Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=\frac{1}{100} \sin 120 \pi t$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=\frac{1}{100} \sin 120 \pi t\) is \(\frac{1}{100}\). The period of the function is \(\frac{1}{60}\).

Step 1: Identify the Amplitude

The amplitude of a function \(y=A\sin(Bx)\) is \(|A|\). For the given function \(y=\frac{1}{100} \sin 120 \pi t\), the amplitude A is \(\frac{1}{100}\), which is a positive number. Therefore, the amplitude of the function is \(\frac{1}{100}\).

Step 2: Identify the Period

The period of a function \(y=A\sin(Bx)\) is \(\frac{2\pi}{|B|}\). For the given function \(y=\frac{1}{100} \sin 120 \pi t\), B is \(120\pi\). Therefore, the period of the function is \(\frac{2\pi}{120\pi}\) that simplifies to \(\frac{1}{60}\).

Step 3: Graph the Function

Please use a suitable graphing utility to plot the function \(y=\frac{1}{100} \sin 120 \pi t\) and include two full periods. Given the amplitude is very small and the period is large, you may need to adjust your viewing window to clearly see the graph.

Key Concepts

Amplitude

When it comes to trigonometric functions, the amplitude determines how far up and down the wave reaches from its midline. It is essentially the height of the wave. For a sine function of the form \( y = A \sin(Bx) \), the amplitude is given by \( |A| \). This coefficient \( A \) will always be a positive number as amplitude is about distance, and distance can't be negative.

In the exercise's function \( y=\frac{1}{100} \sin 120 \pi t \), the amplitude is \( |\frac{1}{100}| \), which simplifies to \( \frac{1}{100} \). This tells us that the sine wave will oscillate 1/100th unit above and below the central horizontal axis, making it a very "flat" looking wave compared to standard sine waves, where the amplitude is usually \( 1 \).

This tiny amplitude results in a graph where the sine wave struggles to lift away from the x-axis. Understanding amplitude is crucial to interpreting how much a wave fluctuates from its equilibrium position, making it an essential characteristic in graphing and analyzing trigonometric functions.

Period of a Function

The period of a trigonometric function tells us about its cycle length — how frequently the wave pattern repeats as we move along the horizontal axis. For the standard sine function \( y = A \sin(Bx) \), the period is calculated using the formula \( \frac{2\pi}{|B|} \).

For the function \( y = \frac{1}{100} \sin 120 \pi t \) in the exercise, we determine \( B \) to be \( 120\pi \). Substituting into the period formula, we find the period is \( \frac{2\pi}{120\pi} \), which simplifies to \( \frac{1}{60} \).

This means that the wave completes one full cycle every \( \frac{1}{60} \) units along the t-axis. Hence, in a span of just \( \frac{1}{60} \) of whatever units we use for time \( t \), the sine wave makes one full transition from its starting point back to the equivalent point, indicating a very rapid oscillation. Understanding the period is critical for predicting the behavior of wave-like phenomena over time.

Sine Function

The sine function is one of the foundational trigonometric functions used to model periodic behavior, such as sound waves, light waves, and tides. It is represented as \( y = A \sin(Bx) \), where \( A \) influences the height of the wave (amplitude) and \( B \) impacts how quickly the wave repeats its pattern (period).

In a standard form, the sine function oscillates between \(-1\) and \(1\) with a default amplitude of \(1\) and a period of \(2\pi\). However, when coefficients are introduced, they modify these characteristics, allowing us to tailor the function to fit specific data or scenarios, as seen in \( y=\frac{1}{100} \sin 120 \pi t \).

Sine functions are known for their smooth, continuous wave patterns which repeat at regular intervals. These regular intervals are exceptionally useful for modeling cyclic phenomena found in nature and engineering. Mastery of the sine function and its parameters can aid in understanding and predicting repetitive processes in various scientific and real-world applications.