Problem 60 - Algebra and Trigonometry

Question

Non-Sinusoidal Waves Both the sawtooth and square waves (see Problems 58 and 59 ) are examples of non-sinusoidal waves. Another type of non-sinusoidal wave is illustrated by the function $$ f(x)=1.6+\cos x+\frac{1}{9} \cos (3 x)+\frac{1}{25} \cos (5 x)+\frac{1}{49} \cos (7 x) $$ Graph the function for \(-5 \pi \leq x \leq 5 \pi\).

Step-by-Step Solution

Answer

Graph \[ f(x) = 1.6 + \cos x + \frac{1}{9} \cos (3x) + \frac{1}{25} \cos (5x) + \frac{1}{49} \cos (7x) \] from \(-5 \pi\) to \(5 \pi\).

Step 1: Understand the Function

The given function is a sum of cosines with different frequencies and coefficients. Each term will contribute to the overall shape of the wave. The function is: \[ f(x) = 1.6 + \cos x + \frac{1}{9} \cos (3x) + \frac{1}{25} \cos (5x) + \frac{1}{49} \cos (7x) \]

Step 2: Identify the Range for x

The function needs to be graphed within the range \(-5 \pi \leq x \leq 5\pi\). Ensure that the graph includes this interval.

Step 3: Choose Graphing Tools

Select a graphing tool or software, such as a graphing calculator, Desmos, or any other plotting program that supports custom functions and can handle trigonometric functions over a large interval.

Step 4: Input the Function

Input the function into your chosen graphing tool exactly as given: \[ f(x) = 1.6 + \cos x + \frac{1}{9} \cos (3x) + \frac{1}{25} \cos (5x) + \frac{1}{49} \cos (7x) \]

Step 5: Set the Interval

Set the interval for the x-axis to be from \(-5 \pi\) to \(5 \pi\). This ensures that the graph displays the function over the specified range.

Step 6: Analyze and Adjust the Graph

Review the graph to ensure it displays correctly. You may need to adjust the y-axis to fit the range of the function's values. The function will generally display oscillations characteristic of the added cosine terms.

Step 7: Interpret the Graph

Observe the overall shape of the wave. Notice how the different terms (each having their cosine) interact to form the final waveform, with higher-frequency terms contributing finer details to the wave.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It describes the x-coordinate of a point on the unit circle as the angle varies. The general form is given by \( \text{cos}(x) \). In our problem, we are dealing with a sum of cosine functions:

• \( \text{cos}(x) \)
• \( \frac{1}{9} \text{cos}(3x) \)
• \( \frac{1}{25} \text{cos}(5x) \)
• \( \frac{1}{49} \text{cos}(7x) \)

Each term has a different frequency and amplitude, which affects the final shape of the wave. The cosine function repeats every \( 2\text{π} \). More complicated waveforms can be built by adding different cosine terms together at various frequencies and amplitudes. This process is known as Fourier series. It's important to recognize how these individual terms combine in the given function \( f(x) = 1.6 + \text{cos}(x) + \frac{1}{9} \text{cos}(3x) + \frac{1}{25} \text{cos}(5x) + \frac{1}{49} \text{cos}(7x) \). Each cosine function has a different coefficient, influencing the wave's amplitude or height at different points.

Graphing

Graphing functions is a crucial skill in understanding trigonometric behavior. The given function needs to be graphed within the interval \( -5\text{π} \leq x \leq 5\text{π} \). This range ensures we capture multiple periods of the oscillating wave, providing a comprehensive view of its behavior. Here's a concise guide to graphing this function:

• Identify the function components and their respective frequencies and amplitudes.
• Choose a suitable graphing tool or software that handles trigonometric functions well. Tools like graphing calculators, Desmos, or other plotting programs are ideal.
• Input the function \( f(x) = 1.6 + \text{cos}(x) + \frac{1}{9} \text{cos}(3x) + \frac{1}{25} \text{cos}(5x) + \frac{1}{49} \text{cos}(7x) \).
• Set the interval for the x-axis from \( -5\text{π} \) to \( 5\text{π} \).
• Review and if necessary, adjust the y-axis to ensure all values of the function are displayed properly.

By following these steps, you ensure that your graph accurately represents the given function and you can analyze how each cosine term affects the overall wave.

Waveform Analysis

Waveform analysis involves examining the shape and characteristics of a wave. In the given function, the waveform is constructed from several cosine terms. Each term introduces specific characteristics:

• The base function \( 1.6 + \text{cos}(x) \) forms the primary structure of the wave.
• Additional terms like \( \frac{1}{9} \text{cos}(3x) \) add more detailed oscillations.
• Higher frequency terms \( \frac{1}{25} \text{cos}(5x) \) and \( \frac{1}{49} \text{cos}(7x) \) introduce smaller, finer details to the waveform.

Each additional cosine term modifies the wave, creating intricate patterns. The higher the frequency of the term, the more frequently it oscillates. This leads to smaller waves superimposed onto the larger wave, enriching the overall waveform's complexity. When analyzing a waveform, it's beneficial to understand how these different terms interact and modify the wave's appearance. For example:

• If you only had \( \text{cos}(x) \), you'd see a simple sinusoidal wave.
• Adding \( \frac{1}{9} \text{cos}(3x) \) means you now see faster oscillations superimposed on that wave.
• Each subsequent term further refines this, adding more oscillations and changing the wave's shape.

Observing how these higher-order terms impact the overall waveform helps in understanding more complex signals and in applications like signal processing.

Frequency

Frequency refers to how often a wave oscillates within a given period. In our problem, different cosine terms oscillate at various rates. The basic term \( \text{cos}(x) \) has a base frequency associated with it. Higher frequency terms introduce more frequent oscillations:

• \( \frac{1}{9} \text{cos}(3x) \): This implies the wave oscillates three times faster than the base term.
• \( \frac{1}{25} \text{cos}(5x) \): This term oscillates five times faster.
• \( \frac{1}{49} \text{cos}(7x) \): This oscillates seven times faster.

The frequency domain view is crucial for understanding these oscillations. Each term’s frequency impacts the resultant wave's form. Higher frequencies often contribute to the fine structure of the wave, introducing small, rapid oscillations. When graphing or analyzing such functions, it's essential to differentiate these frequencies and understand their contributions:

• High-frequency components (terms with greater multipliers of x) produce more oscillations.
• Low-frequency components contribute to the broader shape or outline of the wave.

Understanding frequency and how it manifests in these terms aids in interpreting and analyzing complicated waveforms effectively.