Problem 47 - Algebra and Trigonometry

Question

Identify and graph each polar equation. $$ r=1+2 \sin \theta $$

Step-by-Step Solution

Answer

The given polar equation \( r = 1 + 2 \sin \theta \) represents a dimpled limaçon.

Step 1 Title - Understand the Basic Form

The given polar equation is in the form of a limaçon curve, specifically: \( r = a + b \sin \theta \) Here \( a = 1 \) and \( b = 2 \). This indicates that the equation represents a dimpled limaçon.

Step 2 Title - Determine Key Features

Identify the key points in the polar graph by substituting specific values of \( \theta \). - For \theta = 0 \: \( r = 1 + 2 \sin 0 = 1 \) - For \theta = \frac{\pi}{2} \ \( r = 1 + 2 \sin \frac{\pi}{2} = 1 + 2 \times 1 = 3 \) - For \theta = \pi \: \( r = 1 + 2 \sin \pi = 1 + 2 \times 0 = 1 \) - For \theta = \frac{3\pi}{2} \: \( r = 1 + 2 \sin \frac{3\pi}{2} = 1 + 2 \times (-1) = -1 \) (Ignore negative radius for initial graph)

Step 3 Title - Sketch the Graph

Plot the points obtained in Step 2 and other intermediate values of \( \theta \) to sketch the graph. Note that since \( r \) can be negative, reflect these points across the origin to complete the graph. The final shape should resemble a dimpled limaçon.

Key Concepts

limaçon curve

The limaçon (pronounced 'lee-mah-son') is a type of polar curve that can take on various shapes, including dimpled, looped, or even resembling a cardioid. Derived from the French word for snail, a limaçon has a distinctive circular appearance.

When working with limaçon curves, the general form of the equation you will often encounter is \( r = a + b \, \text{func} \, \theta \), where 'func' typically refers to trigonometric functions like sine or cosine. In this exercise, we have \( r = 1 + 2 \, \text{sin} \, \theta \), making it a dimpled limaçon because \( |b| > |a| \).

This type of curve is fascinating because it can reveal a wide variety of shapes just by altering the values of 'a' and 'b'. Observing its symmetry and patterns helps in understanding polar graph behaviors.

polar equations

Polar equations are equations that express relationships in terms of polar coordinates instead of Cartesian coordinates. Polar coordinates are denoted as \( (r, \, \theta) \), where 'r' represents the radius, or distance from the origin, and \( \theta \) represents the angle from the positive x-axis.

In the given polar equation, \( r = 1 + 2 \, \text{sin} \, \theta \), the relationship between 'r' and \( \theta \) defines the curve. Instead of plotting points as (x, y) pairs, you plot them using these (r, \( \theta \)) pairs.<>ul>

\( r \): represents the radius or distance from the origin

\( \theta \): represents the angle

This polar form allows you to graph circular shapes more naturally and makes it simpler to identify symmetrical properties specific to polar coordinates.

graphing polar coordinates

Graphing polar coordinates entails plotting points in the polar coordinate system. This is different from the Cartesian system and is especially suited for equations like \( r = 1 + 2 \, \text{sin} \, \theta \).

To graph a polar equation:

• Calculate 'r' for various values of \( \theta \) (like 0, \( \frac{\text{π}}{2} \), \( \text{π} \), etc.)
  
• Plot each point using the radial distance 'r' and the angle \( \theta \)
• If 'r' is negative, reflect the point across the origin

For instance, in the given exercise:

• When \( \theta = 0, \, r = 1 \)
• When \( \theta = \frac{\text{π}}{2} \, r = 3 \)

Connect these points smoothly to visualize the complete curve, ensuring to reflect any points where 'r' comes out negative across the origin.

trigonometric functions

Trigonometric functions like sine (sin) and cosine (cos) are fundamental in understanding polar equations. They express how the radius 'r' changes with the angle \( \theta \).

For the equation \( r = 1 + 2 \, \text{sin} \, \theta \), the function sin(\( \theta \)) can vary between -1 and 1. Consequently, multiplying it by 2 and adding 1 alters the radius 'r' as \( \theta \) changes, creating the dimpled limaçon.

Important points to remember about sin(\( \theta \)):

• sin(\( \theta \)) is 0 at \( \theta = 0, \, \text{π}, 2\text{π} \)
• sin(\( \theta \)) is 1 at \( \frac{\text{π}}{2} \)
• sin(\( \theta \)) is -1 at \( \frac{3\text{π}}{2} \)

These properties help us determine the key points needed to plot and understand the full graph of the polar equation.