Problem 42 - 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. $$r=2-\cos \theta$$

Step-by-Step Solution

Answer

The graph is a limaçon with a dimple, described by \( r = 2 - \cos \theta \).

Step 1: Understand the Equation

The given polar equation is \( r = 2 - \cos \theta \). This is a type of limaçon equation, a general form of \( r = a - b \cos \theta \). This specific form, where \( a > b \), typically results in a limaçon with an inner loop.

Step 2: Determine Key Points

Determine some key points by substituting values for \( \theta \):ewline- When \( \theta = 0^{\circ} \), \( r = 2 - \cos 0^{\circ} = 1 \).ewline- When \( \theta = 90^{\circ} \), \( r = 2 - \cos 90^{\circ} = 2 \).ewline- When \( \theta = 180^{\circ} \), \( r = 2 - \cos 180^{\circ} = 3 \).ewline- When \( \theta = 270^{\circ} \), \( r = 2 - \cos 270^{\circ} = 2 \).ewline- When \( \theta = 360^{\circ} \), \( r = 2 - \cos 360^{\circ} = 1 \).

Step 3: Analyze the Graph Shape

Since \( r = 2 - \cos \theta \) is of the form \( r = a - b \cos \theta \) with \( a = 2 \) and \( b = 1 \), the graph will be a limaçon without an inner loop but rather a dimple, since \( a > b \).

Step 4: Draft the Polar Graph

1. Start at the initial point where \( \theta = 0^{\circ} \) and \( r = 1 \).2. At \( \theta = 90^{\circ} \), the point \( r = 2 \) lies further out from the pole.3. The maximum value occurs at \( \theta = 180^{\circ} \) with \( r = 3 \).4. At \( \theta = 270^{\circ} \), \( r \) returns to 2.5. Complete the cycle as \( \theta \) approaches \( 360^{\circ} \), going back to \( r = 1 \).Join these points in a continuous curve, reflecting the shape of a dimpled limaçon.

Step 5: Confirm Type of Polar Graph

By observing the points and continuous path, it's clear that the graph is a limaçon with a dimple, which is typical for the cases where \( a > b \) in the equation \( r = a - b \cos \theta \).

Key Concepts

Understanding Limaçons

Limaçons are intriguing shapes formed by certain polar equations. A polar equation like \( r = 2 - \cos \theta \) creates a limaçon. These curves can take various forms depending on the relation between the constants \( a \) and \( b \) in the general equation \( r = a - b \cos \theta \) or \( r = a - b \sin \theta \).
One common feature of limaçons is that they can appear as a dimpled shape, have an inner loop, or be convex.

• If \( a > b \), as in our equation with \( a = 2 \) and \( b = 1 \), the limaçon will have a dimple but no inner loop.
• If \( a = b \), you'll get a cardioid shape.
• If \( a < b \), the curve features an inner loop.

These characteristics help in identifying and differentiating various types of limaçons.

Role of Trigonometric Functions

Trigonometric functions like sine and cosine are foundational in polar equations. In the equation \( r = 2 - \cos \theta \), the trigonometric function \( \cos \theta \) dictates how the radius \( r \) changes as the angle \( \theta \) varies.
Cosine function is periodic, ranging from -1 to 1. This periodicity causes the radius \( r \) to vary depending on the angle \( \theta \), leading to the cyclical nature of polar graphs.

• When \( \theta = 0^{\circ} \), \( \cos \theta \) is at its maximum (1), so r is smaller.
• As \( \theta \) approaches \( 180^{\circ} \), \( \cos \theta \) reaches its minimum (-1), maximizing \( r \).
• when \( \theta = 90^{\circ}\) or \( 270^{\circ} \), \( \cos \theta \) is zero, reflecting the mid-range value of \( r \).

Understanding these relationships between \( \theta \) and \( r \) through trigonometric functions is key to predicting and analyzing the shape of polar curves.

Graphing on the Polar Coordinate System

Polar graphing requires plotting points based on their angle and distance from the origin, called the pole. It's different from Cartesian graphing, which uses x and y coordinates.
For the limaçon equation \( r = 2 - \cos \theta \):

• You start by plotting points where \( \theta \) is at key angles, such as \( 0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ}, \) and \( 360^{\circ} \).
• With \( \theta = 0^{\circ} \), the point is closer to the pole because \( r = 1 \).
• At \( \theta = 180^{\circ} \), the point is farthest from the pole with \( r = 3 \).
• As you map these points, draw a smooth curve connecting them to reflect the continuous nature of the limaçon.

This method not only helps in plotting limaçons but also aids in graphing any type of polar equation with precision and visual clarity.