Problem 33 - Calculus Early Transcendentals

Question

\(29-48\) Sketch the curve with the given polar equation. $$r=2(1-\sin \theta), \theta \geqslant 0$$

Step-by-Step Solution

Answer

The curve is a cardioid with a cusp at the origin and opens downward.

Step 1: Understanding the Polar Equation

We start with the given polar equation \( r = 2(1 - \sin \theta) \). This type of equation is in the form of a Limaçon, which typically takes the form \( r = a + b \cdot \sin \theta \) or \( r = a + b \cdot \cos \theta \). Here, \( a = 2 \) and \( b = -2 \).

Step 2: Identifying Key Features

For \( r = a + b \cdot \sin \theta \), the Limaçon can take different shapes depending on the relationship between \( a \) and \( b \). Since \( a = b \), this particular equation represents a cardioid without an inner loop. For \( r = 2(1 - \sin \theta) \), we expect a heart-shaped curve.

Step 3: Creating a Table of Values

To sketch the cardioid, calculate \( r \) values for assorted \( \theta \). Starting with \( \theta = 0 \), \( r = 2(1-\sin 0) = 2 \). Continuing, calculate for \( \theta = \frac{\pi}{2} \) where \( \sin \frac{\pi}{2} = 1\), thus \( r = 2(1-1) = 0 \). Similarly, compute values for \( \theta = \pi, \frac{3\pi}{2}, 2\pi \).

Step 4: Plotting the Points

Using the table of values, plot the points calculated in polar coordinates. For instance, when \( \theta = 0 \), plot the point at \( (2, 0) \). At \( \theta = \frac{\pi}{2} \), plot the point at the origin \( (0, \frac{\pi}{2}) \). Repeat for other values: \( (2, \pi), (4, 3\pi/2), (2, 2\pi) \).

Step 5: Drawing the Curve

Connect the plotted points, ensuring the curve passes smoothly through each, mimicking the cardioid shape. The graph will have a cusp at the origin when \( \theta = \frac{\pi}{2} \), looping outwards and forming the heart shape typical of a cardioid, opening towards the negative y-axis.

Key Concepts

Limaçon

In polar coordinates, a Limaçon is a fascinating type of curve that arises from specific polar equations. It appears in the form of either \( r = a + b \cdot \sin \theta \) or \( r = a + b \cdot \cos \theta \), where \( a \) and \( b \) are constants. The shape of a Limaçon can vary greatly:

• If \( |a| > |b| \), the Limaçon has a dimple.
• If \( |a| = |b| \), it forms a cardioid, which is a special Limaçon.
• If \( |a| < |b| \), the Limaçon has an inner loop.

These variations give the Limaçon its versatility and beauty in polar graphing. It showcases how the balance between \( a \) and \( b \) influences the curve's shape and nature. This concept is crucial in understanding the diversity of polar curves.

Cardioid

A cardioid is a special type of Limaçon with a distinct heart-like shape. This curve emerges when \( a = b \) in the Limaçon's equation, specifically in forms like \( r = a + a \cdot \sin \theta \) or \( r = a + a \cdot \cos \theta \):

• The point of the heart, known as the cusp, often lies at the origin when expressed as \( r = a(1 - \sin \theta) \) or elsewhere depending on trigonometric function used.
• The cardioid curves smoothly around, indicative of a smooth, continuous loop, unlike other Limaçon forms with loops or dimples.

The cardioid is significant due to its simplicity and elegance, commonly appearing in acoustic and optical contexts like microphones and antennas. Understanding a cardioid's behavior helps in graphing and recognizing these distinct patterns in applied mathematical fields.

Graphing Polar Curves

Polar coordinates provide a unique method of graphing that focuses on angles and radii rather than traditional x and y axes. When graphing polar curves, follow a step-by-step approach:

• Start by understanding the polar equation and identifying its type, such as a Limaçon or cardioid.
• Calculate the radius \( r \) for various angles \( \theta \), creating a table of values.
• Plot these points on polar graph paper, respecting the angle and distance from the pole (origin) as defined by \( r \).
• Connect these plotted points smoothly to reveal the curve's shape.

This method transforms complex equations into visually intuitive shapes. It aids in predicting and analyzing the properties of functions in a two-dimensional space, showcasing the beauty and complexity of mathematical concepts in a straightforward and visual manner.