Problem 16 - Algebra 2 and Trigonometry

Question

Find the period of each function. \(y=\sin \frac{1}{3} x\)

Step-by-Step Solution

Answer

The period of \( y = \\sin \frac{1}{3} x \) is \( 6\\pi \).

Step 1: Understand the General Sine Function

The general form of the sine function is \( y = \sin(bx) \). The period of the sine function \( \sin(x) \) is \( 2\pi \). The period is affected by the coefficient \( b \).

Step 2: Formula for Period

The period of the sine function \( y = \sin(bx) \) is given by \( \frac{2\pi}{|b|} \). This formula adjusts the standard period \( 2\pi \) according to the value of \( b \).

Step 3: Identify the Coefficient

In the function \( y = \sin \frac{1}{3} x \), the coefficient \( b \) is \( \frac{1}{3} \).

Step 4: Calculate the Period

Using the formula \( \frac{2\pi}{|b|} \), substitute \( b = \frac{1}{3} \). This gives the period as \[ \frac{2\pi}{|\frac{1}{3}|} = \frac{2\pi}{\frac{1}{3}} = 2\pi \times 3 = 6\pi \].

Step 5: Conclusion

The period of the function \( y = \sin \frac{1}{3} x \) is \( 6\pi \). This means the function completes one full cycle every \( 6\pi \) units along the x-axis.

Key Concepts

Understanding the Sine Function

The sine function, represented as \( y = \sin(x) \), is a fundamental trigonometric function that describes wave-like oscillations. It's important to note that the sine function operates within a cycle or "period". A standard sine wave completes one oscillation between 0 and \( 2\pi \) radians. This cyclical nature defines how the function repeats its values over and over.

The graph of \( y = \sin(x) \) is shaped like a smooth, continuous wave, where:

• The peaks (maxima) are at \( y = 1 \).
• The troughs (minima) are at \( y = -1 \).
• The function crosses the x-axis at 0, \(\pi\), and \(2\pi\).

The "period" of this function marks the distance along the x-axis where the wave pattern repeats itself. Mastering the basic sine function is essential to understanding more complex trigonometric functions.

Formula for Period

To determine the period of any sine function beyond the basic \( y = \sin(x) \), you can use a specific formula. This formula is designed to adjust the standard period of a sine wave, \( 2\pi \), based on the presence of a coefficient affecting the variable within the sine function.

The formula is: \[Period = \frac{2\pi}{|b|}\]Here, \( b \) represents the coefficient of the variable within the sine function as in \( y = \sin(bx) \). The absolute value \( |b| \) ensures that the period is always a positive number, as distances cannot be negative.

For instance, if \( b = \frac{1}{3} \), which means that our function is \( y = \sin(\frac{1}{3}x) \), the period would be calculated as:\[\frac{2\pi}{\frac{1}{3}} = 6\pi\]This signifies that the sine wave will complete one full cycle every \( 6\pi \) units.

Coefficient of Trigonometric Function

In trigonometry, the coefficient of the trigonometric function directly influences how the graph of the function behaves. Specifically, in the function \( y = \sin(bx) \), the coefficient \( b \) affects the period.

Here's how it works:

• If \( |b| \) is greater than 1, the sine wave will "squish" together, completing more cycles over the same distance.
• If \( |b| \) is less than 1, as in \( b = \frac{1}{3} \), the waveform will expand, taking longer to complete each cycle.
• A negative value of \( b \) would flip the waveform horizontally, but the absolute value ensures the period stays positive.

Understanding \( b \) helps predict how and when the function's cycles will repeat, ultimately aiding in graphing and problem-solving scenarios. The modulus \(|b|\) ensures that we only deal with positive periods, aligning with the concept that distances cannot be intrinsically negative.