Problem 50 - Algebra and Trigonometry

Question

Identify and graph each polar equation. $$ r=2+4 \cos \theta $$

Step-by-Step Solution

Answer

The graph is a limaçon with an inner loop.

Step 1: Understand the polar equation

The given polar equation is \( r = 2 + 4 \cos \theta \). This equation defines the radius \( r \) in terms of the angle \( \theta \) and suggests it is a limaçon with a cosine term.

Step 2: Identify the type of limaçon

A polar equation of the form \( r = a + b \cos \theta \) represents a limaçon. Here, \( a = 2 \) and \( b = 4 \). Since \( b > a \), the limaçon has an inner loop.

Step 3: Determine key points

To graph, determine key points by plugging in specific values of \( \theta \). For example:- For \( \theta = 0 \), \( r = 2 + 4 \cos 0 = 6 \)- For \( \theta = \frac{\pi}{2} \), \( r = 2 + 4 \cos \frac{\pi}{2} = 2 \)- For \( \theta = \pi \), \( r = 2 + 4 \cos \pi = -2 \)- For \( \theta = \frac{3\pi}{2} \), \( r = 2 + 4 \cos \frac{3\pi}{2} = 2 \)

Step 4: Sketch the graph

Plot the points obtained in the previous step and connect them in the order of increasing angle \( \theta \). Recognize the shape as a limaçon with an inner loop. The inner loop indicates negative values of \( r \), so adjustments must be made accordingly.

Step 5: Verify the graph

Double-check the plotted points and the shape to ensure it matches the characteristics of a limaçon with an inner loop. Ensure that the loop is appropriately placed and that the outer points are correctly plotted.

Key Concepts

Polar Coordinates

Polar coordinates are a way to describe locations in a plane using a distance and an angle. Unlike Cartesian coordinates, which use an \(x\) and a \(y\) axis, polar coordinates use:

• \(r\) - the radial distance from the origin (center of the plane).
• \(\theta\) - the angle from the positive \(x\) axis (measured counterclockwise).

These coordinates are very useful for dealing with curves and shapes that have a circular symmetry. For example, in our exercise, we'll deal with a type of shape called a limaçon, described using polar coordinates. This makes it easier to understand and plot complex shapes, especially those involving circles or spirals.

Limaçon

A limaçon is a type of curve with interesting properties, often studied in polar coordinates. It is defined by equations of the form \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\).
In our case, the equation given is \(r = 2 + 4 \cos \theta\), which tells us:

• \(a = 2\)
• \(b = 4\)

Because \(b > a\), this limaçon has an inner loop. The loop feature occurs because at certain angles, \(r\) becomes negative, indicating points on the opposite side. To visually identify these features, we plot key points and look at changes in \(r\) as \(\theta\) changes.
Understanding these values helps in sketching the shape accurately, revealing both the inner loop and the general outward appearance of the limaçon.

Trigonometric Functions

Trigonometric functions like \(\cos \theta\) and \(\sin \theta\) are central when dealing with polar coordinates. In our polar equation \(r = 2 + 4 \cos \theta\):

• \(\cos \theta\) affects the distance \(r\) from the origin.
• As \(\theta\) changes, \(\cos \theta\) oscillates between \(-1\) and \(1\).
• These oscillations modify \(r\) and result in rising and falling shapes.

For example, when \(\theta = 0\), \(\cos \theta\) is maximum (\