Problem 127 - Precalculus

Question

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

Step-by-Step Solution

Answer

The graphs are identical, showing that \(\sin x = \cos\left(x-\frac{\pi}{2}\right)\).

Step 1: Understand the Functions

We have two functions to graph: \(Y_1=\sin x\) and \(Y_2=\cos\left(x-\frac{\pi}{2}\right)\). The first function is the sine function, and the second is a cosine function that is shifted by \(\frac{\pi}{2}\) radians.

Step 2: Graph the Sine Function

Using a graphing calculator, enter the function \(Y_1=\sin x\). Plot this function across a range of at least one full period (0 to \(2\pi\)) to observe its characteristic wave-like pattern.

Step 3: Adjust the Cosine Function

Enter the second function \(Y_2=\cos\left(x-\frac{\pi}{2}\right)\) into the graphing calculator. This expression represents a cosine function, but due to the \(\frac{\pi}{2}\) horizontal shift, it will initially look like a sine function.

Step 4: Graph the Cosine Function

Plot \(Y_2=\cos\left(x-\frac{\pi}{2}\right)\) on the same graph as \(Y_1=\sin x\). Make sure to use the same x-axis range to directly compare the two functions.

Step 5: Analyze the Graphs

Observe the two graphs. Notice that \(Y_1=\sin x\) and \(Y_2=\cos\left(x-\frac{\pi}{2}\right)\) lie on top of each other. This confirms that \(\sin x\) is equivalent to \(\cos\left(x-\frac{\pi}{2}\right)\) graphically.

Key Concepts

Sine Function

The sine function, represented as \( y = \sin x \), is one of the fundamental trigonometric functions. It plays a crucial role in mathematical modeling of periodic phenomena. The graph of the sine function is a smooth, continuous wave that oscillates between -1 and 1.

• It has a period of \( 2\pi \), meaning it repeats every \( 2\pi \) units.
• It starts at the origin (0,0) and when x increases from 0 to \( \frac{\pi}{2} \), the sine value rises from 0 to 1.
• From \( \frac{\pi}{2} \) to \( \pi \), it decreases back to 0, reaching -1 at \( \frac{3\pi}{2} \) and completes the cycle returning to 0 at \( 2\pi \).

The wave-like pattern of the sine function makes it ideal for representing cyclical behavior such as sound waves or tides.

Cosine Function

The cosine function, expressed as \( y = \cos x \), is closely related to the sine function. Similar to sine, it is used to model periodic behavior and shares a number of characteristics with sine.

• The period of \( \cos x \) is also \( 2\pi \), providing a repetitive wave pattern.
• Unlike sine, the cosine function starts at its peak value of 1 when \( x = 0 \).
• As \( x \) increases to \( \pi \), the value of \( \cos x \) decreases to -1, and it returns to 1 at \( 2\pi \).

In essence, the graphs of cosine and sine are similar, but start at different points on the y-axis. This phase difference is crucial in understanding the relationship between the two functions.

Phase Shift

Phase shift occurs when a trigonometric function is horizontally shifted along the x-axis. For the function \( y = \cos(x - \frac{\pi}{2}) \), a phase shift is applied.

• A phase shift of \( \frac{\pi}{2} \) to the right modifies the cosine function to start at the same point as the sine function.
• This specific shift transforms \( \cos x \) into \( \sin x \), effectively aligning the graphs of the two functions.
• Understanding phase shift is key in analyzing oscillations and patterns, as it allows us to predict and manipulate the timing of cycles.

In this particular exercise, observing the phase shift of \( \frac{\pi}{2} \) demonstrates the equivalency of the sine function to a shifted cosine function, reinforcing their interconnectedness in trigonometry.