Problem 1 - Algebra and Trigonometry

Question

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=2 \sin 3 t$$

Step-by-Step Solution

Answer

Amplitude: 2, Period: \(\frac{2\pi}{3}\), Frequency: \(\frac{3}{2\pi}\).

Step 1: Identify the Function Structure

The given function for the displacement is \( y = 2 \sin 3t \). This is in the form \( y = A \sin(Bt + C) + D \), where \( A \) is the amplitude, \( \frac{2\pi}{B} \) is the period, and the function is shifted by \( D \) units vertically.

Step 2: Determine the Amplitude

The amplitude \( A \) of the function is the absolute value of the coefficient in front of the sine function. Thus, the amplitude is \( |2| = 2 \).

Step 3: Calculate the Period

The period \( T \) of a sine function is given by \( \frac{2\pi}{B} \).In this function, \( B = 3 \).\[T = \frac{2\pi}{3}\]

Step 4: Calculate the Frequency

The frequency \( f \) is the reciprocal of the period, i.e., \( f = \frac{1}{T} \).\[f = \frac{3}{2\pi}\]

Step 5: Sketch the Graph

A sketch of the graph of \( y = 2 \sin 3t \) should be drawn, showing its oscillation from \(-2\) to \(2\) over one complete period from \( t = 0 \) to \( t = \frac{2\pi}{3} \). The curve starts at the origin, reaches its maximum at a quarter period, crosses zero at half period, reaches its minimum at three-quarters, and returns to zero at the full period.

Key Concepts

Amplitude

The amplitude of a wave describes how far the wave oscillates from its central position. Think of it as the height of the wave from its middle point to its top, or from the middle down to its bottom. For the function

• \(y = 2 \sin 3t\)

we can see the number 2 right before the sine function. This number tells us the amplitude.
The amplitude is always a positive number, so we take the absolute value of 2, which is simply 2.
Therefore, the wave swings 2 units up and 2 units down from the center line. This is why you will see the wave move between -2 and 2 on the graph.

Period

The period of a trigonometric function is the time it takes for the wave to complete one full cycle before starting over. To determine the period of our function

• \(y = 2 \sin 3t\)

look at the coefficient inside the sine function, which is 3 in this case.
The period \(T\) for a sine function is found using the formula

• \(T = \frac{2\pi}{B}\)

where \(B\) is the coefficient of \(t\).
For our function, this means the period is

• \(T = \frac{2\pi}{3}\)

This tells us that every \(\frac{2\pi}{3}\) time units, the sine wave pattern repeats.

Frequency

Frequency refers to how often the wave cycles occur in a specific interval. It is the inverse of the period, and tells us how many cycles happen per unit of time. For our sine function,

• \(y = 2 \sin 3t\)

we already found the period \(T\) to be

• \(\frac{2\pi}{3}\).

The frequency \(f\) is calculated as

• \(f = \frac{1}{T}\).

So,

• \(f = \frac{3}{2\pi}\)

This means that in one time unit, the sine wave completes

• \(\frac{3}{2\pi}\)

cycles.

Sine Function

The sine function is one of the most common trigonometric functions and involves wave-like oscillations. It is periodic, meaning it repeats its pattern away from the source over regular intervals.
The function

• \(y = \sin t\)

describes a wave starting at 0, reaching a peak at \(\frac{\pi}{2}\), descending back through zero to -1 at \(\frac{3\pi}{2}\), and coming back to zero at \( 2\pi \).
This makes the sine function an excellent model for circular motion or oscillations, like swinging or sound waves.
In our equation

• \(y = 2 \sin 3t\)

the sine function is adjusted by the amplitude and frequency to fit different real-life scenarios.

Graphing Trigonometric Functions

Graphing trigonometric functions like sine requires understanding of how the amplitude, period, and frequency alter the curve. To graph the function

• \(y = 2 \sin 3t\)

follow these steps:

• Identify the amplitude (2 in this case), showing the wave height from the middle to the peak.
• Calculate the period \(\frac{2\pi}{3}\), demonstrating how long before the wave pattern repeats.
• Plot points along the curve starting at the origin
  • (0, 0)
  reach the highest amplitude at one-quarter \(\frac{2\pi}{12}\).
• Cross zero at halfway period
  • \(\frac{\pi}{3}\).
• Reach the lowest amplitude at three-quarters
  • \(\frac{\pi}{2}\).
• Return to zero at full period
  • \(\frac{2\pi}{3}\).

Seeing this on the graph shows how effectively waves are modeled using sine functions.