Problem 39 - Algebra 2

Question

Compare the graphs of \(y=-\sin \left[\frac{1}{4}\left(\theta-\frac{\pi}{2}\right)\right]\) and \(y=\cos \left[\frac{1}{4}\left(\theta+\frac{3 \pi}{2}\right)\right]\)

Step-by-Step Solution

Answer

The graphs have the same amplitude and period but differ in phase shift and vertical reflection.

Step 1: Identify the Standard Form

The standard form for the sine function is \( y = -rac{1}{4} imes heta \), and for the cosine function, it's \( y = rac{1}{4} imes heta \) with respective translations and reflections. This will help identify the key features of each function.

Step 2: Analyze the Sine Function Transformations

The sine function provided is \( y = -\sin\left[\frac{1}{4}\left(\theta - \frac{\pi}{2}\right)\right] \). Here:- Amplitude: 1 (multiplied by -1 so inverted)- Period: \( 2\pi \times 4 = 8\pi \) due to the \(\frac{1}{4}\) factor.- Phase shift: \(+\frac{\pi}{2}\) to the right due to \(-\frac{\pi}{2}\).

Step 3: Analyze the Cosine Function Transformations

The cosine function is \( y = \cos\left[\frac{1}{4}\left(\theta + \frac{3\pi}{2}\right)\right] \). Here:- Amplitude: 1- Period: \( 8\pi \) due to the \( \frac{1}{4} \) factor.- Phase shift: \(-\frac{3\pi}{2}\) to the left due to \(+\frac{3\pi}{2}\).

Step 4: Compare the Amplitudes

Both functions have an amplitude of 1, meaning they oscillate between -1 and 1 on the y-axis. However, the sine function is inverted.

Step 5: Compare the Periods

Both functions have the same period of \( 8\pi \), so they complete one full cycle over the same interval on the x-axis.

Step 6: Compare the Phase Shifts

The sine function has a phase shift to the right by \(\frac{\pi}{2}\), while the cosine function shifts to the left by \(\frac{3\pi}{2}\). This means their starting points are displaced with respect to each other.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. Its standard form is expressed as \( y = \sin(\theta) \). In the given exercise, the sine function is transformed into \( y = -\sin\left[\frac{1}{4}\left(\theta - \frac{\pi}{2}\right)\right] \). Let's break down this transformation:

• **Amplitude:** The amplitude is the maximum value the function reaches. Here, it is 1, but the "-" sign indicates the function is inverted. The graph reflects about the x-axis.
• **Period:** The period represents how long it takes the graph to complete one cycle. It can be calculated using \( 2\pi \times \frac{1}{\frac{1}{4}} = 8\pi \). This means that the complete wave stretches over an interval of \( 8\pi \).
• **Phase Shift:** This refers to a horizontal translation from the typical start. A shift of \(+\frac{\pi}{2}\) moves the entire graph to the right.

Understanding these parameters helps to sketch how the function behaves on a graph.

Cosine Function

The cosine function is closely related to the sine function, and its general form is \( y = \cos(\theta) \). The function given in the problem is \( y = \cos\left[\frac{1}{4}\left(\theta + \frac{3\pi}{2}\right)\right] \). Here's a breakdown of the transformation:

• **Amplitude:** Similar to the sine function, the amplitude is 1, showing the graph oscillates between -1 and 1.
• **Period:** With the modification, the period is also \( 8\pi \), matching the sine function's period.
• **Phase Shift:** The positive \(\frac{3\pi}{2}\) inside the cosine function indicates a shift to the left by \(\frac{3\pi}{2}\).

This transformation fully explains the altered graph's appearance compared to a standard cosine waveform.

Phase Shift

Phase shift in trigonometric functions involves shifting the graph horizontally. It plays an essential role in how the graph is positioned along the x-axis. For both the sine and cosine functions in this exercise:

• **Sine Function:** The phase shift of \(+\frac{\pi}{2}\) indicates a movement to the right by \(\frac{\pi}{2}\).
• **Cosine Function:** The shift for this function is \(-\frac{3\pi}{2}\), which means it moves to the left by the same amount.

This horizontal translation changes the starting point of the graph without affecting its shape or period. Recognizing phase shifts is crucial for aligning trigonometric functions, particularly when analyzing or comparing them.

Amplitude

Amplitude controls the height of the wave formed by trigonometric functions, measured from the center line (known as the axis of the wave) to its peak. In both the sine and cosine functions from the exercise:

• Each function has an amplitude of 1, indicating the wave fluctuates between -1 and 1.
• The sine function has an additional factor of -1, which reflects it across the x-axis. It's like turning the wave upside-down while keeping its height the same.

Amplitude helps in determining how "tall" the wave is and affects its visual representation on a graph.

Periodic Functions

Periodic functions repeat their values in regular intervals or cycles, critical in trigonometry for describing numerous natural phenomena.

• The **sine** and **cosine** functions are classic examples of periodic functions due to their repeating wave patterns.
• The **period** of a function, such as the assigned \( 8\pi \) here, is the length of one complete cycle before starting over. Every point on the graph at \( \theta \) will recur at \( \theta + 8\pi \).
• Despite differences in phase shifts and reflections, both functions in the problem have the same period, meaning they complete their oscillations in the same span on the x-axis.

Understanding periodicity is fundamental for analyzing and predicting wave behavior over time.