Problem 17 - College Trigonometry

Question

In Exercises \(1-20\), plot the graph of the polar equation by hand. Carefully label your graphs. Limaçon: \(r=3-5 \sin (\theta)\)

Step-by-Step Solution

Answer

Plot the graph of the limaçon and label it, ensuring the inner loop is shown with symmetry about the polar axis.

Step 1: Understanding the Limaçon Curve

The given polar equation is a limaçon, which can be written as \(r = a - b \sin(\theta)\). In this case, \(a = 3\) and \(b = 5\). The coefficient \(b\) is greater than \(a\), indicating that the graph will have an inner loop.

Step 2: Identify Key Angles

To plot polar graphs, it is helpful to identify key angles where the behavior changes. Generally, we consider angles like \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\). We will calculate \(r\) for these angles to understand the graph's shape.

Step 3: Calculate \(r\) for Key Angles

- At \(\theta = 0\), \(r = 3 - 5 \sin(0) = 3\).- At \(\theta = \frac{\pi}{2}\), \(r = 3 - 5 \sin\left(\frac{\pi}{2}\right) = 3 - 5 = -2\).- At \(\theta = \pi\), \(r = 3 - 5 \sin(\pi) = 3\).- At \(\theta = \frac{3\pi}{2}\), \(r = 3 - 5 \sin\left(\frac{3\pi}{2}\right) = 3 + 5 = 8\).

Step 4: Analyze the Graph

With the points calculated, sketch the circle in the polar coordinate system. At \(\theta = \frac{\pi}{2}\), the negative \(r\) indicates a point on the opposite side of the origin compared to \(\theta = \frac{3\pi}{2}\). Starting from the origin at \(\theta = 0\), the graph loops inside creating an inner loop, and extends outwards to \(\theta = \frac{3\pi}{2}\) showing the curve with an inner loop.

Step 5: Connect the Points

Draw a smooth curve beginning at the initial point \((3, 0)\) as \(\theta\) moves. Move through the inner loop as you reach the \(\theta = \frac{\pi}{2}\) with a radius \(-2\). The graph then continues to extend outwards to \((8, \frac{3\pi}{2})\). Ensure the graph looks symmetrical about the polar axis.

Key Concepts

Limaçon Curve

The limaçon is a fascinating type of polar curve derived from equations of the form \( r = a - b \sin(\theta) \) or \( r = a - b \cos(\theta) \). In the equation \( r = 3 - 5 \sin(\theta) \), it's clear that we have a specific kind of limaçon because the coefficient \(b\) is greater than \(a\). This characteristic tells us that the graph will feature an inner loop. When \(b\) is greater than \(a\), such as in \( r = 3 - 5 \sin(\theta) \), the distance from the pole (origin in the polar system) can actually become negative for some angles. This directs us to plot points that might loop inside the main curve, creating the hallmark inner loop shape of the limaçon.

• The values \(a\) and \(b\) control the size and shape of the limaçon.
• The use of sine or cosine affects the orientation of the inner loop.
• Typically, a limaçon with a sine function is oriented vertically.

Understanding limaçon curves enhances our appreciation of polar coordinates and the beautiful loops they can create.

Polar Graphs

Polar graphs offer a unique way to visualize equations by plotting points based on their distance from the pole and their angle \(\theta\). Unlike Cartesian plots where coordinates are found at intersects of \(x\) and \(y\), polar graphs map points based on a circle's radius and angle from a central point. This creates visually appealing shapes and curves, such as limaçons, roses, and spirals.

• Key angles \(\theta\), like \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\), help define the structure of polar graphs.
• Values of \(r\) can be positive and negative, affecting how points are plotted relative to the pole.
• Understanding the behavior at key angles simplifies the sketching of complex polar graphs.

By plotting based on angle and radius, polar graphs let us explore dynamic shapes and patterns not possible in rectangular coordinate systems, enriching our mathematical toolkit with elegant visual expressions.

Trigonometric Functions

Trigonometric functions like sine and cosine are crucial for manipulating and understanding polar coordinates. In the limaçon equation \( r = 3 - 5 \sin(\theta) \), the sine function dictates the curve's fluctuations as \(\theta\) varies. This variation permits the creation of intricate shapes, such as loops and petals in polar graphs.

• Sine and cosine control the vertical and horizontal alignment of curves.
• The amplitude (coefficient of sine or cosine) influences how far the curve extends from the pole.
• The period determines the frequency of loops or petals in circular paths.

Trigonometric functions are not only fundamental for defining polar equations but also vital for unraveling the underlying geometric patterns they generate. They are keys to unlocking the relationships between angles, radii, and the mesmerizing shapes of polar curves.