Problem 10 - College Trigonometry

Question

Graph the following equations. $$ r=\frac{3}{2+\sin (\theta)} $$

Step-by-Step Solution

Answer

The graph is a dimpled limaçon symmetric about the vertical axis.

Step 1: Identify the Equation Type

The polar equation given is in the form \( r = \frac{k}{a + b \sin(\theta)} \). This form represents a conic section, specifically a limaçon, which is a type of polar graph with distinctive loops or dimples depending on the parameters.

Step 2: Determine Limaçon Characteristics

For the equation \( r = \frac{3}{2 + \sin(\theta)} \), we identify the constants: \( k = 3 \), \( a = 2 \), and \( b = 1 \). Since \( a > b \), the graph will be a dimpled limaçon without an inner loop.

Step 3: Find Symmetry and Key Angles

Because the function has a \( \sin(\theta) \) component, the graph is symmetric about the vertical axis (θ = π/2). Evaluate the equation at key angles: \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) to understand the graph's key distances from the origin.

Step 4: Calculate Polar Coordinates

Use the polar equation to calculate specific points:- \( \theta = 0 \): \( r = \frac{3}{2 + 0} = \frac{3}{2} \)- \( \theta = \frac{\pi}{2} \): \( r = \frac{3}{2 + 1} = 1 \)- \( \theta = \pi \): \( r = \frac{3}{2 - 0} = \frac{3}{2} \)- \( \theta = \frac{3\pi}{2} \): \( r = \frac{3}{2 - 1} = 3 \)

Step 5: Plot the Graph

Using the calculated points, plot them in polar coordinates. Join the points smoothly to reflect the dimpled characteristic of the limaçon. The graph will have a larger radius at \( \theta = \frac{3\pi}{2} \) and symmetric dimpled shape about the vertical axis.

Key Concepts

Limaçon

A limaçon is a fascinating type of graph in polar coordinates. It appears when you graph polar equations in the form \( r = \frac{k}{a + b \sin(\theta)} \) or \( r = \frac{k}{a + b \cos(\theta)} \). The shape of a limaçon can vary widely:

• It can have a loop (when \( b > a \))
• Be dimpled, resembling a heart-like shape (when \( a > b \))
• Or form a perfect circle in some cases

In our given equation \( r = \frac{3}{2 + \sin(\theta)} \), the shape will be dimpled, as \( a \) is greater than \( b \). Limaçons can be playful on the graph, changing their appearance dramatically with different parameters.

Conic Sections

Conic sections are shapes created by slicing a cone with a plane. The most common conic sections include circles, ellipses, parabolas, and hyperbolas. In polar coordinates, these sections appear when equations take specific forms, predicting beautiful and varied shapes.

• Circles have the simplest form.
• Ellipses are stretched circles.
• Parabolas open in one direction.
• Hyperbolas open in two directions.

However, the limaçon is a special type of conic section derived from speficic polar equations related to hyperbolas and ellipses. They display unique traits like dimples or loops, adding to the interesting patterns of symmetry and design in this category.

Graphing Polar Equations

Graphing polar equations starts with converting equations into polar form, like our given equation \( r = \frac{3}{2 + \sin(\theta)} \). Polar graphs are plotted using angles \( \theta \) and radius \( r \).

• Identify the form: Check if it's a known shape like a limaçon.
• Analyze the constants to determine the key graph characteristics.
• Calculate key points by substituting angles \( \theta \) to find corresponding \( r \) values.

For our limaçon, calculate coordinates at crucial angles like \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \). Map these points into the polar plane where each set of \( (\theta, r) \) represents a unique location, creating the renowned dimpled shape.

Symmetry in Polar Graphs

Symmetry plays a critical role in understanding and sketching polar graphs effectively. It simplifies the graphing process, providing a balanced design.

• Analyze symmetry in our equation which involves \( \sin(\theta) \), implying vertical axis symmetry (along \( \theta = \frac{\pi}{2} \)).
• Symmetric properties reduce computation by confirming that behavior on one side reflects that of another.
• This helps in rapidly plotting and checking your graph shape.

By understanding symmetry, you predict how the graph unfolds around its axes. With this knowledge, connect points to see how traits like dimples and loops are reflected across the axis.