Problem 47 - Thomas Calculus

Question

Graph the lines and conic sections in Exercises \(47-56\) $$ r=3 \sec (\theta-\pi / 3) $$

Step-by-Step Solution

Answer

Plot the line by treating \( r = 3 \sec(\theta - \frac{\pi}{3}) \) as a vertical line with angle translation \( \theta = \frac{\pi}{3} \).

Step 1: Understand the Polar Equation

The given equation is in polar form: \( r = 3 \sec(\theta - \frac{\pi}{3}) \). This equation represents a vertical line in polar coordinates, shifted by an angle. To graph it, we need to convert it into Cartesian coordinates or understand its behavior in polar form.

Step 2: Convert Secant to Cartesian Form

In polar coordinates, \( \sec \theta = \frac{1}{\cos \theta} \), so we can rewrite the equation as \( r = \frac{3}{\cos(\theta - \frac{\pi}{3})} \). Knowing that \( x = r \cos \theta \) and \( y = r \sin \theta \), the equation implies that the line is vertical in the direction it points, translated by \( \theta = \frac{\pi}{3}\).

Step 3: Transform into Rectangular Coordinates

To transform into rectangular coordinates, realize that \( \sec \theta \equiv \frac{1}{\cos(\theta)} \), thus translating to \( x = 3 \cos(\theta - \frac{\pi}{3}) \). By identity transformation, derive that this represents the modified direction and position according to the polar definition.

Step 4: Analyze Symmetry and Translation

The shift by \( -\frac{\pi}{3} \) indicates a horizontal translation in the direction of \( \theta \), specifically a translation affecting the angle from standard position, hinting a vertical shift considering the pole as origin.

Step 5: Plotting with Polar Understanding

Since the line can extend infinitely when \( \cos \theta \to 0 \), you plot it considering it a vertical line for polar graphing, but understanding transposing its line direction due to the translation in \( \theta \).

Step 6: Finalizing the Graph

With the understanding that a generic sec of form \( r = a \sec(\theta) \) transforms to vertical lines, plot accordingly ensuring visualization according to negative denotation for angles \( \pm \pi \) to correctly depict the scenario.

Key Concepts

Polar Equations

In the realm of mathematics, polar equations like the one presented here, \( r = 3 \sec(\theta - \frac{\pi}{3}) \), are used to represent relationships where the position is defined via a radius \( r \) and angle \( \theta \). Instead of the traditional x and y axes, polar coordinates utilize the distance from a fixed point, known as the pole, and an angle from a fixed direction, typically the positive x-axis.
These equations allow us to portray curves and shapes, like lines and conic sections, in a rotational frame rather than a strictly horizontal and vertical perspective.
Such forms simplify the depiction of periodic phenomena and symmetric properties. Polar equations often require a deep understanding of trigonometric identities, and conversion to Cartesian coordinates for comprehensive graphing insights.

Cartesian Coordinates

Converting polars into Cartesian coordinates helps visualize plots on a traditional grid. The given equation \( r = \frac{3}{\cos(\theta - \frac{\pi}{3})} \), once translated, points to a significant aspect of vertical alignment, by transforming the equation, revealing its true geometry.
Understanding that in Cartesian planes, via transformations, a polar equation can become \( x = r \cos \theta \) and \( y = r \sin \theta \). This enables plotting trajectory by acknowledging vector pointing alignment and directionality.
The advantage of this conversion lies in offering intuitive spatial relationships, enhancing mathematical perception.

Graphing

Graphing polar equations like \( r = 3 \sec(\theta - \frac{\pi}{3}) \) requires a distinct approach different from Cartesian systems. It's vital to first comprehend the behavior and implications within the polar framework.

• Understand the circle foundation and radial extension.
• Use polar grids that accentuate symmetry.
• Focus on shifts indicated by angle translations, such as \( -\frac{\pi}{3} \).

Preparing a graph involves considering the full cycle of the angle \( \theta \) from 0 to \( 2\pi \), marking significant points where tradtional functions (like cos and secant) equate to zero or shift undefined states, creating vertical representations.
Visualization in this manner aids memorization of geometrical attributes and transformations through depiction.

Conic Sections

Conic sections manifest in various forms, including lines, circles, parabolas, ellipses, and hyperbolas, all of which can be represented with polar equations. The exercise illustrates the conversion of a secant-based equation into a vertical line, a basic yet essential feature within the broader category of conic sections.
Acknowledging how rotations and translations affect these sections in polar graphs affirms the understanding. For example, the vertical nature of the secant transformation establishes key linear characteristics within the framework.
By exploring transformations like \( \theta = \theta - \frac{\pi}{3} \), one can visualize the plane shift and capture the subtleties involved in transitioning between polar and Cartesian representations of conic figures.