Problem 34 - Calculus Early Transcendentals

Question

\(29-48\) Sketch the curve with the given polar equation. $$r=1-3 \cos \theta$$

Step-by-Step Solution

Answer

The polar curve described by \(r = 1 - 3 \cos \theta\) is a limaçon with an inner loop.

Step 1: Understand Polar Equations

In polar coordinates, each point on the plane is defined by a distance, \(r\), from the origin and an angle, \(\theta\), from the positive x-axis. The equation \(r = 1 - 3 \cos \theta\) describes a specific type of polar curve.

Step 2: Identify the Type of Curve

The given equation, \(r = 1 - 3 \cos \theta\), is of the form \(r = a + b \cos \theta\). This is a limaçon. Specifically, when \(b > a\), the curve has an inner loop.

Step 3: Determine Key Values and Plot Points

Calculate points by substituting different values of \(\theta\) from 0 to \(2\pi\) into the equation:- \(\theta = 0 : r = 1 - 3 \cdot 1 = -2\)- \(\theta = \pi/2 : r = 1 - 3 \cdot 0 = 1\)- \(\theta = \pi : r = 1 - 3 (-1) = 4\)- \(\theta = 3\pi/2 : r = 1 - 3 \cdot 0 = 1\)Plot these points and find additional points by calculating \(r\) for other \(\theta\) values.

Step 4: Sketch the Curve

Use the key points and symmetry of polar graphs to sketch the limaçon. Start at the inner loop and sketch the continuous curve around these points, ensuring the curve reflects the negative values where necessary, which represent points on the opposite side of the origin.

Step 5: Verify Features of the Limaçon

Check the curve for an inner loop, as indicated by the condition \(b > a\). The lowest value of \(r\) (most negative), when \(\theta = 0\), reflects the presence of this loop. Ensure the curve aligns with the limaçon characteristics.

Key Concepts

Polar Equation

Polar equations offer a unique way to represent curves on a plane using polar coordinates. These differ from the Cartesian coordinate system, where we describe locations using (x, y) pairs.
In polar coordinates, the position of a point is determined by:

• the distance, represented by \(r\), from the origin, known as the pole,
• and the angle, \(\theta\), measured from the positive x-axis.

For the equation \(r = 1 - 3 \cos \theta\), it defines how the distance \(r\) varies as the angle \(\theta\) changes. Such equations help us explore polar curves' symmetry and shapes.
This particular equation forms a distinct shape and belongs to a family of curves based on a general formula that can describe circles, spirals, and other intriguing shapes based on the interplay between \(r\) and \(\theta\).

Limaçon

A limaçon is a special type of polar curve that takes the form \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\). It is named after the French word for "snail" due to its distinctive shape. Limaçons can exhibit a variety of forms depending on the relative values of \(a\) and \(b\):

• If \(b < a\), the limaçon is a dimpled curve.
• If \(b = a\), a limaçon becomes a cardioid, a heart-like shape.
• When \(b > a\), like in our example \(r = 1 - 3 \cos \theta\), the limaçon features an inner loop.

This inner loop occurs because the negative values for \(r\) signify points that loop back toward the origin on the opposite side. Understanding these basic variations in the limaçon's form is essential for identifying and sketching these curves accurately on a polar graph.

Graphing Polar Curves

Graphing polar curves involves plotting points calculated from the polar equation, like the example \(r = 1 - 3 \cos \theta\). Follow these steps for a clear graph:

• Identify points by substituting values for \(\theta\) in the equation, ranging from \(0\) to \(2\pi\).
• For \(\theta = 0\), calculate \(r\) to find points like \((-2, 0)\); for \(\theta = \pi/2\), \(r = 1\), giving \((1, \pi/2)\) and so on.
• Keep in mind symmetry. Polar curves often repeat or mirror themselves, easing your task of predicting additional points.
• Carefully join these points, respecting the inner loop if present, as they determine the path the curve takes.

By sketching these shapes, students gain hands-on insight into characteristics like loops and symmetries of polar curves. It becomes visible how the interplay of \(a\) and \(b\) in a limaçon reflects directly in the visual representation.