Problem 68 - Thomas Calculus

Question

Graph the lines and conic sections in Exercises \(65-74.\) $$r=-2 \cos \theta$$

Step-by-Step Solution

Answer

The graph is a circle with a radius of 1, centered at \((-1, 0)\).

Step 1: Understand the Polar Equation

The given polar equation is \( r = -2 \cos \theta \). This equation is a type of conic section known as a circle, but since the coefficient of cosine is negative, it is a circle centered somewhere on the negative x-axis relative to the pole.

Step 2: Convert to Cartesian Coordinates

To better understand the graph, convert the polar equation into Cartesian coordinates. Use the relationships: \( x = r \cos \theta \) and \( y = r \sin \theta \).Substituting \( r = -2 \cos \theta \) into \( x = r \cos \theta \), we get:\[x = (-2 \cos \theta) \cos \theta = -2 \cos^2 \theta\]And \( y = r \sin \theta = (-2 \cos \theta) \sin \theta \) gives:\[y = -2 \cos \theta \sin \theta\]

Step 3: Simplify Using Trigonometric Identities

Use the cosine double angle identity: \( \cos 2\theta = 2 \cos^2 \theta - 1 \). Therefore,\[\cos^2 \theta = \frac{1+ \cos 2\theta}{2}\]Substituting into the expression for \( x \), we get:\[x = -2 \left(\frac{1 + \cos 2\theta}{2}\right) = -(1 + \cos 2\theta)\]For \( y \), use the identity for \( \sin 2\theta = 2 \sin \theta \cos \theta \), so:\[y = -\sin 2\theta\]

Step 4: Recognize the Conic - A Circle

By recognizing the transformations, we realize the effect of the negative sign and trigonometric conversion means a reflection over the pole. The graph traces a circle with radius 1, centered at \((-1,0)\) on the Cartesian plane.

Step 5: Graph the Polar Equation

Plot the polar equation by calculating values for \( r \) and corresponding rotational positions \( \theta \). A plot from \( \theta = 0\) to \( \theta = 2\pi\) will complete the circle, maintaining a radius of 1.

Key Concepts

Polar to Cartesian Conversion

In the realm of mathematics, converting between polar and Cartesian coordinates is an essential skill. Polar coordinates

• are denoted by \( r, \theta \), where \( r \) represents the radial distance from the origin (or pole),
• and \( \theta \) is the angular coordinate.

On the other hand, Cartesian coordinates utilize \( x \) and \( y \) to indicate positions on a plane. The conversion between these two systems hinges on the trigonometric relationships:

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

For the equation \( r = -2 \cos \theta \), apply these conversions to translate it into a Cartesian form.
By inserting \( r \) in the expressions for \( x \) and \( y \), one obtains the transformed coordinates. This process of translation unveils the geometric interpretations that might be obscured in polar form, providing clarity and a different perspective for analysis.

Trigonometric Identities

Trigonometric identities are powerful tools in mathematics that allow us to simplify complex expressions and make connections between functions. One valuable identity is the cosine double angle identity, which is:

• \( \cos 2\theta = 2 \cos^2 \theta - 1 \)

This identity aids in expressing \( \cos^2 \theta \) in terms of \( \cos 2\theta \), allowing for simplification when handling expressions like \( -2 \cos^2 \theta \).
Additionally, the sine double angle identity, \( \sin 2\theta = 2 \sin \theta \cos \theta \), permits the conversion of \( -2 \cos \theta \sin \theta \) into a more recognizable form: \( -\sin 2\theta \).
These identities act as bridges, transforming polar equations into Cartesian equations, and simplifying the task of sketching or analyzing graphs. They play a pivotal role in the exploration and understanding of conic sections and their attributes.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane in various ways. Depending on the angle of the slice, they can manifest as circles, ellipses, parabolas, or hyperbolas. The polar equation \( r = -2 \cos \theta \) defines a conic section—in this case, a circle. The equation translates to a circle on the Cartesian plane through conversion.
A circle centered at a point \( (-1,0) \) with a radius of 1 emerges when the equation is interpreted correctly. The negative sign in \( -2 \cos \theta \) reflects it over the pole in polar terms, altering its orientation on the Cartesian plane.
Conic sections have distinct properties linked to their geometric forms, offering intriguing insights and applications in real-world scenarios, ranging from architecture to astrophysics. Understanding the intricacies of conic sections enriches your mathematical perspective and enhances your problem-solving acumen.