Problem 128 - Precalculus

Question

Use a graphing calculator to graph \(Y_{1}=\cos x\) and \(Y_{2}=\sin \left(x+\frac{\pi}{2}\right) .\) What do you notice?

Step-by-Step Solution

Answer

The graphs of \( Y_1 = \cos x \) and \( Y_2 = \sin(x + \frac{\pi}{2}) \) coincide.

Step 1: Understand the Functions

The functions given are trigonometric functions. The first function, \( Y_1 = \cos x \), is the cosine function. The second function, \( Y_2 = \sin\left(x + \frac{\pi}{2}\right) \), is the sine function with a phase shift of \( \frac{\pi}{2} \) to the left.

Step 2: Analyze the Phase Shift

The sine function, \( \sin(x) \), becomes \( \sin(x + \frac{\pi}{2}) \) when shifted left by \( \frac{\pi}{2} \). In trigonometric terms, sinsine function with this shift is equivalent to the cosine function: \( \sin(x + \frac{\pi}{2}) = \cos(x) \).

Step 3: Graph the Functions

Use a graphing calculator or software to plot both functions on the same set of axes. Input \( Y_1 = \cos x \) and \( Y_2 = \sin(x + \frac{\pi}{2}) \) to create the graphs.

Step 4: Observe the Graphs

Upon graphing, you will observe that the graphs of \( Y_1 \) and \( Y_2 \) coincide. They appear as a single graph because \( \sin(x + \frac{\pi}{2}) \) is equivalent to \( \cos(x) \).

Key Concepts

Phase Shift

When we talk about phase shift in trigonometric functions, we're delving into how a graph is moved horizontally along the x-axis. This shift can either be toward the right or the left depending on the sign of the phase shift value.

• A positive phase shift, such as in the function \( f(x) = \, \sin(x + \frac{\pi}{2}) \), indicates a movement to the left.
• A negative phase shift moves the graph to the right.

For \( Y_2 = \sin(x + \frac{\pi}{2}) \), the entire sine wave moves \( \frac{\pi}{2} \) units to the left. This movement results in a sine graph that aligns with the cosine graph. The sine function naturally repeats every \( 2\pi \), so this shift effectively changes its phase, making it match with another periodic expression— the cosine function in this case.
Understanding phase shifts helps predict and visualize how trigonometric graphs transform and assists greatly in comparing functions to identify equivalences.

Cosine Function

The cosine function is a fundamental function in trigonometry, represented by \( Y = \cos(x) \). It displays some unique properties that are helpful in understanding wave-like behaviors.

• The cosine graph creates a wave that begins at a peak when \( x = 0 \), descending down the y-axis, hitting a trough, then ascending back to a peak forming a full cycle over \( 2\pi \).
• It is considered an even function, meaning it is symmetric about the y-axis. So, \( \cos(-x) = \cos(x) \).

Its entire waveform completes one cycle in the same length of \( 2\pi \). This feature of the cosine function becomes very useful when comparing it with other trigonometric functions, like the sine function. In particular, it helps in understanding why \( \sin(x + \frac{\pi}{2}) \) appears identical to \( \cos(x) \) when plotted, highlighting the impact of phase shifts and function transformations.

Sine Function

The sine function, \( Y = \sin(x) \), forms another essential part of trigonometric graphs. It is closely intertwined with its counterpart, the cosine function, by its properties and transformations.

• The sine graph starts at the origin \((0,0)\), then moves upwards to hit a maximum at \( \frac{\pi}{2} \), descends through zero at \( \pi \), and hits a minimum at \( \frac{3\pi}{2} \) before returning to zero at \( 2\pi \).
• Unlike the cosine function, it is an odd function. This means that \( \sin(-x) = -\sin(x) \).

Interestingly, when the sine graph undergoes a horizontal shift of \( \frac{\pi}{2} \) units to the left, it transforms into the cosine graph. This relationship can greatly simplify graphing transformations and can solidify the understanding of trigonometric functions working together. Recognizing this harmony between sine and cosine is not only mathematically elegant but also practically useful in fields such as physics and engineering, where harmonic motions are studied.