Problem 68 - Precalculus

Question

Many microphone manufacturers advertise that their microphones' exceptional pickup capabilities isolate the sound source and minimize background noise. These microphones are described as cardioid microphones because of the pattern formed by the range of the pickup. Graph the cardioid curve \(r=-4-4 \sin \theta\) to see what the range looks like.

Step-by-Step Solution

Answer

The cardioid curve for the equation is symmetric and shaped like an upside-down heart, with a loop on the left side.

Step 1: Interpret the Equation

The given polar equation is \( r = -4 - 4 \sin \theta \). This defines a cardioid, which is a specific type of polar curve. The term \(-4 \sin \theta\) affects how the curve is plotted with respect to the angle \( \theta \).

Step 2: Set up the Polar Axes

To graph this cardioid, start by setting up polar axes on a coordinate plane. The center of the polar graph is the pole, and the angles \( \theta \) increase counterclockwise from the positive x-axis.

Step 3: Plot Key Points

Choose various values for \( \theta \) (e.g., \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \ldots, 2\pi\)) and calculate the corresponding \(r\) values using the equation \( r = -4 - 4 \sin \theta \). For example:- When \( \theta = 0 \), \( r = -4 \).- When \( \theta = \frac{\pi}{2} \), \( r = -8 \).- When \( \theta = \pi \), \( r = -4 \).- When \( \theta = \frac{3\pi}{2} \), \( r = 0 \).

Step 4: Draw the Cardioid

Using the points plotted from the previous step, connect them smoothly to draw the complete cardioid shape. Notice the loop toward the left of the pole due to negative \(r\) values when \( \theta \) is close to \( \pi \) and \( 0 \). This shape resembles an upside-down heart, typical for cardioids with negative coefficients.

Step 5: Check Symmetry

The graph \( r = -4 - 4 \sin \theta \) should be symmetric about the horizontal line through the pole because it's an even function in terms of \(r\). Verify this symmetry by reflecting points across this line.

Key Concepts

Cardioid Curve

A cardioid is a heart-shaped curve that appears in the realm of polar graphing. Specifically characterized by the equation form \( r = a - a \sin \theta \) or \( r = a + a \sin \theta \), the cardioid gets its name from the Greek word "cardio," meaning heart. The unique shape is due to one loop of the curve turning inward.

In this particular exercise, we explore a cardioid described by the equation \( r = -4 - 4 \sin \theta \). Notice the negative coefficients "-4" and "-4 \sin \theta", which lead to the cardioid being directed downward. This form generates a cardioid with a cusp pointing away from the pole or origin. Cardioid curves are not only significant in mathematics but have practical applications too, often used to describe the pickup patterns of certain types of microphones.

Polar Coordinates

Polar coordinates provide a way of representing points on a plane using a radius and angle, known as \( r \) and \( \theta \), respectively. While Cartesian coordinates use \( x \) and \( y \) coordinates to describe a point, polar coordinates give us the distance from the origin (known as the pole) and the angle from a fixed direction (usually the positive x-axis).

The benefits of using polar coordinates become apparent when dealing with curves like the cardioid, which involve circular or rotational elements. With polar coordinates:

• \( r \) represents the radius or the length of the line segment from the origin to the point.
• \( \theta \) (theta) is the angular measurement from the polar axis.

For instance, in the cardioid equation \( r = -4 - 4 \sin \theta \), \( r \) changes based on the sine of the angle \( \theta \). This dynamic reflects how far each point is from the center as the angle varies.

Graphing Techniques

Graphing in polar coordinates requires understanding how to translate angles and radial distances into positions on a plane. To accurately plot a cardioid:

• Set up a polar grid, dividing the plane into sectors based on angles, typically starting from the positive x-axis moving counterclockwise.
• Calculate \( r \) for a series of angles \( \theta \) to find location points. It's good practice to choose angles like \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \ldots, 2\pi \).
• Plot these (\( r, \theta \)) as coordinates where the angle gives direction and \( r \) provides the magnitude of distance from the pole.
• Connect these points smoothly, showing the full cardioid shape.

Negative \( r \) values flip the point to the opposite direction of the initial angle, useful in comprehending the full cardioid structure. This approach ensures that you capture the essential heart-like curve.

Symmetry in Graphs

Symmetry is a key feature in understanding and verifying polar graphs like the cardioid. Our cardioid \( r = -4 - 4 \sin \theta \) shows symmetry about the horizontal axis through the pole, meaning that the graph on one side mirrors the other side.

Symmetry can simplify graphing by reducing the points you need to calculate and plot. For the cardioid, this symmetry indicates an even distribution of points around the symmetric axis. Checking symmetry involves visually inspecting the graph to confirm reflective similarity across the expected line or axis. If you find symmetry, it ensures that the graph is not skewed or misaligned, and all points are accurately reflected in the correct location.

In polar graphs, examining for symmetry can include checking how \( r \) behaves when \( \theta \) changes. The sine function in \( r = -4 - 4 \sin \theta \) hints at straightforward symmetry about horizontal lines through its periodic behavior and array of values that each have counterparts across this line.