Problem 13 - Precalculus

Question

Graph the polar equations. $$r=2-2 \sin \theta$$

Step-by-Step Solution

Answer

The graph of \( r = 2 - 2 \sin \theta \) is a cardioid.

Step 1: Understanding Polar Coordinates

Polar coordinates represent points in a plane by an angle and a radius. Here, \( r \) is the radius, and \( \theta \) is the angle (in radians) from the positive x-axis. The equation \( r = 2 - 2 \sin \theta \) defines the radius as a function of the angle \( \theta \).

Step 2: Identifying the Type of Graph

The given equation \( r = 2 - 2 \sin \theta \) often represents a limacon. Based on the form \( r = a - b \sin \theta \), where \( a = 2 \) and \( b = 2 \), the graph will be a cardioid because \( a = b \).

Step 3: Finding Key Points

To graph the equation, calculate \( r \) for key values of \( \theta \). For example, when \( \theta = 0 \), \( \sin 0 = 0 \), so \( r = 2 \). When \( \theta = \frac{\pi}{2} \), \( \sin \frac{\pi}{2} = 1 \), hence \( r = 0 \).

Step 4: Plotting the Points

Plot the points corresponding to the calculated \( r \) for given \( \theta \) values on polar axis. Extend calculations for points like \( \theta = \pi, \theta = \frac{3\pi}{2} \) and so on.

Step 5: Drawing the Graph

Join the plotted points smoothly, considering the symmetry. The graph will form a cardioid which looks somewhat like a heart, starting from the pole, looping outwards, and returning to the pole.

Key Concepts

Polar Equations

Polar equations are a way to define curves on a two-dimensional plane using a radius and an angle. This approach is different from the Cartesian coordinate system, which uses x and y coordinates. In polar coordinates, each point is defined by:

• Radius (\( r \)) - the distance from the origin (pole).
• Angle (\( \theta \)) - the angular direction from the positive x-axis.

For example, the equation \( r = 2 - 2 \sin \theta \) describes a curve based on how the radius changes with the angle. As \( \theta \) takes on different values, \( r \) will vary accordingly, helping us trace out the curve.

Graphing Polar Curves

Graphing polar curves involves plotting points on a polar coordinate grid by calculating values for \( r \) for various angles \( \theta \). Here are the steps:

• Start by choosing key angles such as \( 0, \frac{\pi}{2}, \pi, \text{and} \frac{3\pi}{2} \).
• Calculate \( r \) for each angle using your polar equation.
• Plot each calculated point on a polar grid where \( \theta \) is the angle line, and \( r \) is the distance from the origin.

By connecting these points smoothly, you'll see the shape of the curve emerge. This visual representation helps in understanding the behavior of polar functions.

Cardioid Graph

A cardioid is a specific type of polar curve that resembles the shape of a heart. Polar equations of the form \( r = a - b \sin \theta \) or \( r = a - b \cos \theta \) often generate this shape. When \( a = b \), the equation describes a perfect cardioid.
In our example, \( r = 2 - 2 \sin \theta \) fulfills this condition because \( a = b = 2 \). This means the curve will:

• Begin at a point on the polar axis, loop outward, creating a cusp at the pole,
• and return back to the starting point.

Cardioids are interesting because they demonstrate symmetry and a unique closed-loop pattern that is easily identified.

Understanding Polar Coordinates

Understanding polar coordinates is crucial for navigating problems involving circular and spiraled shapes. Unlike Cartesian coordinates, polar coordinates provide a different perspective:

• Origin-Centric: Points revolve around an origin, making it suitable for circular patterns.
• Angle and Radius: Defining points by angle (\( \theta \)) and distance (\( r \)) offers unique insights into radial symmetry and rotations.

This system is particularly useful in fields such as physics and engineering, where motion and waves are often described in circular terms. Polar coordinates offer a straightforward approach to understanding and visualizing rotational systems.