Problem 57 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-78: Sketch the graph of the polar equation. $$ r=2+4 \sin \theta $$

Step-by-Step Solution

Answer

Sketch a limaçon with an inner loop, based on the values of \( r = 2 + 4 \sin \theta \).

Step 1: Understand the Polar Equation

We have the polar equation \( r = 2 + 4 \sin \theta \). \( r \) represents the distance from the origin, and \( \theta \) is the angle with respect to the positive x-axis. This is a limaçon with inner loops.

Step 2: Identify the Maximum and Minimum Values

Identify the maximum and minimum values of \( r \) based on \( \sin \theta \). The maximum value of \( \sin \theta \) is 1, giving \( r = 2 + 4(1) = 6 \). The minimum value is -1, leading to \( r = 2 + 4(-1) = -2 \), but in polar coordinates, we consider \( r = 2 - 4 = -2 \) as having a magnitude of 2 with the opposite direction.

Step 3: Plot Key Points

Plot points for key values of \( \theta \), such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \( 2\pi \). At \( \theta = 0 \), \( r = 2 \); at \( \theta = \frac{\pi}{2} \), \( r = 6 \); at \( \theta = \pi \), \( r = 2 \); at \( \theta = \frac{3\pi}{2} \), \( r = -2 \); and at \( \theta = 2\pi \), \( r = 2 \).

Step 4: Sketch the Graph

Use the plotted points to sketch the graph. Visualize how the graph moves with varying \( \theta \) values, understanding that \( r \) represents the radial distance at each \( \theta \). From \( \theta = 0 \) to \( \theta = \pi \), \( r \) creates the outer part of the limaçon, and from \( \theta = \pi \) to \( \theta = 2\pi \), it creates the part where the curve loops towards the origin.

Key Concepts

Limaçon

A limaçon is a fascinating shape in polar coordinates, formed in the plane by equations of the form \(r = a + b \sin \theta\) or \(r = a + b \cos \theta\). These curves create a variety of interesting patterns depending on the relationship between \(a\) and \(b\). For instance, when \(|a| < |b|\), a limaçon will have an inner loop, as seen in the exercise equation \(r = 2 + 4 \sin \theta\).

To understand limaçons better, it's helpful to visualize how they expand and contract with varying values of \(\theta\). When sin or cos reaches its peak or nadir, you get the maximum or minimum length in the radial direction. The inner loop appears when the equation results in negative radial values, indicating a curve portion that stretches inward.

Graph Sketching

Graph sketching in polar coordinates is all about understanding how the radius \(r\) changes with the angle \(\theta\). Begin by identifying crucial angles, such as \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \(2\pi\). These angles represent key points where trigonometric functions take on extreme values, often simplifying calculations.

• \(\theta = 0\): Start at the positive x-axis, ensuring you know \(r\)'s value here.
• At \(\theta = \frac{\pi}{2}\): What does \(r\) become at 90 degrees, a critical angle for sinusoidal adjustments?
• As you reach \(\theta = \pi\), you pass through the negative x-axis, noting further changes in \(r\).

Connecting these points provides a directional guide to sketch the limaçon by tracking how \(r\) behaves over increments of \(\theta\). Matching the sketch to your calculations solidifies your understanding. This process not only creates graphs but a tangible intuition about the dynamics of polar functions.

Trigonometric Functions

Trigonometric functions like \(\sin\) and \(\cos\) are the backbone of understanding polar equations. In limaçons, the function component governs how the radius \(r\) oscillates as \(\theta\) progresses. It's crucial to remember:

• \(\sin \theta\) cycles between -1 and 1.
• This cycling directly affects \(r\), contributing to dynamic graph changes.

In the equation \(r = 2 + 4 \sin \theta\), observe how the sine component modifies the radial distance, causing it to expand or contract based on \(\theta\)'s value. When understanding polar equations, you should always consider the maximum and minimum effects of these trigonometric functions.

Mathematical Graphing

Math graphing with polar equations is an exciting exploration of geometry and dynamic shapes. When working with an equation like \(r = 2 + 4 \sin \theta\), the aim is to visualize it on the polar plane. Here, \(\theta\) represents the angle we measure from the positive x-axis, while \(r\) denotes the distance from the origin.

The process involves several steps:

• Calculate \(r\) for important values of \(\theta\).
• Plot these coordinates on polar graph paper, with each point represented by a distance and angle.

Continuous plotting reveals the complete shape, like drawing the limaçon, showcasing the intriguing nature of mathematical graphing. This approach connects abstract equations to visual patterns, deepening comprehension through structured sketches and insights.