Problem 41 - Algebra and Trigonometry

Question

Sketch a graph of the polar equation. $$ r=2+\sec \theta \quad(\text { conchoid }) $$

Step-by-Step Solution

Answer

The graph of \( r = 2 + \sec \theta \) is a conchoid with loops and asymptotes at \( \theta = \frac{\pi}{2} + k\pi \).

Step 1: Understand the Polar Equation

The given equation is in polar form: \( r = 2 + \sec \theta \). The equation describes how the distance \( r \) from the origin varies with the angle \( \theta \). \( \sec \theta \) is the secant of the angle, which is the reciprocal of \( \cos \theta \), defined as \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Analyze Characteristics of Secant Function

The secant function \( \sec \theta = \frac{1}{\cos \theta} \) has vertical asymptotes where \( \cos \theta = 0 \), i.e., at \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. Because of these asymptotes, \( r = 2 + \sec \theta \) also will have similar characteristics and exhibit undefined points at these angles.

Step 3: Identify Invariant Points

Substitute several values of \( \theta \) where \( \cos \theta \) is defined to find specific points. For example, for \( \theta = 0 \), \( \sec 0 = 1 \), so \( r = 2 + 1 = 3 \). Similarly, for \( \theta = \pi \), \( \sec \pi = -1 \), hence \( r = 2 - 1 = 1 \). These calculations give polar coordinates (0, 3π) and (π, 1π).

Step 4: Sketch the Graph

Plot the invariant points, respecting vertical asymptotes. In quadrants where \( \cos \theta \) is positive, the function \( r \) increases. As \( \theta \) nears \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), \( \sec \theta \) increases to infinity, making \( r \to \infty \), showing upward or downward trends. Combine these plotted points and characteristics to sketch smooth segments of the conchoid, looping into asymptotes.

Key Concepts

Understanding Polar Graphs

A polar graph is a way of plotting curves using polar coordinates. In polar coordinates, each point on the plane is defined by a distance from the origin, denoted as \( r \), and an angle \( \theta \) measured from the positive x-axis.
You can imagine these coordinates as a combination of how far you go out in a straight line (the radius \( r \)) and the direction that line extends (the angle \( \theta \)).

• Polar graphs are particularly useful in plotting circular and spiral patterns, which are not easily represented in rectangular coordinates.
• The same point on the graph can be reached with different combinations of \( r \) and \( \theta \), providing more flexibility.

To sketch a polar graph like \( r = 2 + \sec \theta \), begin by analyzing how \( r \) changes as \( \theta \) varies, particularly focusing on important angles such as \( 0 \), \( \frac{\pi}{2} \), \( \pi \), and so on. Due to the nature of polar graphs, certain angles will reveal unique traits like loops or cusps.

Exploring the Secant Function

The secant function, \( \sec \theta \), is central to understanding our equation \( r = 2 + \sec \theta \). This function is the reciprocal of the cosine function, \( \sec \theta = \frac{1}{\cos \theta} \).
Because it involves division, the secant function tends to infinity wherever \( \cos \theta \) equals zero.

• The secant function has vertical asymptotes at these undefined points, specifically \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.
• Between these asymptotes, the secant function oscillates, causing dramatic increases or decreases in the value of \( r \).

When \( \theta \) approaches \( \frac{\pi}{2} \) or \( \frac{3\pi}{2} \), \( r \) becomes extremely large, moving towards infinity or negative infinity. Understanding these behaviors is crucial when sketching the graph, as they indicate the way the conchoid arcs around the pole (origin) and approaches these critical angles.

What is a Conchoid?

In the context of polar coordinates and our equation \( r = 2 + \sec \theta \), a conchoid is a specific type of curve. This conchoid arises when you add the secant function to a constant.
Conchoids have distinct shapes often characterized by loops or breaks.

• The structure appears as a looped curve around vertical asymptotes due to the sudden increase in \( r \).
• In our equation, as \( \theta \) nears the vertical asymptotes of the secant function, \( r \) increases sharply, creating unique "waves" or "wings" around those lines.

When exploring conchoids, it's vital to recognize these patterns because they provide insights into the behavior of curves in nature and mechanical systems. For our equation, plotting points and observing the transitions around critical angles help visualize the full beauty of polar graphs.