Finding The Period Of Y = Sin(3x) A Trigonometric Exploration

In the fascinating world of trigonometry, understanding the period of trigonometric functions is crucial. The period, in essence, tells us how often a function's graph repeats itself. This concept is particularly important when dealing with transformations of basic trigonometric functions like sine, cosine, and tangent. Let's delve into the specific case of y = sin(3x) to illustrate this concept. The function y = sin(3x) represents a horizontal compression of the standard sine function, y = sin(x). The key to finding the period lies in the coefficient of x inside the sine function, which in this case is 3. This coefficient affects the horizontal scaling of the graph. To determine the period, we use the formula: Period = (2π) / |B|, where B is the coefficient of x. In our example, B = 3. Thus, the period of y = sin(3x) is (2π) / 3. This means that the graph of y = sin(3x) completes one full cycle in an interval of 2π/3 units along the x-axis. 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. Imagine taking the sine wave and squeezing it horizontally – this is precisely what the transformation sin(3x) does. This understanding of periods is not just an academic exercise. It has practical applications in various fields, including physics, engineering, and music. For instance, in physics, periodic functions are used to model oscillations, waves, and other cyclical phenomena. In electrical engineering, alternating current (AC) is described by sinusoidal functions, and the period represents the time it takes for one complete cycle of the current. Furthermore, understanding the period helps in analyzing and predicting the behavior of systems that exhibit periodic motion. Therefore, mastering the concept of periods and trigonometric transformations is essential for a solid foundation in mathematics and its applications. By recognizing the relationship between the coefficient of x and the period, we can effectively analyze and manipulate trigonometric functions to solve a wide range of problems. In summary, the period of y = sin(3x) is 2π/3, a direct consequence of the horizontal compression caused by the coefficient 3. This principle applies to other trigonometric functions as well, making it a fundamental concept in trigonometry.

The Impact of the Coefficient on the Period: Visualizing the Transformation

To further solidify our understanding, let's visualize the impact of the coefficient 3 on the period of y = sin(3x). Picture the graph of the standard sine function, y = sin(x), which oscillates between -1 and 1 and completes one full cycle over an interval of 2π. Now, consider what happens when we replace x with 3x. The function y = sin(3x) still oscillates between -1 and 1, but the rate of oscillation is increased. This means that the function completes one full cycle in a shorter interval. Specifically, it completes three cycles in the same interval where y = sin(x) completes only one cycle. This is why the period of y = sin(3x) is 2π/3 – it is one-third of the period of the standard sine function. We can observe this compression visually by plotting both functions on the same graph. The graph of y = sin(3x) appears to be squeezed horizontally compared to the graph of y = sin(x). This visual representation helps to reinforce the concept that the coefficient of x inside the sine function directly affects the period. A larger coefficient leads to a shorter period, indicating a faster oscillation, while a smaller coefficient would result in a longer period, indicating a slower oscillation. This concept extends beyond sine functions. For any trigonometric function of the form y = A sin(Bx + C) + D, the coefficient B determines the period. The constant A affects the amplitude, C causes a horizontal shift, and D causes a vertical shift. By understanding the individual effects of these parameters, we can analyze and manipulate trigonometric functions with greater confidence. Moreover, the ability to visualize these transformations is crucial for problem-solving in various contexts. For example, in signal processing, understanding the period of a sinusoidal signal is essential for analyzing its frequency content. Similarly, in mechanical systems, the period of oscillation is a key parameter in determining the system's stability and response. In conclusion, the coefficient of x plays a pivotal role in determining the period of a trigonometric function. By visualizing the transformation and understanding the relationship between the coefficient and the period, we can gain a deeper appreciation for the behavior of these functions and their applications in various fields. The compression of the sine wave in y = sin(3x) illustrates this principle effectively, highlighting the importance of horizontal scaling in trigonometric transformations.

Generalizing the Concept: Periods of Other Trigonometric Functions

While we've focused on the sine function, the concept of the period and the impact of coefficients extend to other trigonometric functions as well. Let's briefly consider cosine and tangent functions to generalize our understanding. The cosine function, y = cos(x), is closely related to the sine function. In fact, it is simply a horizontal shift of the sine function by π/2 units. The standard cosine function also has a period of 2π, meaning it completes one full cycle over an interval of 2π along the x-axis. Similar to the sine function, if we have a function of the form y = cos(Bx), the period is given by (2π) / |B|. For example, the period of y = cos(2x) would be π, indicating a horizontal compression compared to the standard cosine function. The tangent function, y = tan(x), behaves differently from sine and cosine. It has a period of π, which is half the period of sine and cosine. This is because the tangent function repeats its pattern every π units along the x-axis. The tangent function is defined as tan(x) = sin(x) / cos(x), and its graph has vertical asymptotes at points where cos(x) = 0. If we have a function of the form y = tan(Bx), the period is given by π / |B|. For instance, the period of y = tan(x/2) would be 2π, indicating a horizontal stretch compared to the standard tangent function. Understanding the periods of these basic trigonometric functions is crucial for analyzing more complex trigonometric expressions and equations. When dealing with combinations of trigonometric functions or transformations, knowing the individual periods helps in determining the overall behavior of the function. Moreover, this knowledge is essential in fields like Fourier analysis, where complex periodic signals are decomposed into sums of sine and cosine functions with different frequencies and periods. In summary, the concept of the period applies to all trigonometric functions, with the coefficient of x playing a key role in determining the period. While sine and cosine have a period of 2π, the tangent function has a period of π. By understanding how coefficients affect the period, we can effectively analyze and manipulate trigonometric functions in various mathematical and scientific contexts. The generalization of this concept to other trigonometric functions solidifies our understanding of periodic behavior and its applications.

The function given is y = sin(3x). To find the period, we use the formula: Period = (2π) / |B|, where B is the coefficient of x. In this case, B = 3. Period = (2π) / |3| = (2π) / 3 Therefore, the period of y = sin(3x) is (2π) / 3.

Answer

The correct answer is B. (2π) / 3