Problem 5 - Basic Engineering Mathematics

Question

Sketch \(y=3 \sin 2 A\) from \(A=0\) to \(A=2 \pi\) radians Amplitude \(=3\) and period \(=2 \pi / 2=\pi\) rads (or \(180^{\circ}\) ) A sketch of \(y=3 \sin 2 A\) is shown in Fig. 20.18.

Step-by-Step Solution

Answer

The graph of \(y = 3 \sin 2A\) is sinusoidal, with an amplitude of 3, a period of \(\pi\), repeating cycles from \(0\) to \(2\pi\).

Step 1: Understand the function

The function given is \(y = 3 \sin 2A\). Here, 3 is the amplitude, and the coefficient 2 inside the sine function affects the period of the function.

Step 2: Determine the amplitude

The amplitude is the maximum value the sine wave will reach from its equilibrium (zero) position. For this function, the amplitude is 3, meaning the wave will peak at +3 and trough at -3.

Step 3: Calculate the period

The period of a sine function \(y = a \sin bA\) is given by \(\frac{2\pi}{|b|}\). Here, \(b = 2\), so the period is \(\frac{2\pi}{2} = \pi\). This means the function will complete one full cycle over an interval of \(\pi\).

Step 4: Identify key points for plotting

For sine functions, key points occur at intervals of a quarter period (\frac{\pi}{4} in this case): \(0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \) and \(\pi\). Evaluate these points: \(y(0) = 0\), \(y(\frac{\pi}{4}) = 3\), \(y(\frac{\pi}{2}) = 0\), \(y(\frac{3\pi}{4}) = -3\), \(y(\pi) = 0\).

Step 5: Plot the sine curve

On a graph, plot the points computed from Step 4 along the horizontal axis from \(A = 0\) to \(A = 2\pi\). Repeat the pattern for the interval from \(A = \pi\) to \(2\pi\) to show one complete cycle from \(A = 0\) to \(2\pi\).

Step 6: Complete the sketch

Draw a smooth, sinusoidal curve connecting the plotted points that peaks at +3, goes through zero, troughs at -3, and back through zero across one cycle from \(0\) to \(\pi\). Repeat this pattern for \(\pi\) to \(2\pi\) to complete the sketch.

Key Concepts

Amplitude of a Sine Wave

The amplitude of a sine wave is essential to understand because it tells us how far the function's values can spread from the midpoint of the cycle.
The amplitude is the height of the wave from the center line (equilibrium position) to the peak (or the trough).
In the equation of the form \(y = a \sin bA\), the amplitude is represented by the coefficient \(a\).

Here, you’re looking at the function \(y = 3 \sin 2A\).
The amplitude is 3, which means the wave reaches a maximum vertical distance of 3 units above or below the line \(y = 0\).
This dictates the wave’s largest and smallest values being +3 and -3, respectively.

• Important: The amplitude only affects the vertical stretch or shrink of the sine wave.
• Observe that different numbers will only change how tall or short the curve appears.

Period of the Sine Function

The period of a sine function refers to the distance along the x-axis before the function starts repeating itself.
It's like one full wave cycle.
For a sine function \(y = a \sin bA\), the period can be calculated using the formula \(\frac{2\pi}{|b|}\).

In the example of \(y = 3 \sin 2A\), \(b = 2\), which means the period is \(\frac{2\pi}{2} = \pi\).
This tells us that the function completes its cycle in an interval of \(\pi\) radians, or 180 degrees.

• Key insight: A higher \(b\) value decreases the period, causing the sine wave to cycle more frequently.
• This value defines how squished or stretched the wave looks horizontally.
• Period is all about repetition along the horizontal axis, not height.

Graph Plotting of a Sine Function

Graph plotting of a sine function involves identifying key points that form the characteristic sinusoidal wave shape on a graph.
For the function \(y = 3 \sin 2A\), these points occur at important intervals across its period.

Here’s a step-by-step guide on plotting:
1. **Identify key points:** For a half cycle, these are 0, \(\frac{\pi}{4}\), \(\frac{\pi}{2}\), \(\frac{3\pi}{4}\), and \(\pi\), which provide a smooth curve when plotted. - At \(A = 0\): \(y = 0\) - At \(A = \frac{\pi}{4}\): \(y = +3\) - At \(A = \frac{\pi}{2}\): \(y = 0\) - At \(A = \frac{3\pi}{4}\): \(y = -3\) - At \(A = \pi\): \(y = 0\)2. **Extend to a full cycle:** Repeat the shape for the second half from \(\pi\) to \(2\pi\) to complete a full cycle.3. **Draw a smooth curve:** Connect these points with a flowing curve to form the sinusoidal graph.

• Visualization: Use graph paper for accuracy or graph tools for precision.
• Remember, each plot and connection captures the wave's rise, peak, decline, and trough perfectly.