Problem 65 - Thomas Calculus

Question

Graph the lines and conic sections in Exercises \(65-74.\) $$r=3 \sec (\theta-\pi / 3)$$

Step-by-Step Solution

Answer

The graph is a straight line in polar coordinates shifted by \(\pi/3.\)

Step 1: Understand the Polar Equation

The given equation is in polar form: \( r = 3 \sec(\theta - \pi/3) \). Here, \(r\) represents the radius or the distance from the origin, and \(\theta\) is the angle from the positive x-axis. The secant function, \(\sec\), is the reciprocal of the cosine function, meaning \(\sec(\alpha) = \frac{1}{\cos(\alpha)}\).

Step 2: Convert the Equation

To graph it, consider rewriting \(\sec(\theta - \pi/3)\) as \(\frac{1}{\cos(\theta - \pi/3)}\). Thus, \( r = \frac{3}{\cos(\theta - \pi/3)} \). This implies that \(r \cos(\theta - \pi/3) = 3 \), which can be a hint that this is related to a vertical line in Cartesian coordinates when \(\theta - \pi/3\) is considered.

Step 3: Identify the Graph Type

The presence of \(\sec\) or \(\cos\) indicates a vertical or horizontal line rather than a conic section. In this case, consider values of \(\theta\) where \(\cos(\theta - \pi/3)\) equals zero. These values create vertical lines in the plane. Hence, this represents a line equation in polar coordinates.

Step 4: Graph the Line

To graph this in a polar coordinate system, note that the relationship implies a line that is shifted by \(\pi/3\) from the standard polar axis. For each \(\theta\), calculate \(\cos(\theta - \pi/3) \), and plot the resulting \(r\). You will notice that for points where \(\cos(\theta - \pi/3) = 0\), the radius \(r\) tends to infinity, showing that the line extends infinitely in the plane.

Key Concepts

Conic Sections

Conic sections are curves obtained by slicing a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas. In mathematics and geometry, they are essential because they model many real-world phenomena and have distinct properties that make them applicable in various fields of study.

In the context of polar coordinates, conic sections can be represented by equations that describe their shapes and orientations in a plane. Although the given equation \( r = 3 \sec(\theta - \pi/3) \) doesn't directly represent a conic section, understanding conic sections helps in identifying when a given equation could be graphing these familiar shapes instead of lines. This knowledge can also facilitate the transition from polar to Cartesian coordinates by allowing the transformation and interpretation of equations with trigonometric functions like secant and cosine.

Every conic section can be represented using polar coordinates, typically by equations involving sine or cosine functions, adjusted with additions and multiplications. Knowing the form and properties of conic sections aids in distinguishing them from other types of curves or lines in polar equations.

Secant Function

The secant function is a trigonometric function representing the reciprocal of the cosine function. Specifically, for an angle \(\alpha\), the secant is defined as \( \sec(\alpha) = \frac{1}{\cos(\alpha)} \). This means it is undefined when the cosine of the angle is zero, leading to vertical asymptotes in its graph.

Understanding the secant function is vital for interpreting polar equations like \( r = 3 \sec(\theta - \pi/3) \). When the secant function is used in a polar equation, it shapes the nature of the graph significantly. For instance, values of \(\theta\) where \( \cos(\theta - \pi/3) = 0 \) result in the secant function tending towards infinity, highlighting potential lines or asymptotes in the plot.

The secant function, due to its reciprocal nature, frequently crops up in equations where infinite or undefined values suggest lines rather than closed curves. Its integration into polar equations often signals infinite extensions along certain angles, influenced by shifts like \( -\pi/3 \) from the equation, causing unique rotations or shifts in the graph.

Graphing in Polar Coordinates

Graphing in polar coordinates involves plotting points based on their distance, \(r\), from the origin and an angle, \(\theta\), from the positive x-axis. This system offers a unique perspective on plotting shapes and lines, particularly those that are symmetric or circular.

The given equation \( r = 3 \sec(\theta - \pi/3) \) highlights particular aspects of polar graphs. By converting or manipulating the equation \( \sec \) term, calculations of \(r\) for different \(\theta\) values showcase points that form a line rather than a classic conic section. This method reveals certain geometric entities when graphing equations involving secant or cosine functions.

When graphing with polar coordinates, specific attention is focused on points where functions may become undefined, such as \(\theta\) values that cause \( \cos(\theta - \pi/3) = 0 \). These values lead to infinite plots and should be noted to visualize vertical or horizontal lines in the graph. Understanding this framework opens the door to graphing complex equations and visualizing them intuitively.