\cos(2\theta) \), the equation becomes:\[ r^2 (-\cos(2\theta)) = 4 \] Simplifying further:\[ r^2 \cos(2\theta) = -4 \]

Key Concepts

Cartesian Coordinates

The concept of Cartesian coordinates is fundamental in mathematics, helping us to understand and plot points on a plane. In simple terms, the Cartesian coordinate system uses two perpendicular axes, traditionally labeled as the x-axis (horizontal) and y-axis (vertical). These axes intersect at a point called the origin. Any point in the plane can be described by a pair of numbers, \(x, y\), which are its Cartesian coordinates.

Here are some key points about Cartesian coordinates:

• The x-coordinate tells you how far left or right a point is from the origin.
• The y-coordinate tells you how far up or down a point is from the origin.
• Together, these numbers (x, y) uniquely identify a position in the plane.

In the context of our exercise, the original equation \(y^2 - x^2 = 4\) involves Cartesian coordinates. It plots a hyperbola, which we'll get into next.

Hyperbola

A hyperbola is a type of smooth curve lying in a plane. It is defined by its relation to conic sections and can be thought of as two mirrored curves. The standard equation for a hyperbola in Cartesian coordinates typically looks like \((\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1)\) or vice versa.In our case, the equation \(y^2 - x^2 = 4\) is a basic form of a hyperbola equation. Key characteristics include:

• Centered at the origin with axes aligned with the Cartesian axes.
• The equation might be rewritten as \((y^2/4) - (x^2/4) = 1\), illustrating that this hyperbola opens upwards and downwards.
• Hyperbolas always consist of two separate "arms," which means they resemble a pair of mirrored, open-ended curves.

Understanding the hyperbola's features is essential before converting it into polar coordinates.

Trigonometric Identity

Trigonometric identities are equations that hold true for all angles. They help simplify expressions and solve trigonometric equations. One widely used identity in mathematics is \(\sin^2\theta + \cos^2\theta = 1\), which expresses a basic relationship in trigonometry.In our problem-solving, we encountered the identity \(\sin^2\theta - \cos^2\theta = -\cos(2\theta)\). This identity arises from the double-angle formulae:

• It enables the simplification of trigonometric expressions involving squared terms.
• Such identities are pivotal when converting Cartesian equations into polar equations, streamlining the transformations.

Understanding these identities helps in recognizing patterns and simplifying complex trigonometric expressions, aiding us in the conversion step.

Coordinate Conversion

Coordinate conversion is the process of translating coordinates from one system to another. In mathematics, two common coordinate systems are Cartesian and polar. Each system has its own advantages for representing certain equations.To convert from Cartesian to polar coordinates, you make use of these relationships:

• \(x = r \cos\theta\)
• \(y = r \sin\theta\)
• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

In our exercise, we replaced \(x\) and \(y\) in the Cartesian equation with their polar counterparts to start solving. This allowed us to factor and simplify, ultimately transforming the Cartesian hyperbola into a polar equation, which is presented as \(r^2 \cos(2\theta) = -4\).

Learning how to efficiently change between these systems is fundamental as it leverages the best of both representations to solve problems seamlessly.