Problem 28 - The Calculus with Analytic Geometry

Question

Draw a sketch of the graph of the given equation.\(r=3-4 \cos \theta\) (limaçon)

Step-by-Step Solution

Answer

Start with \(r=3-4\cos(\theta)\) in polar form, plot key points for \(\theta=0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\), then draw the limaçon shape with an inner loop.

Step 1: Understand the Equation

The given equation is in polar form, where the radius, r, is expressed as a function of the angle, \(\theta\). Specifically, \(r=3-4 \cos(\theta)\) describes a type of limaçon. Note that the form is similar to \(r=a-b \cos(\theta)\).

Step 2: Identify Key Features

Identify the parameters a and b in the equation. In this case, \(a=3\) and \(b=4\). Since \(a < b\), the limaçon will have an inner loop.

Step 3: Determine Important Points

To sketch the graph, calculate values of \(r\) for important angles \(\theta\). Consider \(\theta=0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\). For example, when \(\theta=0, r=3-4\cos(0)=3-4=-1\); \(\theta=\pi, r=3-4\cos(\pi)=3-4(-1)=7\); and so forth.

Step 4: Plot Points

Plot the calculated points from the previous step on polar coordinates. For each angle \(\theta\), use the corresponding value of \(r\) to plot the points (e.g., for \(\theta=0\), plot \((r,\theta)=(-1,0)\)).

Step 5: Draw the Sketch

Use the plotted points and knowledge of the limaçon shape to draw the curve. Start at \(\theta=0, r=-1\), curve outwards, looping inward for the characteristic inner loop, then back out. Ensure symmetry as \(r(\theta)\) will repeat periodically.

Key Concepts

polar coordinates

Polar coordinates allow us to describe the location of a point in a plane using the distance from a reference point and an angle from a reference direction. Unlike Cartesian coordinates (x, y), which use horizontal and vertical distances, polar coordinates use:

• r: the distance from the origin (also called the 'pole')
• θ (theta): the angle measured from the positive x-axis (also called the 'initial line')

Any point in a plane can be represented as (r, θ). The angle θ is typically measured in radians for mathematical convenience, where 2π radians corresponds to a full circle (360 degrees). Understanding polar coordinates is essential for graphing in this system, particularly when equations describe complex shapes like the limaçon.

graphing polar equations

Graphing polar equations involves plotting a set of points defined by their polar coordinates (r, θ). Here's a simplified method to graph polar equations:

• Identify the form of the equation (e.g., r=a+bcos(θ)).
• Calculate values for r at specific angles θ (such as 0, π/2, π, 3π/2).
• Plot the points (r, θ) on polar graph paper or using polar coordinate plotting tools.
• Connect the points smoothly, considering the known shape of the graph (e.g., a circle, spiral, or limaçon).

For the equation r=3-4cos(θ), plotting begins with calculating r for basic angles:

• At θ=0: r=3-4= -1
• At θ=π/2: r=3-4cos(π/2)=3-4(0)=3,
• At θ=π: r=3-4cos(π)=3-4(-1)=7
, etc.

These points (and others if necessary) are then plotted on the polar coordinate plane and connected to sketch the complete curve.

limaçon curve

The limaçon curve, given by equations of the form r=a±bcos(θ) or r=a±bsin(θ), produces distinct shapes based on the relationship between a and b:

• When a > b: The curve resembles a dimpled limaçon.
• When a = b: The curve takes the shape of a cardioid (heart-shaped).
• When a < b: The curve has an inner loop.

In our example, the equation is r=3-4cos(θ) with a=3 and b=4. Here, a < b, indicating an inner loop in the resulting graph. By plotting key points and drawing the curve as described, students observe this unique inner loop shape, characterizing the curve’s distinct look. Understanding the limaçon’s modifications based on its equation parameters helps students visualize and graph these intriguing shapes accurately.