Question

Draw a graph over a whole period of each of the following combinations of a fundamental musical tone and some of its overtones: $$ \cos 2 \pi t+\cos 4 \pi t+\frac{1}{2} \cos 6 \pi t $$

Short answer

Plot \(\cos 2\pi t + \cos 4\pi t + \frac{1}{2} \cos 6\pi t\) over the interval \(0 \leq t \leq 1\).

Step-by-step solution

- Identify the Functions Involved

Identify each trigonometric function in the given combination. The functions are: 1. \(\cos 2\pi t\) 2. \(\cos 4\pi t\) 3. \(\frac{1}{2} \cos 6\pi t\)

- Determine Periods of Each Function

Find the period of each individual function. The period of \(\cos(kt)\) is given by \(\frac{2\pi}{k}\): 1. Period of \(\cos 2\pi t\) is \(\frac{2\pi}{2\pi} = 1\) 2. Period of \(\cos 4\pi t\) is \(\frac{2\pi}{4\pi} = \frac{1}{2}\)3. Period of \(\frac{1}{2} \cos 6\pi t\) is \(\frac{2\pi}{6\pi} = \frac{1}{3}\)

- Determine the Fundamental Period of the Combination

The fundamental period of the combination is the least common multiple (LCM) of the periods of the individual functions. Here, the LCM of 1, \(\frac{1}{2}\), and \(\frac{1}{3}\) is 1.

- Generate Data Points

Generate data points for the combination function \(\cos 2\pi t + \cos 4\pi t + \frac{1}{2} \cos 6\pi t\) over one period from t=0 to t=1.

- Plot the Graph

Plot the graph of the combination function using the generated data points over the range \(0 \leq t \leq 1\). The x-axis represents the time \(t\) and the y-axis represents the function value \(f(t)\).

Key concepts

Cosine Function

The cosine function is a fundamental trigonometric function. It is represented as \(\text{cos}(x)\). The graph of the cosine function is a wave that oscillates between -1 and 1.
The general form is \(\text{cos}(kt)\), where k affects the frequency of the oscillation. The cosine function is periodic, meaning it repeats its values in regular intervals.
This periodic nature is useful in modeling many real-world phenomena, such as sound waves and tides.
When plotting the cosine function, you’ll notice it starts at 1 when t=0, drops to -1, and then returns to 1 in a smooth wave.
This behavior continues indefinitely.
The cosine function is often combined with other trigonometric functions to form more complex waves. In our exercise, we see a combination of \(\text{cos}2\pi t\), \(\text{cos}4\pi t\), and \(\frac{1}{2} \text{cos}6\pi t\).

Fundamental Period

The fundamental period of a function is the smallest positive interval over which the function repeats itself. For the cosine function \(\text{cos}(kt)\), the period is \(\frac{2\pi}{k}\).
Each component function in our exercise has a different period: \(\text{cos}2\pi t\) has a period of 1, \(\text{cos}4\pi t\) has a period of \(\frac{1}{2}\), and \(\frac{1}{2} \text{cos}6\pi t\) has a period of \(\frac{1}{3}\).
To find the fundamental period of the combined function, we need to determine the least common multiple, or LCM, of these periods. The LCM of 1, \(\frac{1}{2}\), and \(\frac{1}{3}\) is 1.
This means that the combination function repeats itself every 1 unit of time.

Least Common Multiple

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. Finding the LCM is crucial when dealing with periodic functions to determine the overall period of their combination.
For the periods 1, \(\frac{1}{2}\), and \(\frac{1}{3}\), the process involves finding the smallest number that all these periods can divide evenly into.
Let's determine the least common multiple for our exercise:

• The LCM of 1 and \(\frac{1}{2}\) is 1 because 1 is already a multiple of \(\frac{1}{2}\).
• Next, we find that 1 is also a multiple of \(\frac{1}{3}\) since 1 can exactly accommodate both \(\frac{1}{2}\) and \(\frac{1}{3}\).

In summary, the LCM of 1, \(\frac{1}{2}\), and \(\frac{1}{3}\) turns out to be 1. This is our fundamental period for the combination function.

Data Points

Generating data points is an important step when plotting graphs of functions. Data points allow us to visualize the function’s behavior over a specified range.
For our combination function \(\text{cos}2\pi t + \text{cos}4\pi t + \frac{1}{2} \text{cos}6\pi t\), we need to generate data points from \(t=0\) to \(t=1\), because we found the fundamental period to be 1.
To generate these data points:

• Choose several values of \(t\) between 0 and 1.
• Calculate the value of \(\text{cos}2\pi t + \text{cos}4\pi t + \frac{1}{2} \text{cos}6\pi t\) for each \(t\).

By plotting these points on a graph, where the x-axis represents time \(t\) and the y-axis represents the function value, we create a visual representation of the function over one period. This graph helps us understand the behavior of the combined trigonometric functions.