Problem 58 - Algebra and Trigonometry

Question

Graph \(f, g,\) and \(f+g\) on a common screen to illustrate graphical addition. $$ f(x)=\sin x, \quad g(x)=\sin 2 x $$

Step-by-Step Solution

Answer

Graph \( \sin x \), \( \sin 2x \), and \( \sin x + \sin 2x \) to show interference patterns.

Step 1: Understand the Functions

In this exercise, you need to graph three separate functions: \( f(x) = \sin x \), \( g(x) = \sin 2x \), and their sum \( f(x) + g(x) = \sin x + \sin 2x \). These functions represent trigonometric waves with different frequencies.

Step 2: Plot \( f(x) = \sin x \)

Plot the function \( f(x) = \sin x \). This function is a standard sine wave with a period of \( 2\pi \) and amplitude 1. It completes one full cycle as \( x \) goes from \( 0 \) to \( 2\pi \).

Step 3: Plot \( g(x) = \sin 2x \)

Plot the function \( g(x) = \sin 2x \). This function is also a sine wave, but with a higher frequency. It's period is \( \pi \) which means it completes two full cycles from \( x = 0 \) to \( x = 2\pi \).

Step 4: Plot \( f(x) + g(x) = \sin x + \sin 2x \)

Plot their sum \( f(x) + g(x) = \sin x + \sin 2x \). This is the result of adding the values of \( \sin x \) and \( \sin 2x \) for each \( x \). You should observe an interference pattern because of the different frequencies.

Step 5: Compare the Graphs

On the same set of axes, observe how the composite wave \( \sin x + \sin 2x \) behaves in relation to the individual waves \( \sin x \) and \( \sin 2x \). Notice the beats and how the wave peaks and troughs change frequency and amplitude due to constructive and destructive interference.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in describing oscillations like sound waves and tides. These functions include sine, cosine, and tangent, but for this exercise, we focus on sine functions.

• **Sine Function**: The sine function, noted as \( \sin x \), gives the y-coordinate of a point on the unit circle. It varies smoothly and periodically, forming a wave-like pattern called a sine wave.
• **Frequency and Period**: The frequency of a sine wave describes how often the wave oscillates over a fixed period, while the period is the length over which the wave repeats. In the function \( \sin x \), the period is \(2\pi\), meaning one complete cycle occurs from \(0\) to \(2\pi\).
• **Amplitude**: This is the height from the center line (or the x-axis) to the peak of the wave. For the standard sine wave, \( \sin x \), the amplitude is 1.

Understanding these properties is crucial for graphing the functions and seeing how they interact when added together.

Exploring Sine Waves

Sine waves are visually recognized by their smooth, continuous wave shape. Let's break down the characteristics of the waves we are graphing in this exercise.

• For \( f(x) = \sin x \): - **Period**: \(2\pi\) - **Frequency**: Lower - **Shape**: Standard Sine Wave
• For \( g(x) = \sin 2x \): - **Period**: \(\pi\) - **Frequency**: Double that of \( \sin x \)

The increase in frequency for \( g(x) \) means it oscillates more quickly and completes two cycles for every single cycle of \( f(x) \). When graphed, these visual differences help in understanding frequency changes and wave behavior.

Graphical Addition of Waves

Graphical addition involves adding two or more waveforms to produce a composite wave. This results in new patterns that exhibit interference effects.

• **Constructive Interference**: Occurs when wave peaks align, leading to an increased amplitude.
• **Destructive Interference**: Happens when wave peaks and troughs coincide, potentially reducing amplitude.

When graphing \( f(x) + g(x) = \sin x + \sin 2x \), you witness these interference patterns. The composite wave shows changes in frequency and amplitude depending on how the waves \( \sin x \) and \( \sin 2x \) interact with each other. Observing these changes provides insight into how complex waveforms are formed and manipulated.