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Magazine Chapter 8: Problem 41
Graph the family of polar equations \(r=1+\sin n \theta\) for \(n=1,2,3,4,\) and \(5 .\) How is the number of loops related to \(n ?\)
Short Answer
Expert verified
The number of loops equals the value of \( n \) for odd \( n \), and \( n \) times two for even \( n \).
Step by step solution
01
Understand the Polar Equation
The polar equation given is \( r = 1 + \sin n \theta \). This equation describes a family of curves where \( n \) is a positive integer. In polar coordinates, \( r \) is the radius from the origin, and \( \theta \) is the angle.
02
Determine the Shape for Different n Values
For each value of \( n \), the equation \( r = 1 + \sin n \theta \) influences the number of loops. Calculate and sketch the polar graph for each \( n \) value individually to identify the number of loops.
03
Graph for n=1
For \( n = 1 \), the polar equation becomes \( r = 1 + \sin \theta \). This will yield a "one-looped" curve, which is known as a cardioid.
04
Graph for n=2
For \( n = 2 \), the polar equation is \( r = 1 + \sin 2\theta \). This results in a graph with two loops, known as a 2-leaved rose.
05
Graph for n=3
For \( n = 3 \), the equation is \( r = 1 + \sin 3\theta \). The graph will depict three loops, forming a 3-leaved rose.
06
Graph for n=4
For \( n = 4 \), the equation is \( r = 1 + \sin 4\theta \). The graph will have four loops, which is a 4-leaved rose.
07
Graph for n=5
For \( n = 5 \), the equation is \( r = 1 + \sin 5\theta \). This graphs as a 5-leaved rose, showing five loops.
08
Relate Number of Loops to n
By examining the graphs for each \( n \), it can be seen that the number of loops in each graph is directly equal to the \( n \) value when \( n \) is odd, and twice the value of \( n \) for even \( n \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cardioid
The cardioid is a special type of curve on a polar graph that looks similar to a heart shape, hence its name. In polar coordinates, a cardioid is defined by the equation \( r = a + a \sin \theta \) or \( r = a + a \cos \theta \), where \( a \) is a constant. In this exercise, when \( n = 1 \), the equation \( r = 1 + \sin \theta \) represents a cardioid. This means that for every angle \( \theta \), the radius length \( r \) changes based on the function \( \sin \theta \).
- A cardioid forms a single loop and is symmetric about the x-axis or y-axis depending on the sine or cosine function used.
- It will often intersect the pole (origin) at one point.
- This simple loop makes it one of the most easily recognizable polar curves.
Rose Curve
Rose curves are fascinating polar graphs that resemble the petals of a rose. These are represented by the polar equation \( r = a + \sin n\theta \) or \( r = a + \cos n\theta \), where \( n \) determines the number of petals. In the exercise, for \( n = 2, 3, 4, 5 \), we draw rose curves with different numbers of loops.
- When \( n \) is even, the number of loops (or petals) is \( 2n \), due to symmetry in polar coordinates.
- When \( n \) is odd, the curve has exactly \( n \) loops.
- The case \( n = 2 \) presents a 2-leaved rose, while \( n = 3 \) shows a 3-leaved rose, and so on.
Polar Coordinates
Polar coordinates are a two-dimensional coordinate system, helpful for plotting points on a plane, using a radius and an angle. Unlike the Cartesian system that uses x and y coordinates, the polar system describes a point's distance from the origin and the angle from the positive x-axis.
- The primary components are \( r \), the radius, which measures how far the point is from the origin.
- \( \theta \), the angle, which measures the direction from the positive x-axis, typically in radians.
Number of Loops
Understanding the number of loops in polar graphs, specifically in rose curves, involves examining the parameter \( n \). The parameter \( n \) in \( r = 1 + \sin n\theta \) directly influences how many loops the curve will feature.
- For an odd \( n \), the curve has exactly \( n \) loops.
- When \( n \) is even, the curve has \( 2n \) loops due to symmetry.
