Problem 72 - Precalculus: Functions and Graphs

Question

Sketch the graph of the polar equation. $$r=2(1+\sin \theta)$$

Step-by-Step Solution

Answer

The graph of \( r = 2(1 + \sin \theta) \) is a Limaçon with an inner loop.

Step 1: Identify the Polar Equation Type

The given polar equation is \( r = 2(1 + \sin \theta) \). This is similar to the standard form of a Limaçon with an inner loop, which is expressed as \( r = a(1 + \sin \theta) \). In this case, \( a = 2 \).

Step 2: Determine the General Shape

Since the equation is a Limaçon and \( a > 0 \), it has the general shape of a heart where an inner loop is formed around the pole. The inner loop will form because \( |a| = 2 \), confirming that this is a Limaçon with an inner loop.

Step 3: Calculate Values at Key Angles

Calculate \( r \) at key angles. At \( \theta = 0 \), \( r = 2(1 + \sin 0) = 2 \). At \( \theta = \pi/2 \), \( r = 2(1 + \sin(\pi/2)) = 4 \). At \( \theta = \pi \), \( r = 2(1 + \sin \pi) = 2 \). And at \( \theta = 3\pi/2 \), \( r = 2(1 + \sin(3\pi/2)) = 0 \). These values suggest a loop at the bottom of the graph.

Step 4: Sketch the Graph Using Key Points

Begin at the pole since \( r = 0 \) at \( \theta = 3\pi/2 \), draw the inner loop forming between \( 3\pi/2 \) and \( \pi/2 \). From \( \theta = 0 \) (aligned on the positive x-axis) to \( \theta = \pi \) (aligned on the negative x-axis), sketch the outer heart shape. Lastly, this symmetry suggests the graph will also look responsible over negative angles, reinforcing the heart-shaped appearance with a loop.

Key Concepts

Limaçon

A Limaçon is a unique type of polar graph that resembles something between a circle and a distorted heart shape. Its polar equation is generally represented as \( r = a + b\sin \theta \) or \( r = a + b\cos \theta \). The shape of a Limaçon depends heavily on the constant values of \( a \) and \( b \).
- If \( |a| = |b| \), the Limaçon will form a cardioid, which resembles a heart without the inner loop. - If \( |a| < |b| \), like in our case where \( a = 2 \) and the factor of the trig function is also 2, the Limaçon will sport an inner loop.
- Conversely, if \( |a| > |b| \), there will be no loop, and the Limaçon will take on a dimpled form.
Understanding these differences is key to unlocking the various shapes Limaçons can form on a polar plane. It helps predict the general layout of the graph before computing individual points.

Inner Loop

The inner loop of a Limaçon occurs when the absolute value of \( a \) is less than \(|b| \), as in \( r = 2(1 + \sin \theta) \). The loop signifies that there are points where the graph folds over itself.
This results in a smaller, distinct loop that "dips" into the center, or the pole, of the polar coordinate system. This can be mathematically validated through specific angle calculations, as seen in our exercise where:

• At \( \theta = 0 \), \( r = 2 \) indicates the outer portion.
• At \( \theta = \frac{3\pi}{2} \), \( r = 0 \) aligns right at the pole, suggesting the tip of the inner loop.

By understanding when and how the radians in the polar equation form specific \( r \) values, students can accurately map this inner loop on paper. This intricate loop is significant because it alters the graph's symmetry and general appearance.

Graphing Polar Coordinates

Graphing polar coordinates can seem daunting at first, but it becomes manageable with practice and the right approach. Instead of using two axes parallel to each other like in Cartesian coordinates, polar graphs operate on a radial system centered around a pole. Here, points are plotted using a radius (\( r \)) and an angle (\( \theta \)), and each point’s exact position is determined by rotating from the positive x-axis by the angle \( \theta \).
Key points gathered from calculations of the polar equation provide a roadmap to sketch the graph accurately. In our example:

• The key angles \( \theta = 0, \pi/2, \pi, \text{and } 3\pi/2 \) determine the shape, focusing especially on points where the radius changes, such as \( r = 0 \) and \( r = 4 \).
• Mapping these values helps in visualizing the full Limaçon, especially its inner loop.

By connecting these key values with smooth curves, we depict the graph's structure step-by-step. This systematic plotting practice solidifies the concept of graphing in polar coordinates, enhancing the understanding of polar systems overall.