Problem 60 - Algebra and Trigonometry

Question

\(59-60\) \(f, g,\) and \(f+g\) on a common screen to illustrategraphical addition. $$ f(x)=\sin x, \quad g(x)=\sin 2 x $$

Step-by-Step Solution

Answer

Graph \(f(x) = \sin x\), \(g(x) = \sin 2x\), and \(f+g(x) = \sin x + \sin 2x\) on the same plane.

Step 1: Understand the functions

We have two functions, \(f(x) = \sin x\) and \(g(x) = \sin 2x\). The function \(f(x)\) is the sine function, which has a period of \(2\pi\). The function \(g(x)\) is the sine function with the argument \(2x\), which means it has a period of \(\pi\).

Step 2: Define the function \(f+g\)

The function \(f+g\) is defined as the sum of \(f(x)\) and \(g(x)\): \((f+g)(x) = \sin x + \sin 2x\). This function will combine the properties of both \(f(x)\) and \(g(x)\).

Step 3: Graph \(f(x) = \sin x\)

Plot \(f(x) = \sin x\) on the graph. The sine function starts at 0, reaches its peak at \(\frac{\pi}{2}\), goes back to 0 at \(\pi\), reaches -1 at \(\frac{3\pi}{2}\), and returns to zero at \(2\pi\). Repeat this pattern to cover the desired range.

Step 4: Graph \(g(x) = \sin 2x\)

Plot \(g(x) = \sin 2x\) on the same graph. This function completes a full cycle between 0 and \(\pi\) due to its period of \(\pi\). Plot it by starting at 0, reaching 1 at \(\frac{\pi}{4}\), back to 0 at \(\frac{\pi}{2}\), going to -1 at \(\frac{3\pi}{4}\), and returning to 0 at \(\pi\).

Step 5: Graph \((f+g)(x) = \sin x + \sin 2x\)

Finally, graph \((f+g)(x) = \sin x + \sin 2x\) by adding the values of \(f(x)\) and \(g(x)\) at each corresponding \(x\)-value. At \(x=0\), \(f(0) = \sin 0 = 0\) and \(g(0) = \sin 0 = 0\), so \(f+g(0) = 0\). Continue this addition throughout the cycle to illustrate how the two functions combine.

Key Concepts

Sine Function

The sine function, often written as \(f(x) = \sin x\), is one of the fundamental trigonometric functions and is essential for understanding waves and oscillations. It produces a smooth and repeating wave that is symmetric about the origin.
- The function starts at zero when \(x = 0\).- It reaches its maximum value of 1 at \(x = \frac{\pi}{2}\).- It descends back to zero at \(x = \pi\), and touches its minimum value of -1 at \(x = \frac{3\pi}{2}\).- The function then returns to zero at \(x = 2\pi\), completing one cycle of its wave.
This pattern repeats indefinitely. Sine waves are used to model many natural phenomena, from sound waves to tides. Understanding its graph is crucial when analyzing more complex functions.

Function Period

The period of a function is the distance required for it to complete one full cycle before its pattern begins to repeat. The period of the standard sine function \(\sin x\) is \(2\pi\), as it completes one oscillation over this interval.
However, altering the function's argument affects its period. For \(g(x) = \sin 2x\), the argument '2x' indicates that the function's period is half the standard sine period, reducing it to \(\pi\). This means \(\sin 2x\) completes a full cycle in the span from 0 to \(\pi\), making it repeat twice as fast.
Recognizing how to determine the period is vital when dealing with trigonometric graphs, as it helps predict how the function behaves over time.

Trigonometric Graphs

Graphing trigonometric functions like \(\sin x\) and \(\sin 2x\) helps visualize their behavior and relationships. Starting from the segment definitions for these sines:

• For \(\sin x\), the graph is a smooth wave oscillating between -1 and 1 over an interval of \(2\pi\).
• For \(\sin 2x\), it oscillates within the smaller interval \(\pi\).

Plotting these on the same graph highlights their respective patterns and how they combine, showing each function's periodic nature and symmetry. Graphs are powerful tools for gaining intuition about these functions and aid significantly in understanding the effects of their combination.

Function Addition

Function addition, particularly of trigonometric functions, involves creating a new function by summing the values of two or more functions for each point on the interval. For example, \((f+g)(x) = \sin x + \sin 2x\).
To perform this addition graphically:

• Identify each function's value at a given \(x\)-point.
• Add these values together to find the resultant point for \((f+g)(x)\).

By graphing the resulting points, you obtain a new wave pattern that incorporates features of both \(\sin x\) and \(\sin 2x\). The graphical addition visually demonstrates how the peaks and troughs of the two constituent waves interact, creating a more complex waveform. This concept is extensively used in physics and engineering, especially for analyzing wave interference and combining signals.