Finding The Period Of Y = Sin(3x) A Step-by-Step Guide

Introduction: Unraveling Trigonometric Periods

In the vast realm of mathematics, trigonometric functions hold a position of paramount importance. They serve as the bedrock for understanding periodic phenomena, from the rhythmic swing of a pendulum to the cyclical patterns of sound waves. Among these functions, the sine function, denoted as sin(x), stands out as a fundamental building block. Its elegant wave-like graph encapsulates the essence of periodicity, a concept crucial in various fields, including physics, engineering, and even music. Understanding the period of a trigonometric function is key to unlocking its behavior and predicting its values. In this article, we embark on a comprehensive exploration of the period of the sine function, with a specific focus on the function y = sin(3x). We will dissect the concept of periodicity, delve into the mechanics of how transformations affect the period, and ultimately arrive at a clear understanding of the period of y = sin(3x).

To begin, let's establish a firm grasp on the concept of a period. The period of a trigonometric function is the horizontal distance over which the function's graph completes one full cycle. In simpler terms, it's the length of the interval after which the function's values repeat themselves. For the standard sine function, y = sin(x), this period is 2π. This means that the graph of sin(x) repeats its pattern every 2π units along the x-axis. This fundamental understanding of the sine function's period is the cornerstone for analyzing more complex variations, such as y = sin(3x). The coefficient of x within the sine function plays a crucial role in determining the period. It essentially compresses or stretches the graph horizontally, thereby altering the period. In the case of y = sin(3x), the coefficient 3 has a compressive effect on the graph, leading to a shorter period compared to the standard sin(x) function. To precisely calculate the period, we need to consider how this coefficient modifies the fundamental period of 2π.

The General Form and Period Calculation

The general form of a sine function that allows us to easily determine the period is given by y = A sin(Bx + C) + D, where A represents the amplitude, B affects the period, C introduces a phase shift, and D causes a vertical shift. While these parameters each contribute to the function's overall characteristics, it is the coefficient B that directly influences the period. The period of the function y = A sin(Bx + C) + D is calculated by dividing the standard period of the sine function, which is 2π, by the absolute value of B. Mathematically, this relationship is expressed as Period = 2π / |B|. This formula is the key to unlocking the period of any sine function, regardless of its specific parameters. Applying this understanding to our function of interest, y = sin(3x), we can readily identify that B = 3. Therefore, to find the period, we substitute this value into the formula: Period = 2π / |3| = 2π / 3. This calculation reveals that the period of y = sin(3x) is 2π/3, a value significantly shorter than the standard period of 2π. This compression of the period is a direct consequence of the coefficient 3 multiplying the x term within the sine function. The graph of y = sin(3x) completes three full cycles within the same interval where y = sin(x) completes only one cycle. This visual representation further solidifies our understanding of the period's reduction.

Understanding the impact of the coefficient B is crucial for analyzing various trigonometric functions. When |B| is greater than 1, the graph of the function is horizontally compressed, resulting in a shorter period. Conversely, when |B| is less than 1, the graph is horizontally stretched, leading to a longer period. This principle applies not only to sine functions but also to other trigonometric functions like cosine, tangent, and their reciprocals. For instance, the period of y = sin(0.5x) would be 2π / 0.5 = 4π, which is twice the period of the standard sine function. The coefficient B acts as a scaling factor that directly affects the horizontal dimension of the graph, thereby influencing the period. The formula Period = 2π / |B| provides a simple yet powerful tool for determining the period of any sine function expressed in the general form. By carefully identifying the value of B, we can accurately predict the period and understand the function's cyclical behavior.

Applying the Concept to y = sin(3x)

In our specific case of y = sin(3x), the coefficient of x is 3. This means B = 3. Using the formula Period = 2π / |B|, we substitute B = 3 to get Period = 2π / |3| = 2π / 3. Therefore, the period of y = sin(3x) is 2π/3. This signifies that the graph of y = sin(3x) completes one full cycle in an interval of length 2π/3. Compared to the standard sine function, y = sin(x), which has a period of 2π, the graph of y = sin(3x) is compressed horizontally by a factor of 3. This compression is evident when visualizing the graphs of both functions. The graph of y = sin(3x) oscillates more rapidly, completing three full cycles within the interval where y = sin(x) completes only one. The coefficient 3 effectively squeezes the sine wave, resulting in a shorter period.

To further illustrate this, consider the key points of the sine function within one period. For y = sin(x), the key points are at x = 0, π/2, π, 3π/2, and 2π, corresponding to the values 0, 1, 0, -1, and 0, respectively. For y = sin(3x), these key points are compressed by a factor of 3. Thus, they occur at x = 0, π/6, π/3, π/2, and 2π/3. This means that the function y = sin(3x) reaches its maximum value at x = π/6, its zero crossing at x = π/3, its minimum value at x = π/2, and completes one full cycle at x = 2π/3. The compression of these key points provides a tangible understanding of how the period is affected by the coefficient of x. Understanding the period of a trigonometric function is essential for various applications. In signal processing, the period represents the duration of a single cycle of a signal. In physics, it can represent the time taken for an object to complete one oscillation in simple harmonic motion. In music, the period is related to the frequency of a sound wave, which determines the pitch of the sound. Therefore, the ability to accurately determine the period of trigonometric functions is crucial in numerous scientific and engineering disciplines.

Conclusion: Summarizing the Period of y = sin(3x)

In summary, the period of the function y = sin(3x) is 2π/3. This value is obtained by dividing the standard period of the sine function, 2π, by the absolute value of the coefficient of x, which is 3 in this case. The coefficient 3 compresses the graph of the sine function horizontally, resulting in a shorter period compared to the standard sine function y = sin(x). Understanding the relationship between the coefficient of x and the period is crucial for analyzing and interpreting trigonometric functions. The formula Period = 2π / |B| provides a powerful tool for determining the period of any sine function expressed in the general form y = A sin(Bx + C) + D. By applying this formula and carefully identifying the value of B, we can accurately predict the period and understand the function's cyclical behavior. This knowledge is essential in various fields, including mathematics, physics, engineering, and signal processing, where trigonometric functions are widely used to model periodic phenomena. Mastering the concept of periodicity and the factors that influence it allows for a deeper understanding of the behavior of trigonometric functions and their applications in the real world. The ability to determine the period is a fundamental skill for anyone working with trigonometric functions, and it forms the basis for more advanced concepts in trigonometry and calculus.

Therefore, the correct answer is B. 2π/3.