Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. (Include two full periods.) Make a conjecture about the functions. $$\begin{aligned} &f(x)=\sin x\\\ &g(x)=\cos \left(x+\frac{3 \pi}{2}\right) \end{aligned}$$

Step-by-Step Solution

Answer

By examining the graphs of the functions \(f(x) = \sin{x}\) and \(g(x) = \cos{(x + \frac{3\pi}{2})}\), it can be conjectured that these two functions are equivalent due to their same periodicity, maxima and minima, and the effects of phase shifting.

Step 1: Graph the function \(f(x) = \sin{x}\)

The sine function has a period of \(2\pi\) which means it repeats every \(2\pi\) units. The function goes from 0 to 1 at \(\pi/2\), back to 0 at \(\pi\), down to -1 at \(3\pi/2\), and back to 0 at \(2\pi\). It continues this pattern indefinitely. Graph this function using your graphing utility.

Step 2: Graph the function \(g(x) = \cos{(x + \frac{3\pi}{2})}\)

The cosine function also has a period of \(2\pi\), but the given function has a phase shift due to \(x + \frac{3\pi}{2}\). This means the graph of the function is shifted \(\frac{3\pi}{2}\) units in the negative x-direction. At \(x=-\frac{3\pi}{2}\), the function is 1, at \(x=-\pi\), the function is 0, at \(x=-\frac{\pi}{2}\), the function is -1, and at \(x=0\), the function is 0 again. The pattern then repeats every \(2\pi\) units. Graph this function using your graphing utility.

Step 3: Make a conjecture

Look at the plotted functions together in the same viewing window. Notice that both functions have the same period of \(2\pi\) and the same maximum and minimum values of 1 and -1, respectively. However, they are phase-shifted relative to each other. Based on this inspection, you might conjecture that \(g(x) = \cos{(x + \frac{3\pi}{2})}\) is equivalent to \(f(x) = \sin{x}\) due to the properties and shapes of the sine and cosine functions.

Key Concepts

Sine Function

The sine function, denoted as \(f(x) = \sin{x}\), is one of the fundamental trigonometric functions found in mathematics. It is a periodic function with a period of \(2\pi\), which means its pattern repeats every \(2\pi\) units on the x-axis. The sine wave smoothly oscillates between its peak values of 1 and -1.
Some key characteristics of the sine function include:

• It passes through the origin, meaning \(f(0) = 0\).
• Reaches its maximum value of 1 at \(x = \frac{\pi}{2} + 2k\pi\), where \(k\) is any integer.
• Reaches its minimum value of -1 at \(x = \frac{3\pi}{2} + 2k\pi\).

These points form part of the repeating cycle known as the "waveform." Understanding these patterns aids in predicting the behavior of the sine function at any given angle.

Cosine Function

The cosine function, represented by \(g(x) = \cos{(x + \frac{3\pi}{2})}\), is another crucial trigonometric function. Like the sine function, its period is also \(2\pi\), allowing it to complete a full cycle every \(2\pi\) units.
For the standard \cos{x}\ function, you will find the wave shifting away from the origin:

• The maximum value of 1 occurs at \(x = 2k\pi\).
• The minimum value of -1 at \(x = \pi + 2k\pi\).
• It starts at its maximum value when \(x=0\).

With a phase shift present, as in the exercise \(g(x) = \cos{(x + \frac{3\pi}{2})}\), the entire graph is shifted along the x-axis. This shift influences how the cosine function relates to the sine function, often leading to interesting observations when plotted together.

Phase Shift

A phase shift in trigonometric functions refers to the horizontal shifting of the graph along the x-axis. In our exercise, we observe the function \(g(x) = \cos{(x + \frac{3\pi}{2})}\), which appears shifted due to the term \(x + \frac{3\pi}{2}\).
Key Points on Phase Shift:

• A positive \(\frac{3\pi}{2}\) shift means the graph is moved \(\frac{3\pi}{2}\) units to the left.
• This shift does not affect the period, just the starting position of the curve.
• The graph's amplitude and frequency remain unchanged with a phase shift.

Understanding phase shifts is crucial for accurately graphing and interpreting trigonometric functions, particularly when studying how different trigonometric graphs relate to each other.

Graphing Utility

A graphing utility is a tool that simplifies the process of plotting complex mathematical functions such as trigonometric functions. Whether you use graphing calculators, software applications, or online tools, they can visualize functions in the same viewing window, offering a clear understanding of their relationships.
Benefits of Graphing Utilities:

• Quickly render graphs for functions like sine and cosine.
• Visually compare multiple functions and observe their properties, such as phase shifts.
• Explore and confirm conjectures by observing patterns and cycles.

The ability to visualize functions like \(f(x) = \sin{x}\) and \(g(x) = \cos{(x + \frac{3\pi}{2})}\) together helps students deeply understand the influence of shifts and other modifications on their graphical representation.