Problem 53 - Algebra 2

Question

Use a graphing calculator to graph each function in the interval from 0 to 2\(\pi .\) Then sketch each graph. $$ y=\sin (x+\cos x) $$

Step-by-Step Solution

Answer

The graph of \(y=\sin (x+\cos x)\) within the interval 0 to \(2\pi\) depicts a sine wave with elasticities due to the phase shift introduced by the addition of \(\cos x.\) These variations demonstrate how the phase shift alters the basic sine function.

Step 1: Recall the graph of the sine function

In the interval from 0 to \(2\pi ,\) the sine function, \(y=\sin x,\) makes one complete wave starting from zero, reaching a maximum value of 1 at \(\pi /2,\) returning to zero at \(\pi ,\) reaching a minimum value of -1 at \(3\pi /2,\) and returning to zero at \(2\pi .\) This is important to note as it serves as a basis for graphing more complex sine functions.

Step 2: Understand the phase shift

In the given function, \(y=\sin (x+\cos x),\) the term \(x+\cos x\) indicates a phase shift in the sine function, as this term is in place of where normally just \(x\) would be in the sine function. The function \(\cos x\) varies from -1 to 1 in the interval 0 to \(2\pi .\) This phase shift makes the computation of the graph more advanced than the basic sine function as the values of \(x+\cos x\) will vary for the same \(x\) values.

Step 3: Graph the function

With a graphing calculator, input the function \(y=\sin (x+\cos x)\) and set the interval from 0 to \(2\pi .\) Elasticities of the sine wave will be present due to the phase shift of \(\cos x.\) These variations demonstrate the effect of the phase shift on the sine wave.

Key Concepts

Understanding the Sine Function

The sine function is one of the fundamental building blocks in trigonometry. It describes how a wave oscillates through a cycle. Let's look closer at the basic properties of the sine function:

• The function is periodic, meaning it repeats its values in regular intervals known as cycles.
• The typical cycle for \( y = \sin x \) spans from \( x = 0 \) to \( x = 2\pi \).
• Within this interval, the sine wave
  • starts from zero, goes up to a maximum of 1 at \( \pi/2 \),
  • returns back to zero at \( \pi \),
  • reaches a minimum of -1 at \( 3\pi/2 \), and finally
  • back to zero at \( 2\pi \).
• This cycle is unique to the "pure" sine function, or \( y = \sin x \).

By understanding this basic waveform, grasping more complex sine functions becomes easier. Alterations like phase shifts or amplitude changes are just variations on this fundamental pattern.

Exploring Phase Shifts in Trigonometric Functions

A phase shift involves moving the wave horizontally along the x-axis and it's an adjustment in the argument of the sine function.

• The general sine function prefers its argument in the form of \( x \), without additional modifications for straightforward oscillations.
• However, when the argument takes the form \( x + \cos x \), it introduces a phase shift.
• This phase shift is special because the \( \cos x \) portion adds variability, ranging between -1 and 1, which modifies how the sine graph begins and ends its cycles.

In our specific function \( y = \sin(x + \cos x) \), each cycle's start and end will not occur uniformly, due to this added cosine term. This creates a more intricate sine wave, changing more dynamically over the defined interval from 0 to \( 2\pi \). Understanding this helps with the expectation of a more fluid and winding graph, rather than a smooth and predictable sine wave.

Using a Graphing Calculator for Complex Functions

Graphing calculators are invaluable tools for visualizing complex trigonometric functions.

• To see how \( y = \sin(x + \cos x) \) behaves, start by inputting the function into the calculator.
• Adjust the viewing window to match the interval of interest, here from 0 to \( 2\pi \).
• The graph will show how the sine function adjusts due to the phase shift, presenting more elasticity in the waveform.

While observing on a graphing calculator:

• Note how the additional \( \cos x \) in the phase shift causes peaks and troughs to occur unpredictably over each cycle.
• Use the graph's visual features to study these anomalies in detail, such as maxima, minima, and intercepts.

Graphing calculators allow experimentation, enabling you to change parts of the function easily and witness how these adjustments impact the graph visually.