Problem 25 - The Calculus with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The graph of the cardioid \( r=4-4 \cos \theta \) is heart-shaped, starting at the origin, reaching 8 units at \( \theta = \pi \), and returning to the origin at \( \theta = 2 \pi \).

Step 1 - Understand the Equation

The given equation is in the polar form: \[ r = 4 - 4 \, \cos \, \theta \] This represents a cardioid, which is a heart-shaped curve.

Step 2 - Determine Key Points

Calculate the values of \( r \) for key angles \( \theta \): 1. \( \theta = 0 \): \( r = 4 - 4 \cos 0 = 0 \) 2. \( \theta = \frac{\pi}{2} \): \( r = 4 - 4 \cos \frac{\pi}{2} = 4 \) 3. \( \theta = \pi \): \( r = 4 - 4 \cos \pi = 8 \) 4. \( \theta = \frac{3\pi}{2} \): \( r = 4 - 4 \cos \frac{3\pi}{2} = 4 \)

Step 3 - Plot Key Points

On polar graph paper, plot the calculated points: 1. At \( \theta = 0 \), radius \( r = 0 \). 2. At \( \theta = \frac{\pi}{2} \), radius \( r = 4 \). 3. At \( \theta = \pi \), radius \( r = 8 \). 4. At \( \theta = \frac{3\pi}{2} \), radius \( r = 4 \).

Step 4 - Draw the Graph

Connect the plotted points smoothly, considering the shape of the cardioid. Note that the graph should come back to the origin at \( \theta = 2\pi \).

Key Concepts

cardioid

A cardioid is a heart-shaped curve that appears frequently in polar coordinates. The term 'cardioid' originates from the Greek word for heart (\(\text{καρδία}\)). In mathematics, a cardioid is described by polar equations where the radius depends on the trigonometric functions of the angle. Notably, a cardioid has the key property of being symmetric about the horizontal axis when the equation is in the form \(r = a + a \, \text{cos} \, \theta \). This curve turns up in many real-world applications, including acoustics and antenna design, due to its unique reflective properties.

polar equations

Polar equations represent curves using the polar coordinate system, where each point on a plane is determined by an angle and a distance from a reference point (the pole). For the given equation \(r = 4 - 4 \, \text{cos} \, \theta\), the radius \(r\) changes as the angle \(\theta\) changes. It's important to understand that instead of representing a curve using \(x\) and \(y\) coordinates (Cartesian coordinates), polar equations relate the radius directly to the angle. This approach can simplify the understanding and sketching of certain curves, such as cardioids, which may be more complex in Cartesian form.

plotting key points

Plotting key points involves determining specific values of \(r\) at significant angles \(\theta\), which help in sketching the graph accurately. For the cardioid \(r = 4 - 4 \, \text{cos} \, \theta\):

• At \(\theta = 0\), \(r = 4 - 4 \, \text{cos}0 = 0\)
• At \(\theta = \frac{\pi}{2}\), \(r = 4 - 4 \, \text{cos} \frac{\pi}{2} = 4\)
• At \(\theta = \pi\), \(r = 4 - 4 \, \text{cos} \pi = 8\)
• At \(\theta = \frac{3\pi}{2}\), \(r = 4 - 4 \, \text{cos} \frac{3\pi}{2}= 4\)

By calculating and plotting these points on polar graph paper, you provide a framework for drawing the curve. These key points inform how the cardioid will look and guide the smooth connection of points to visualize the overall shape.

graphing curves

Graphing curves in polar coordinates involves connecting the key points calculated from the polar equation. For the cardioid \(r = 4 - 4 \, \text{cos} \, \theta\), you'll plot the key points:

• \(\theta = 0\), \(r = 0\)
• \(\theta = \frac{\pi}{2}\), \(r = 4\)
• \(\theta = \pi\), \(r = 8\)
• \(\theta = \frac{3\pi}{2}\), \(r = 4\)

After plotting, connect these points smoothly, ensuring the correct shape. Notice how the curve enters and exits the origin. Polar graphing can seem tricky, but it often simplifies complex relationships seen in Cartesian coordinates. Always consider the symmetry and cyclical nature of polar curves when graphing.