Problem 84 - Precalculus

Question

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=\sin (2 x)+\sin (3 x),-\pi \leq x \leq \pi$$

Step-by-Step Solution

Answer

Graph \(y = \sin(2x) + \sin(3x)\) by summing the y-values of \(\sin(2x)\) and \(\sin(3x)\) for points in \([-\pi, \pi]\).

Step 1: Understand the Problem

We need to graph the function given by the sum of two sine functions, \(y = \sin(2x) + \sin(3x)\), over the interval \(-\pi \leq x \leq \pi\). This means we will evaluate and add the ordinates, i.e., the y-values, of \(\sin(2x)\) and \(\sin(3x)\) at various x-values over the specified interval.

Step 2: Graph \(\sin(2x)\) and \(\sin(3x)\) Separately

Graph the functions \(y_1 = \sin(2x)\) and \(y_2 = \sin(3x)\) on separate axes over the interval \(-\pi \leq x \leq \pi\). Note the periodicity: \(\sin(2x)\) has period \(\pi\) and \(\sin(3x)\) has period \(\frac{2\pi}{3}\). Sketch their sine waves accordingly.

Step 3: Calculate and Add Ordinates at Key Points

Pick key points within the interval (e.g., \(x = -\pi, -\frac{3\pi}{4}, -\frac{\pi}{2}, ..., \pi\)). For each point, compute \(y_1 = \sin(2x)\) and \(y_2 = \sin(3x)\), then sum these values to get \(y = \sin(2x) + \sin(3x)\).

Step 4: Plot the Summed Function

Using the computed ordinate values from the previous step, plot the points on a graph. Connect these points to see the overall shape of the summed function \(y = \sin(2x) + \sin(3x)\). Since it is a sum of periodic functions, expect a pattern of oscillations.

Step 5: Analyze the Graph

Examine the graph to understand the overall behavior of the summed function \(y = \sin(2x) + \sin(3x)\). The waveform should show varying amplitudes depending on whether \(\sin(2x)\) and \(\sin(3x)\) are in phase or out of phase, showing constructive and destructive interference.

Key Concepts

Graphing

When you're working with trigonometric functions, graphing is a key skill that helps visualize how these functions behave. It's especially important when you are combining functions, such as the sum of \( \sin(2x) \) and \( \sin(3x) \). To graph these, you start by plotting each function separately. This involves selecting various x-values within the given interval, \( -\pi \leq x \leq \pi \), and calculating the corresponding y-values. By carefully plotting these y-values against the x-values, you begin to see the curve that each function takes.

• \( y_1 = \sin(2x) \) stretches and compresses as it completes oscillations every \( \pi \) units.
• \( y_2 = \sin(3x) \) demonstrates its oscillations within a shorter frame of \( \frac{2\pi}{3} \) units.

After graphing these individual functions, carefully add their ordinates – the y-values for the same x-values across these graphs. By plotting these summed values, you can create the graph of \( y = \sin(2x) + \sin(3x) \). Connect these plotted points smoothly to form the resultant wave.

Periodicity

Periodicity refers to the repeating nature of trigonometric functions. Each function returns to the same value after a fixed interval, known as its period. In the equation \( y = \sin(2x) + \sin(3x) \), each sine component has its own period.

• The function \( \sin(2x) \) completes one full cycle in \( \pi \) units.
• The function \( \sin(3x) \) completes its cycle in \( \frac{2\pi}{3} \) units.

By understanding the periodicity, you can predict where these functions will repeat their shapes and values. This helps in plotting because you only need to compute and plot values over one full cycle and can then repeat this pattern across the interval. When these variable periodicities combine, they may result in a complex, non-repeating pattern within the given interval, adding variety to the resulting summed function graph.

Sine Function

The sine function is one of the most fundamental trigonometric functions. It describes how the amplitude, or height, of a wave changes as you move along the x-axis. With sine functions, the wave-like pattern arises from their periodic nature.

• For \( \sin(2x) \), the coefficient 2 causes the wave to cycle twice as fast as \( \sin(x) \).
• In \( \sin(3x) \), the coefficient 3 makes it cycle three times as fast as \( \sin(x) \).

The sine function's pattern represents a smooth oscillation that starts at the origin, peaks, comes back down to zero, dips to its minimum, and returns to zero. When working with the sum of sine functions, like \( y = \sin(2x) + \sin(3x) \), consider each function's individual impact on the wave's overall shape. Each influences the amplitude and frequency based on its own parameters.

Interference

Interference occurs when two waves overlap and combine, producing regions of increased or decreased amplitude. This occurs in trigonometric functions when you sum functions like \( \sin(2x) \) and \( \sin(3x) \). Two types of interference affect the shape of the summed function:

• Constructive Interference: When the peaks of \( \sin(2x) \) align with those of \( \sin(3x) \, \) the amplitudes add up, resulting in a larger overall peak.
• Destructive Interference: When the peak of one wave coincides with the trough of another, they partially or completely cancel each other out, resulting in reduced amplitude or even points of zero amplitude.

This alternation between constructive and destructive interference leads to the distinct patterns and varying wave shapes observed in the graph of the function \( y = \sin(2x) + \sin(3x) \.\) While individual wave components have predictable patterns, their combination results in intricate and sometimes unexpected waveforms across \( -\pi \leq x \leq \pi \).