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Magazine Chapter 8: Problem 59
Sketch a graph of the polar equation. $$ r=1+3 \sin (\theta) $$
Short Answer
Expert verified
The graph is a lima莽on with an inner loop.
Step by step solution
01
Understand the Polar Equation Form
The given polar equation is \( r = 1 + 3 \sin(\theta) \). In polar coordinates, \( r \) represents the distance from the origin, and \( \theta \) is the angle from the positive x-axis. This equation resembles the form \( r = a + b \sin(\theta) \) which is commonly known as a lima莽on.
02
Determine 尝颈尘补莽辞苍 Type
In the general form \( r = a + b \sin(\theta) \), if \( b > a \), like here where \( 3 > 1 \), the lima莽on has an inner loop. This indicates that our graph will loop back inside itself.
03
Calculate Key Points
Evaluate \( r \) at special angles: - For \( \theta = 0 \), \( r = 1 + 3 \cdot 0 = 1 \).- For \( \theta = \frac{\pi}{2} \), \( r = 1 + 3 \cdot 1 = 4 \).- For \( \theta = \pi \), \( r = 1 + 3 \cdot 0 = 1 \).- For \( \theta = \frac{3\pi}{2} \), \( r = 1 + 3 \cdot (-1) = -2 \).
04
Plot Points and Sketch the Graph
Plot the calculated points in polar coordinates:- At \( \theta = 0 \), plot point (1, 0掳).- At \( \theta = \frac{\pi}{2} \), plot point (4, 90掳).- At \( \theta = \pi \), plot point (1, 180掳).- At \( \theta = \frac{3\pi}{2} \), plot point (2 units left from origin, as the radius is negative).Join these points to form a lima莽on with an inner loop, noting the symmetry about the line \( \theta = \frac{\pi}{2} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
尝颈尘补莽辞苍
A lima莽on is a type of curve encountered in polar coordinates, and it is often characterized by its distinctive heart-like shape. The polar equation for a lima莽on is typically given in the form \( r = a + b \sin(\theta) \) or \( r = a + b \cos(\theta) \). These shapes can take on various appearances depending on the values of \( a \) and \( b \). 尝颈尘补莽辞苍s can have inner loops, be dimpled, or even convex, based on the relationship between \( a \) and \( b \):
- If \( b = a \), the lima莽on is dimpled.
- If \( b < a \), the lima莽on is convex.
- If \( b > a \), as in our exercise with \( r = 1 + 3 \sin(\theta) \), the lima莽on will have an inner loop.
Graphing Polar Equations
Graphing polar equations such as lima莽ons can seem daunting initially, but it becomes much more digestible with a step-by-step approach. Polar graphs plot points using a combination of angles and radii, rather than the Cartesian method of using x and y coordinates. Here鈥檚 how to graph a typical polar equation:
- Start by determining key angles at which to evaluate \( \theta \), such as \( 0, \frac{\pi}{2}, \pi, \) and \( \frac{3\pi}{2} \).
- Calculate the radius \( r \) for each of these angles to figure out specific points.
- Plot these points using polar coordinates, where the angle \( \theta \) specifies the direction from the origin, and \( r \) specifies how far from the origin to place the point.
Trigonometric Functions
Trigonometric functions like sine and cosine play a crucial role in polar coordinates. These functions help define the direction and magnitude of the distance from the origin when plotting points. In the context of our polar equation \( r = 1 + 3 \sin(\theta) \), \( \sin(\theta) \) determines how the distance \( r \) changes with the angle \( \theta \). Understanding trigonometric functions allows you to anticipate how the graph will behave:
- At \( \theta = 0 \) and \( \theta = \pi \), \( \sin(\theta) = 0 \), leading to \( r = 1 \), making both points symmetrical about the origin.
- At \( \theta = \frac{\pi}{2} \), \( \sin(\theta) = 1 \), giving the maximum \( r \) value, making it the furthest point from the origin.
- At \( \theta = \frac{3\pi}{2} \), \( \sin(\theta) = -1 \), leading to a negative \( r \), which means the point is plotted in the opposite direction along the line, two units away from the origin.
Symmetry in Polar Graphs
Symmetry is a helpful property in graphing polar equations, as it can provide clues about how a graph behaves without needing to plot every point manually. For the polar equation \( r = 1 + 3 \sin(\theta) \), we observe symmetry relative to certain axes or lines.
- The symmetry about the line \( \theta = \frac{\pi}{2} \) is significant. Since \( \sin(-\theta) = -\sin(\theta) \), the graph reflects upon itself along this line.
- Symmetry can sometimes be about the polar axis (corresponding to the x-axis in Cartesian coordinates), depending on the form of the equation.
