Problem 50 - Calculus Early Transcendentals

Question

Investigate the family of curves defined by the parametric equations \(x=\cos t, y=\sin t-\sin c t,\) where \(c>0 .\) Start by letting \(c\) be a positive integer and see what happens to the shape as \(c\) increases. Then explore some of the possibilities that occur when \(c\) is a fraction.

Step-by-Step Solution

Answer

Curves become more complex as integer \( c \) increases, while fractional \( c \) values create irregular patterns.

Step 1: Understanding the Parametric Equations

The problem gives us parametric equations defined by \( x = \cos t \) and \( y = \sin t - \sin(c t) \). The parameter \( t \) typically ranges from 0 to \( 2\pi \), describing the path of the curve. The parameter \( c \) alters the shape by changing the frequency of the second sine function in the \( y \)-coordinate.

Step 2: Exploring Integer Values of c

Let \( c \) be a positive integer like 1, 2, 3, etc. When \( c = 1 \), the equations simplify to \( x = \cos t, y = \sin t - \sin t = 0 \). For \( c = 2 \), \( y = \sin t - \sin(2t) \), the resulting curve will start to resemble a Lissajous figure. As \( c \) increases, the number of loops or intersections in the curve generally increases and creates more complex patterns.

Step 3: Analyzing with Fractional Values of c

Examine \( c \) as a fraction like 1/2 or 1/3. Taking \( c = 1/2 \), the curve will take a different form compared to integer \( c \) values, often elongating or squashing the loops. For example, \( y = \sin t - \sin(t/2) \) creates curves with non-standard frequencies, leading to unique patterns due to the non-integer periodicity.

Step 4: Observations and Patterns

By experimenting with different values of \( c \), both integers and fractions, it becomes clear that integer \( c \) values lead to more symmetric, periodic patterns while fractional \( c \) values produce more irregular or distorted curves. This is due to the changes in the periodic nature of the sine component frequencies.

Key Concepts

Lissajous figures

Lissajous figures are fascinating, looped patterns that form when you plot a system of parametric equations. These curves arise when two trigonometric functions, often sine and cosine, interact at particular angles and frequencies. When exploring the family of curves given by the parametric equations \( x = \cos t \) and \( y = \sin t - \sin(ct) \), you may notice the resemblance to Lissajous figures, especially for integer values of \( c \).

Lissajous figures have their roots in physics, particularly in the study of harmonic motion. They are used to illustrate the interaction of two orthogonal oscillations, essentially two waves crossing paths. Here, the parameter \( c \) in our equation affects how these oscillations combine:

• Integer values of \( c \) lead to periodic, symmetric loops.
• Fractional \( c \) values introduce distortions, departing from the traditional symmetry.

As we adjust \( c \), these harmonic interplay changes, creating diverse visual patterns often used in signal analysis and waveform studies.

frequency

Frequency is a key player in how curves and waveforms shape up over time. In the context of the given parametric equations \( x = \cos t \) and \( y = \sin t - \sin(ct) \), the frequency is manipulated through the parameter \( c \). Here, frequency pertains to how often the wave of the sine component completes a cycle within a set interval.

By choosing different values for \( c \), you effectively change the frequency of the secondary sine component in the \( y \)-coordinate:

• For \( c = 1 \), the curve aligns with a straight line since the two sine terms cancel out.
• With \( c = 2 \), the secondary wave's frequency doubles, resulting in more complex, looping patterns.
• Fractional \( c \) values, like \( c = 1/2 \), adjust the frequency in non-integer increments, altering the balance between the \( x \) and \( y \) sinusoidal components, yielding stretched or squashed loops.

Understanding frequency changes helps explain the beautiful diversity of patterns, as modifying \( c \) can tune the loop count and symmetry across these curves.

curve analysis

Analyzing the behavior of these curves reveals a lot about the nature of oscillating systems. Within this analysis, our focus shifts to the relationships and transformations governed by the parameter \( c \). By examining integer and fractional values of \( c \), we delve into the nuances of parametric curve behavior.

When \( c \) is an integer:

• Curves manifest clear, symmetric patterns.
• The total number of loops or intersections increases as \( c \) grows, displaying richer and more complex designs.

For fractional \( c \) values, the curves behave differently:

• Patterns often lack the symmetry of integer-based curves.
• The period of the secondary sine wave starts to diverge significantly from the primary wave, introducing asymmetry and distortion.

Curve analysis allows us to appreciate not only the aesthetic complexity of these figures but also their applications in engineering, especially where oscillatory systems and signal processing come into play. By tweaking \( c \), one can predict or engineer specific curve properties, useful for simulations and visualizations in various scientific fields.