Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(r^{2}=6 \cos 2 \theta\)

Step-by-Step Solution

Answer

It's a rotated conic section, related to an ellipse or hyperbola.

Step 1: Identify the Equation Type

The given equation \( r^2 = 6 \cos 2\theta \) is a polar equation that represents a conic section, specifically a circle, ellipse, parabola, or hyperbola. In this case, it resembles the equation of a conic section in polar form.

Step 2: Convert to Cartesian Coordinates

To understand the nature of the conic, it helps to translate the polar equation into Cartesian coordinates. Using \( r = \sqrt{x^2 + y^2} \), \( x = r \cos \theta \), and \( y = r \sin \theta \), substitute into the polar equation and simplify.

Step 3: Simplify the Equation

From \( r = \sqrt{6 \cos 2\theta} \), remember \( \cos 2\theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta \). Use these identities to expand the expression and further simplify it to find its Cartesian form.

Step 4: Analyze the Equation

Through simplification, determine the full nature of the Cartesian form which results from the polar equation. Identify if the resulting equation is centered at the origin and categorize the type of conic section it represents.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. These shapes have specific properties and equations that define their form.

• A **circle** is formed when the intersecting plane is perpendicular to the cone's axis, creating a perfectly round shape.
• An **ellipse** is an elongated circle, resulting from a tilted plane that intersects both nappes of the cone.
• A **parabola** occurs when the plane is parallel to an edge of the cone, creating a mirror-symmetric curve.
• Lastly, a **hyperbola** is produced when the plane intersects both nappes of the cone, resulting in two separate curves.

Each conic section has a standard equation form in both Cartesian and polar coordinates. Identifying these sections in polar coordinates begins with recognizing whether they are complete shapes or degeneracies, like a point or a pair of lines. The given polar equation, depending on its form, helps determine which conic section is represented. Studying these structures in both Cartesian and polar terms allows for a complete understanding of their geometric properties.

Cartesian Coordinates

Cartesian coordinates are used to describe positions in a plane using two perpendicular axes, typically labeled as the x-axis and y-axis. Each point in this system is represented by an ordered pair \((x, y)\), where **x** is the horizontal distance from the origin and **y** is the vertical distance.
These coordinates are crucial in translating polar equations into a more familiar form. Polar coordinates use distance from the origin and angle measurements, which can be transformed using:

• \(x = r \cos \theta\) for the x-coordinate,
• \(y = r \sin \theta\) for the y-coordinate,
• \(r = \sqrt{x^2 + y^2}\) to revert back to radial distance.

The conversion allows us to express equations typically found in a circular or angular system into a linear framework. This transformation is essential for analyzing the conic section directly in Cartesian form, providing clarity in understanding the conic's size, shape, and position relative to the origin. By exploring how polar equations like \(r^{2}=6\cos 2\theta\) become recognizable Cartesian forms, students can more easily identify and work with the specific attributes of each conic section.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. They play a critical role in converting and simplifying equations from polar to Cartesian forms.
Key identities include:

• **Double Angle Formulas**: Such as \(\cos 2\theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta\). These are particularly useful when dealing with conic sections that involve double angles, like the given polar equation \(r^{2} = 6\cos 2\theta\).
• **Pythagorean Identities**: Like \(\cos^2 \theta + \sin^2 \theta = 1\), which help in expressing powers or products in simpler terms.

By employing these identities, we can simplify complex trigonometrical representations and easily discern the conic section's nature. For instance, expanding and simplifying \(\cos 2\theta\) will allow us to see the equation's underlying structure, facilitating its translation and analysis in a more accessible Cartesian format. Understanding and applying these identities are essential for students in navigating between different coordinate systems efficiently.