Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Graph each polar equation for \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\). In Exercises \(39-48\), identify the rype of polar graph. \(r=\frac{\cos 2 \theta}{\cos \theta}\) (This is a cissoid with a loop.)

Step-by-Step Solution

Answer

The polar equation \(r=\frac{\cos 2\theta}{\cos \theta}\) creates a cissoid with a loop, symmetric about the polar axis with vertical asymptotes where \(\cos \theta = 0\).

Step 1: Understand the Polar Equation Structure

The given polar equation is \( r = \frac{\cos 2\theta}{\cos \theta} \). This equation is a ratio of trigonometric functions, resembling a cissoid, which typically has a loop.

Step 2: Identify Key Angles

To graph the polar equation, identify key angles where the sine and cosine functions have notable values, such as \(\theta = 0^{\circ}, 90^{\circ}, 180^{\circ},\) and \(270^{\circ}\). At these angles, calculate \(r\).

Step 3: Handle Undefined Points

The equation becomes undefined where \(\cos \theta = 0\) (e.g., \(\theta = 90^{\circ}\) and \(270^{\circ}\)), as it results in division by zero. Note these as vertical asymptotes in the graph.

Step 4: Calculate \(r\) for Key Angles

For \(\theta = 0^{\circ}, r = \frac{1}{1} = 1\). For \(\theta = 180^{\circ}, r = \frac{-1}{-1} = 1\). Check other angles without causing division by zero.

Step 5: Sketch the Graph

With these points, understand where the loop appears. The graph is symmetric about the line \(\theta = 0^{\circ}\) (the polar axis) and shows a loop near the origin due to the trigonometric function's behavior.

Key Concepts

Cissoid

A cissoid is a type of curve, which often appears in the study of polar coordinates. It is typically recognized for having one or more loops on its graph. In polar equations, a cissoid might not look straightforward at first glance. However, by understanding the behavior of trigonometric functions involved, one can predict these loops. Cissoids in polar form can be described using equations that involve ratios or products of trigonometric functions of the angle \(\theta\). Recognizing a cissoid involves identifying these loops, which are clearer when plotting points from key angles.

Trigonometric Functions

Trigonometric functions include sine, cosine, and tangent, among others. They are crucial in polar coordinates as polar equations often depend on them. These functions help in determining the radius, \(r\), for a given angle \(\theta\). A polar equation like \(r = \frac{\cos 2\theta}{\cos \theta}\) uses trigonometric functions to describe a path. Each function has a pattern that repeats, known as periodicity. For example, the cosine function repeats every \(360^\circ\). At angles where these functions hit specific values (like \(0^\circ\), \(90^\circ\), \(180^\circ\), and \(270^\circ\)), they can help in understanding the curve's behavior at crucial points.

Graphing Polar Equations

Graphing polar equations involves plotting points based on the radius \(r\) and angle \(\theta\). Unlike Cartesian coordinates, where you have \(x\) and \(y\)-axes, polar graphs use circles and angles. To graph a polar equation, one often calculates \(r\) for key angles that provide insights into the curve's formation. For instance, if an equation has a division by \(\cos \theta\), you check angles where \(\cos \theta = 0\) to handle undefined points. Such undefined points often turn into features like vertical asymptotes. By calculating \(r\) for angles without causing division by zero, one visualizes the shape, notably any loops specific to curves like cissoids.