Problem 79

Question

(a) Investigate the family of curves defined by the polar equations $r=\sin n \theta,$ where $n$ is a positive integer. How is the number of loops related to $n ?$ (b) What happens if the equation in part (a) is replaced by $r=|\sin n \theta| ?$

Step-by-Step Solution

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Answer

Part (a): Number of loops is $ n $. Part (b): $ n $ loops for odd $ n $, $ 2n $ for even $ n $.

1Step 1: Introduction to Polar Equations

The given polar equation is $ r = \sin n\theta $, where $ n $ is a positive integer. Polar equations describe curves on a plane, where each point on the curve is determined by an angle $ \theta $ and a radius $ r $. We are tasked to understand how the number of loops formed by this curve is related to $ n $.

2Step 2: Analyzing the Behavior of $ r = \sin n \theta $

For $ r = \sin n \theta $, the radius $ r $ becomes zero whenever $ \sin n \theta = 0 $, which occurs when $ n \theta = k\pi $, where $ k $ is an integer. Each time $ \sin n \theta $ completes one full sine wave (i.e., from 0 to $ 2\pi $), $ \theta $ spans $ 2\pi/n $ radians. Thus, within the range of $ 0 $ to $ 2\pi $, $ \theta $ can complete $ n $ full sine waves, forming exactly $ n $ loops.

3Step 3: Conclusion for Part (a)

The number of loops in the curve described by $ r = \sin n \theta $ is exactly equal to $ n $, regardless of whether $ n $ is odd or even, because each loop corresponds to a complete oscillation of the sine function in $ n \theta $.

4Step 4: Introduction to Part (b) with Absolute Values

Now consider the equation $ r = |\sin n \theta| $. Here, the absolute value ensures that the radius $ r $ is never negative. This modification could potentially change the number of loops as it affects how the curve loops back on itself.

5Step 5: Analyzing $ r = |\sin n \theta| $

With $ r = |\sin n \theta| $, the loop formation is affected because any negative values of $ \sin n \theta $ will become positive. As a result, the curve will mirror itself over the origin when previously $ r $ would have been negative. If $ n $ is even, the function $ |\sin n \theta| $ completes two loops for every full cycle of $ \sin n \theta $, resulting in $ 2n $ loops, double the loops for $ r = \sin n \theta $. If $ n $ is odd, the symmetries do not add extra loops beyond the original $ n $, keeping it at $ n $ loops.

6Step 6: Conclusion for Part (b)

When replacing the equation with $ r = |\sin n \theta| $, the number of loops becomes $ n $ for odd $ n $ and $ 2n $ for even $ n $, due to mirror symmetry over the origin that results from taking the absolute value.

Key Concepts

Loops in Polar CurvesBehavior of Trigonometric FunctionsAbsolute Value Effects

Loops in Polar Curves

Polar curves are represented in a plane using polar coordinates, which means describing points with a radius and an angle. One fascinating aspect of polar equations is how they can create loops. Consider the equation $ r = \sin n\theta $, where $ n $ is a positive integer. This equation showcases loops based on the sine function's nature.

The sine function alternates between positive and negative values.
For $ \sin\ n\theta = 0 $, we observe points at the origin (where the radius $ r $ becomes 0).

Every time the sine wave completes a full cycle, the angle $ \theta $ spans $ \frac{2\pi}{n} $ radians. This results in creating $ n $ complete loops as $ \theta $ ranges from 0 to $ 2\pi $. Each loop represents one complete oscillation of the sine function. This is true for any positive integer $ n $, whether it's odd or even.

Behavior of Trigonometric Functions

Understanding the behavior of trigonometric functions is critical as it guides how polar curves are formed. In the polar curve defined by $ r = \sin n\theta $, the sine function dictates the curve's structure and size of the loops.

The sine function cycles through positive and negative values, which influences the plotted radius $ r $.
When $ \sin n\theta $ is positive, the curve expands outward.
When $ \sin n\theta $ is negative, the curve still fills out in the opposite direction but may not be visible due to the sign change.

This cyclical nature results in symmetrically distributed loops around the origin, as each full cycle of $ \sin n\theta $ results in a loop. It's fascinating how this innate periodicity of sine results in evenly distributed and visually pleasing polar loops.

Absolute Value Effects

Introducing absolute values into polar equations adjusts the curve significantly. For $ r = |\sin n\theta| $, every negative part of $ \sin n\theta $ is flipped to be positive. This alteration changes loop formation.

Different Cases of $ n $

Negative values being made positive mirrors the curve across the origin:

If $ n $ is even, the symmetry results in double the loops, so you get $ 2n $ loops.
If $ n $ is odd, the number of loops stays the same at $ n $, aligning more overlapping paths.

The effect of the absolute value creates a more "folded", visually symmetrical pattern, which can look quite different from the original with unmodified sine. Understanding how absolute values dynamically alter the output can demystify why some polar patterns double in count under these modifications.

Problem 77

Problem 80