Problem 41 - Precalculus : Building Concepts and Connections

Question

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=4-3 \cos \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r = 4 - 3 \cos \theta\) forms a limaçon without a loop.

Step 1: Understand polar coordinates' system

Rewrite the equation \(r = 4 - 3 \cos \theta\) to understand how changes in \(\theta\) will affect \(r\). The polar coordinate system allows us to represent points based on a distance \(r\) and an angle \(\theta\) from the origin rather than traditional Cartesian coordinates \(x, y\).

Step 2: Identify graph's type

Polar equations of the form \(r = a ± b cos θ\) or \(r = a ± b sin θ\), where \(a\) and \(b\) are nonzero constants, generate 'limaçon' type curves. In this case, \(r = 4 - 3 cos θ\) will form a limaçon. Its form will depend on the relationship between the constants.

Step 3: Understand the impact of the constants

Comparing the constants \(a = 4\) and \(b = 3\) we see that \(a > b\). In this case, the limaçon will not have a loop.

Step 4: Calculate and plot key points

A few key points to plot: When \(\theta = 0, r = 4 - 3 \cos 0 = 1\). When \(\theta = \pi / 2, r = 4 - 3 \cos \pi / 2 = 4\). When \(\theta = \pi, r = 4 - 3 \cos \pi = 7\). When \(\theta = 3 \pi / 2, r = 4 - 3 \cos 3\pi / 2 = 4\).

Step 5: Sketch the graph

Plot the computed points in a polar coordinate system. The plot will be a smooth curve that resembles a distorted circle, indicating a limaçon without a loop.

Key Concepts

Polar Coordinates

Imagine trying to tell someone where you are in a vast, empty desert with no landmarks. You could say you're a certain distance from a point you both know and in a certain direction—this is essentially how polar coordinates work. In polar coordinates, we describe the location of a point based on its distance from a central point (the pole) and the angle it forms with a reference direction (usually the positive x-axis).

The formula for a polar coordinate is written as \(r, \theta\), where \(r\) is the radius or the distance from the pole, and \(\theta\) is the angle in radians measured counterclockwise from the reference direction. This system is particularly useful for describing curves that are naturally circular or spiral, which can be trickier to represent with traditional Cartesian coordinates \(x,y\).

For example, the given exercise uses polar coordinates to define a curve. Rewriting the equation to \(r = 4 - 3 \cos \theta\) allows us to plot points corresponding to various angles \(\theta\) by calculating the distance \(r\) from the origin.

Limaçon Curves

Limaçon curves are like the whimsical cousins of circles in the polar coordinate family, displaying shapes that range from heart forms to elongated loops. These curves fall into the category of polar graphs known as conics, each with its unique properties.

A limaçon is described by the polar equation \(r = a ± b \cos \theta\) or \(r = a ± b \sin \theta\), where \(a\) and \(b\) are real numbers that define the shape's specific characteristics. When \(b\) is less than \(a\), the limaçon will not have a loop and will look more like a distorted circle; when \(b\) is greater than \(a\), the limaçon will have a loop. Hence, the relationship between \(a\) and \(b\) is crucial in determining the overall appearance of the curve.

For the exercise \(r = 4 - 3 \cos \theta\), since \(a = 4\) is greater than \(b = 3\), the resulting graph is a limaçon without a loop. This can be visualized as a circle that's been pushed inward on one side.

Polar Graphs

Polar graphs bring the equations tied to polar coordinates to life. They serve as a visual language for understanding complex shapes and movements that would be difficult to describe otherwise. To sketch a graph for a polar equation like \(r = 4 - 3 \cos \theta\), we start by plotting key points based on specified angle values. These points act as guideposts, building the framework for our graph.

The graph of our given equation will feature constant shifts as \(\theta\) changes, forming a unique curve on the polar coordinate plane. To sketch it, you'd mark key points such as when \(\theta = 0\), \(\theta = \pi/2\), \(\theta = \pi\), and \(\theta = 3\pi/2\). These allow you to assess the behavior of the equation at significant intervals and begin to visualize the curve's progression.

Plotting Key Points and Sketching

For instance, when \(\theta = 0\), we find \(r = 1\), which places our first point a unit distance from the origin directly to the right. At \(\theta = \pi/2\), \(r = 4\), positioning another point four units directly above the origin. You'd continue this process for other key angles, gradually connecting these points with a smooth curve until the full limaçon is revealed—a practice marbling precision with intuition.