Question

Identify and graph each polar equation. $$ r=1-\cos \theta $$

Short answer

The graph is a limaçon with an inner loop.

Step-by-step solution

- Understand the Polar Equation

The given equation is in polar form: \[ r = 1 - \cos \theta \].In polar coordinates, \(r\) represents the radius, or the distance from the origin, and \(\theta\) is the angle from the positive x-axis.

- Identify the Type of Graph

Recognize the structure of the equation. The format is similar to the equation of a limaçon. In general, \( r = a + b \cos \theta \) produces different forms of limaçons depending on the values of \(a\) and \(b\).

- Evaluate Specific Points

To accurately plot the graph, calculate the value of \(r\) at various angles \(\theta\):- For \(\theta = 0\), \( r = 1 - 1 = 0 \)- For \(\theta = \pi/2\), \( r = 1 - 0 = 1 \)- For \(\theta = \pi\), \( r = 1 + 1 = 2 \)- For \(\theta = 3\pi/2\), \( r = 1 - 0 = 1 \)

- Plot the Points

Mark the points on polar graph paper: \((0, 0)\), \((1, \pi/2)\), \((2, \pi)\), and \((1, 3\pi/2)\).

- Draw the Limaçon

Connect the plotted points smoothly. The graph should appear as a limaçon with an inner loop.

Key concepts

polar coordinates

In polar coordinates, we describe points on a plane using a radius and an angle instead of traditional Cartesian coordinates (x, y). The radius, denoted as \( r \), measures how far a point is from the origin (the center of the graph). The angle, \( \theta \), is measured from the positive x-axis in a counterclockwise direction. For example:

• If \( r = 3 \) and \( \theta = \frac{\pi}{4} \), the point is 3 units from the origin and at a 45-degree angle from the positive x-axis.
• If \( r = -2 \) and \( \theta = \pi \), the point is 2 units from the origin in the opposite direction of the positive x-axis because of the negative radius.

Understanding polar coordinates is crucial when working with polar equations like \( r = 1 - \cos \theta\). It's different from Cartesian because it focuses on distance and direction instead of just vertical and horizontal positioning.

limaçon

The equation \( r = 1 - \cos \theta \) is a special type of graph called a limaçon. In mathematics, a limaçon (pronounced lee-mah-SAWN) is a type of polar curve. It is generally described by the equation \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). This specific limaçon can have different shapes:

• If \( a < b \), the limaçon has an inner loop.
• If \( a = b \), the limaçon has a dimple on one side, known as a cardioid.
• If \( a > b \), the limaçon looks more like a distorted circle without inner loops.

For our example \( r = 1 - \cos \theta \), we are in the first scenario (\( a < b \)), so there is an inner loop present.

graphing polar equations

When graphing polar equations, it helps to understand both the distance from the origin (\( r \)) and the angle (\( \theta \)). The general steps are:
1. Identify the type of equation you’re working with (e.g., limaçon, rose curve, circle).
2. Calculate \( r \) for specific angles \( \theta \). It’s helpful to choose common angles like 0, \( \frac{\pi}{2} \), \( \pi \), and \( \frac{3\pi}{2} \).
3. Plot the computed points on polar graph paper.
4. Connect the points smoothly, understanding the nature of the curve you're drawing.
In \( r = 1 - \cos \theta \), we can see the importance of identifying key points to reveal the limaçon shape. Graphing takes practice, but breaking it down makes it easier.

cosine function

The cosine function, written as \( \cos \theta \), is an important trigonometric function. It returns the x-coordinate of a point on the unit circle at angle \( \theta \). Here’s what you should know for polar equations:

• \( \cos(0) = 1 \)
• \( \cos(\frac{\pi}{2}) = 0 \)
• \( \cos(\pi) = -1 \)
• \( \cos(\frac{3\pi}{2}) = 0 \)

With those values, you can see how \( r = 1 - \cos \theta \) will change:

• At \( \theta = 0 \), \( r = 0 \)
• At \( \theta = \frac{\pi}{2} \), \( r = 1 \)
• At \( \theta = \pi \), \( r = 2 \)
• At \( \theta = \frac{3\pi}{2} \), \( r = 1 \)

These points help in actually plotting the curve.

plotting points

Once you have computed the values of \( r \) at specific angles \( \theta \), you can plot these points to graph the polar equation. Here's how to do it:
1. Use polar graph paper, which has circles representing different radii and lines radiating from the center to represent angles.
2. Mark points at the intersection of the appropriate radius and angle.
3. For \( r = 1 - \cos \theta \):

• When \( \theta = 0 \), plot (0, 0).
• When \( \theta = \frac{\pi}{2} \), plot (1, \( \frac{\pi}{2} \)).
• When \( \theta = \pi \), plot (2, \( \pi \)).
• When \( \theta = \frac{3\pi}{2} \), plot (1, \( \frac{3\pi}{2} \)).

Connect these points smoothly to draw the exact shape of the limaçon. Practice makes perfect, and these steps will help deepen your understanding of how the curve forms.