Problem 65 - Precalculus Mathematics for Calculus

Question

Graph the three functions on a common screen. How are the graphs related? $$y=\cos 3 \pi x, \quad y=-\cos 3 \pi x, \quad y=\cos 3 \pi x \cos 21 \pi x$$

Step-by-Step Solution

Answer

y = -cos(3πx) is a reflection of y = cos(3πx). The third function shows complex oscillations.

Step 1: Analyze the function y = cos(3πx)

The function \( y = \cos(3\pi x) \) represents a cosine wave with an angular frequency of \( 3\pi \). This modifies the period compared to the standard cosine function \( \cos x \). The period is calculated as \( \frac{2\pi}{3\pi} = \frac{2}{3} \). Hence, this function repeats every \( \frac{2}{3} \) units along the x-axis.

Step 2: Analyze the function y = -cos(3πx)

The function \( y = -\cos(3\pi x) \) is simply the negative of \( \cos(3\pi x) \). Negating the cosine function reflects it across the x-axis. Thus, this graph is a cosine wave with the same frequency and period as \( \cos(3\pi x) \), but it is inverted.

Step 3: Analyze the function y = cos(3πx)cos(21πx)

The function \( y = \cos(3\pi x) \cos(21\pi x) \) is a product of two cosine functions, which can be simplified using the product-to-sum identities: \( \cos A \cos B = \frac{1}{2}(\cos(A-B) + \cos(A+B)) \). Applying this gives: \( \frac{1}{2}(\cos(18\pi x) + \cos(24\pi x)) \). Thus, the graph is a combination of two waves with different frequencies.

Step 4: Graphing the Functions

Graph \( y = \cos(3\pi x) \) as a regular cosine wave with a period of \( \frac{2}{3} \). Then, graph \( y = -\cos(3\pi x) \) as the same cosine wave reflected across the x-axis. Finally, graph \( \frac{1}{2}(\cos(18\pi x) + \cos(24\pi x)) \), observing more rapid oscillations due to higher frequencies.

Step 5: Compare the Graphs

The graph of \( y = -\cos(3\pi x) \) is the vertical reflection of \( y = \cos(3\pi x) \). The function \( y = \cos(3\pi x) \cos(21\pi x) \) has a more complicated structure with smaller oscillations due to the sum of higher frequency components. It results in a beat pattern where one waveform modulates another.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function, often represented as \( y = \cos x \). Its graph is a smooth, periodic wave that oscillates between -1 and 1. This wave-like pattern is due to the nature of the cosine function, which describes the x-coordinate of a point on a unit circle as the angle changes.

The standard cosine wave has certain distinct properties:

• **Amplitude**: This is the height from the center line to the peak of the wave. For \( \cos x \), the amplitude is 1.
• **Period**: The standard cosine function repeats every \( 2\pi \) radians.

When adjustments are made to the function, like \( y = \cos(3\pi x) \), these properties can change. The function \( y = -\cos(3\pi x) \) goes a step further by reflecting the wave across the x-axis.”

Angular Frequency

Angular frequency is a critical concept in understanding trigonometric functions as it indicates how fast a wave oscillates. It is especially relevant for cosine functions like \( y = \cos(3\pi x) \). Angular frequency is denoted by \( \omega \) and is involved in the term \( \cos(\omega x) \).

For \( y = \cos(3\pi x) \), the angular frequency is \( 3\pi \). This means the wave completes 3 full cycles over a span of \( 2\pi \) radians, making the graph appear "squished" compared to the standard cosine graph.

Here's how angular frequency affects a cosine function:

• A higher angular frequency results in more oscillations in a given interval.
• A lower angular frequency results in fewer oscillations.

Understanding angular frequency helps in predicting and graphing waves with precision.”

Period of a Function

The period of a trigonometric function, like the cosine function, is the length over which it repeats itself. This concept is vitally important when dealing with modified cosine functions such as \( y = \cos(3\pi x) \).

The period \( T \) of a standard cosine function is \( 2\pi \). However, when the function is adjusted to \( y = \cos(\omega x) \), the period becomes \( T = \frac{2\pi}{\omega} \).

In our example, with \( y = \cos(3\pi x) \):

• **Angular Frequency**: \( \omega = 3\pi \)
• **Period**: \( \frac{2\pi}{3\pi} = \frac{2}{3} \)

This means that the entire wave completes one full cycle every \( \frac{2}{3} \) units along the x-axis. Adjusting the period can drastically change the look and feel of the cosine wave on a graph, squeezing or stretching its oscillations.”