Problem 39 - 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. $$\begin{aligned} &f(x)=\sin x\\\ &g(x)=\cos \left(x-\frac{\pi}{2}\right) \end{aligned}$$

Step-by-Step Solution

Answer

After graphing both functions, we can conjecture that \(f(x)=\sin{x}\) and \(g(x)=\cos{\left(x-\frac{\pi}{2}\right)}\) are equivalent as they both have the same shape, period, and the latter is shifted \(\frac{\pi}{2}\) to the right of the former.

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

Identify the period of the sine function, which is \(2\pi\). Mark points at intervals corresponding to the period and plot the function for a complete cycle showing two full periods.

Step 2: Graph the function \(g(x)=\cos{\left(x-\frac{\pi}{2}\right)}\)

Identify the period of the cosine function which is \(2\pi\). However, this function has a phase shift of \(\frac{\pi}{2}\) to the right. Mark and plot points for the cosine function taking the phase shift into account. Again, draw the graph showing two full periods.

Step 3: Make a conjecture

Once both functions have been graphed, compare the two graphs. Look at their shapes, periods, and starting points.

Key Concepts

Sine Function

The sine function, represented as \( f(x) = \sin{x} \), is a trigonometric function that describes a smooth, wave-like pattern. It is a periodic function, meaning it repeats its values in regular intervals. The period of the sine function is \( 2\pi \), which signifies the length of one complete cycle of the wave. To graph sine, start at the origin (0,0), rise to a peak at \( \pi/2 \), return to the axis at \( \pi \), reach the lowest point at \( 3\pi/2 \), and then return to the axis at \( 2\pi \). This pattern then repeats. Here are some key features:

• Amplitude: The maximum height of the wave from the origin, usually 1.
• Phase Shift: For purely \( \sin x \), it starts at the origin with no shift.
• Vertical Shift: None in the basic sine function, but it can be shifted by adding a constant.

Understanding the sine graph helps you visualize sound waves, light waves, and mechanical vibrations that are modeled by sinusoidal functions.

Cosine Function

The cosine function, denoted by \( f(x) = \cos{x} \), is another fundamental trigonometric function. Similar to sine, it is also periodic with the same period of \( 2\pi \). However, unlike the sine function, cosine starts at its maximum value. For a basic graph of the cosine function:

• Start at the peak (1) at \( x = 0 \).
• Cross the axis at \( \pi/2 \).
• Reach the minimum (-1) at \( \pi \).
• Return to zero at \( 3\pi/2 \).
• Complete the cycle back to the peak at \( 2\pi \).

Key features include:

• Amplitude: Like sine, typically 1.
• Phase Shift: Begins at a maximum, unlike sine which starts at zero.
• Vertical Shift: None, unless altered by adding a constant.

The cosine function is often used in applications involving oscillations, such as electrical currents and sound waves.

Phase Shift

A phase shift refers to the horizontal movement of a periodic function on the coordinate plane. For trigonometric functions like sine and cosine, phase shifts alter the position of the start of the cycle. A function \( g(x) = \cos{(x - \frac{\pi}{2})} \) illustrates a phase shift:

• Right Shift: \( -\frac{\pi}{2} \) moves cosine function to the right by \( \frac{\pi}{2} \).
• It affects when the peaks, troughs, and zero-crossings occur.
• Formula: Generally expressed as \( f(x) = \cos{(x - c)} \), shifting right if \( c > 0 \).

Phase shifts allow one to precisely align trigonometric functions. They are crucial in synchronizing waves and fitting data in real-world scenarios, such as in signal processing and communications.

Graphing Trigonometric Functions

Graphing trigonometric functions like sine and cosine is a vital skill in understanding oscillatory behavior. When graphing, consider:

• Amplitude: Height of the wave, affecting its vertical stretch.
• Period: Length of one complete cycle, traditionally \( 2\pi \) for both sine and cosine.
• Phase Shift: Horizontal shift of the graph, changing where the cycle starts.
• Vertical Shift: Moves the graph up or down without altering its shape.

To graph a function like \( f(x) = \sin{x} \) or \( g(x) = \cos{(x - \frac{\pi}{2})} \), you intuitively plot key points across at least two periods to fully visualize the function. By comparing their graphs, one can make conjectures about relationships, such as how the graphs align or shift against each other. Graphing software can significantly simplify recognizing these patterns and confirming predictions. This skill is important for physics, engineering, and other fields dealing with waveforms and periodic phenomena.