Problem 67 - 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 \(r=2+2 \sin \theta\) to see what the range looks like.

Step-by-Step Solution

Answer

The cardioid curve \(r = 2 + 2 \sin \theta\) forms a heart-shaped pattern, illustrating a sound pickup range that extends farthest at \(\theta = \pi/2\) and is narrowest near the pole.

Step 1: Understanding the Cardioid Equation

The cardioid is a type of polar curve represented by the equation \(r = a + a \sin \theta\), where \(a\) is a constant. In this problem, the cardioid equation is \(r = 2 + 2 \sin \theta\), which means \(a = 2\). This is a standard cardioid centered at the pole.

Step 2: Determine Values of \(\theta\)

To graph the cardioid, we need to determine the values of \(r\) for various \(\theta\) values. Since this is a periodic function, consider values of \(\theta\) from 0 to \(2 \pi\). Start by calculating \(r\) for key angles such as 0, \(\pi/2\), \(\pi\), \(3\pi/2\), and \(2\pi\).

Step 3: Evaluate \(r\) at Key Angles

Calculate \(r\) for each key angle:- For \(\theta = 0\): \(r = 2 + 2 \cdot 0 = 2\).- For \(\theta = \pi/2\): \(r = 2 + 2 \cdot 1 = 4\).- For \(\theta = \pi\): \(r = 2 + 2 \cdot 0 = 2\).- For \(\theta = 3\pi/2\): \(r = 2 + 2 \cdot (-1) = 0\).- For \(\theta = 2\pi\): \(r = 2 + 2 \cdot 0 = 2\).

Step 4: Plotting the Curve

Using the values of \(r\) calculated, plot the points corresponding to each \(\theta\). Connect these points smoothly to form the cardioid curve. You should see the curve touch the pole at \(\theta = 3\pi/2\) and extend outward the farthest at \(\theta = \pi/2\). The curve is symmetric about the vertical axis due to the \(\sin\) function.

Step 5: Finalizing the Graph

Upon plotting points and drawing the curve, the cardioid shape should be evident. The range of the curve extends from 0 to 4, which corresponds with the sound pickup range of a microphone pattern, highlighting the typical directional sensitivity of cardioid microphones.

Key Concepts

Cardioid Curve

A cardioid curve is a specific type of polar curve known for its distinctive heart shape. The term "cardioid" stems from the Greek word for heart, which is apt given its appearance. Cardioids are defined mathematically by equations of the form \(r = a + a \sin \theta\) or \(r = a + a \cos \theta\), where \(a\) is a constant that determines the size of the curve. In the case of \(r = 2 + 2 \sin \theta\), the cardioid is symmetric about the vertical axis.

Key characteristics of a cardioid:

• Centred at the pole (origin in polar coordinates).
• Symmetric about the axis depending on whether the sine or cosine function is used.
• The cardioid in this form has a single loop.

One fascinating property is that a cardioid appears naturally in many fields such as acoustics, especially in the design of microphones that emphasize sound from a particular direction, reflecting its characteristic pickup pattern.

Polar Equations

Polar equations are used to express curves on a plane using polar coordinates, which are defined by a distance from the origin and an angle from a reference direction. Instead of the usual Cartesian coordinates (\(x, y\)), polar coordinates (\(r, \theta\)) specialize in representing points based on their radial distance \(r\) from the origin and angular direction \(\theta\) from the positive x-axis.

Advantages of polar equations include:

• The ability to describe circular and spiral shapes more naturally than with Cartesian systems.
• Simplified expression of periodic phenomena, as seen in waves or rotations.

Polar equations such as \(r = a + a \sin \theta\) lend themselves elegantly to circular curves like the cardioid, allowing for visualization directly related to the angles involved. These equations help us understand how the measurement changes based on angle, providing a robust framework for analyzing directional data.

Graphing Polar Curves

Graphing polar curves involves plotting points in polar coordinates and connecting them to visualize the shape of the curve. The process starts by selecting various values of \(\theta\), often ranging from 0 to \(2 \pi\), and calculating corresponding \(r\) values from the polar equation. This allows us to plot (\(r, \theta\)) pairs on a polar grid.

Steps to graph polar curves include:

• Selecting key angle values that highlight the characteristics of the curve.
• Calculating the radial distance \(r\) for each \(\theta\).
• Plotting these points and smoothly connecting them to see the overall shape.

The curve \(r = 2 + 2 \sin \theta\) illustrates a cardioid that is symmetric and touches the pole at \(\theta = \frac{3\pi}{2}\). Points corresponding to maximum and minimum distances (\(r\)) greatly help in outlining the overall form of the cardioid. The graphing process provides visual insight into interesting features like symmetry and the range of values across angles, essential for understanding the nature of polar curves.