Hey guys! Today, we're diving into the fascinating world of trigonometric functions, specifically focusing on how to determine the period of a sine function. We'll be dissecting the function y = sin 2(x - π/2), breaking down its components, and figuring out its period. Understanding the period is crucial for grasping the cyclical nature of these functions and their applications in various fields like physics and engineering. So, let's get started and unlock the secrets behind this trigonometric expression!
What is the Period of a Trigonometric Function?
Before we jump into our specific function, let's first define what we mean by the period of a trigonometric function. In simple terms, the period is the horizontal distance it takes for the function's graph to complete one full cycle before repeating itself. Think of it like a wave – the period is the length of one complete wave, from crest to crest or trough to trough. For the basic sine function, y = sin(x), the period is 2π. This means the sine wave repeats itself every 2π units along the x-axis. But what happens when we start tweaking the function, like in our case with y = sin 2(x - π/2)? That's where things get interesting, and we need to understand how these modifications affect the period. We often encounter transformations within trigonometric functions that can either stretch or compress the graph horizontally, consequently altering the period. For instance, multiplying the x inside the sine function by a constant can compress the graph, reducing the period. Conversely, if we were to divide x by a constant, the graph would stretch, leading to an increased period. To effectively calculate the period of a transformed trigonometric function, it's essential to identify these horizontal stretches or compressions. We can then use a simple formula to adjust the standard period of the parent function (2π for sine and cosine) according to the transformation factor. Understanding the concept of period is not just a mathematical exercise; it has significant real-world implications. In physics, the period is directly related to the frequency of oscillations, such as in the motion of a pendulum or the vibration of a string. In signal processing, the period helps us understand the cyclical nature of signals, which is crucial for analyzing and manipulating them. Therefore, mastering the concept of the period allows us to apply trigonometric functions to model and solve a wide range of problems across various disciplines. So, as we delve deeper into the specifics of our function y = sin 2(x - π/2), remember that we're not just finding a number; we're uncovering a fundamental property that governs the behavior of the function and its applications in the real world.
Analyzing the Function y = sin 2(x - π/2)
Okay, let's break down the function y = sin 2(x - π/2) step by step. The first thing to notice is that we have a sine function, which we know has a basic period of 2π. However, this isn't just a plain sin(x). We have two key transformations happening here: a horizontal compression and a horizontal shift. Let's tackle the compression first. The '2' multiplying the x inside the sine function is what causes the horizontal compression. This means the graph is being squeezed along the x-axis. To understand how this affects the period, remember that the period is inversely proportional to this coefficient. In other words, if we multiply x by 2, we're essentially making the function complete its cycle twice as fast. This will result in the period being halved. So, instead of the period being 2π, it will now be 2π / 2 = π. The second transformation is the horizontal shift, represented by the '(x - π/2)' part. This shifts the entire graph π/2 units to the right. However, and this is crucial, horizontal shifts do not affect the period of the function. The period is determined solely by the coefficient of x inside the trigonometric function. Think of it like sliding the wave along the x-axis – it doesn't change the length of the wave itself. Therefore, the horizontal shift of π/2 to the right doesn't impact the period, which we've already determined is π due to the compression factor of 2. To further solidify this understanding, consider comparing the graphs of y = sin(2x) and y = sin 2(x - π/2). You'll observe that both functions complete their cycles within the same interval, confirming that the period remains unaffected by the phase shift. This principle extends to other trigonometric functions as well, such as cosine and tangent, where horizontal shifts only alter the position of the graph without influencing the fundamental period. In summary, when analyzing a trigonometric function to determine its period, the focus should be on the coefficient of x within the function. Horizontal shifts, while changing the graph's position, have no bearing on the period. By identifying and applying the compression or stretching factor, we can easily calculate the transformed period from the standard period of the parent function.
Calculating the Period
Now that we understand the transformations, let's formally calculate the period. The general formula for the period of a sine function of the form y = sin(Bx + C) is given by: Period = 2π / |B| where B is the coefficient of x. In our case, y = sin 2(x - π/2), we can see that B = 2. Plugging this into the formula, we get: Period = 2π / |2| = 2π / 2 = π So, the period of the function y = sin 2(x - π/2) is π. This confirms our earlier analysis where we discussed the horizontal compression caused by the factor of 2. Remember, the absolute value of B is used in the formula because the period is always a positive value, representing a distance. The formula Period = 2π / |B| is not just applicable to sine functions; it can also be used to calculate the period of cosine functions, which share the same fundamental period of 2π. The presence of the absolute value ensures that regardless of the sign of B, we obtain a positive period, reflecting the physical interpretation of the period as a length of an interval. To illustrate this concept further, consider a function like y = cos(-3x). In this scenario, B = -3, and applying the formula yields Period = 2π / |-3| = 2π / 3. This result signifies that the cosine function completes one full cycle within an interval of 2π / 3 units along the x-axis, even though the coefficient of x is negative. Understanding and correctly applying this formula is essential for accurately determining the periods of various trigonometric functions, which in turn is crucial for analyzing and modeling cyclical phenomena in fields like physics, engineering, and music.
The Correct Answer and Why
Based on our analysis and calculation, the function y = sin 2(x - π/2) has a period of π. Therefore, the correct answer is C) π. The other options are incorrect because they don't account for the horizontal compression caused by the factor of 2 multiplying the x. Option A) 2π is the period of the basic sine function, y = sin(x), but doesn't consider the transformation. Option B) 4π is incorrect as it suggests a horizontal stretch, which isn't present in our function. Option D) π/2 is also incorrect, as it would imply a horizontal compression greater than what we have. To further emphasize the importance of accurate period calculation, consider the implications in a practical scenario. Suppose we're modeling the oscillation of a spring using a sine function. The period of the function would directly correspond to the time it takes for the spring to complete one full oscillation. If we incorrectly calculate the period, our model would not accurately represent the spring's motion, leading to potentially flawed predictions. Similarly, in signal processing, the period of a signal is a critical parameter for analyzing its frequency content. An incorrect period calculation could result in misinterpretation of the signal's characteristics and ineffective signal processing techniques. Therefore, understanding the underlying principles and applying the correct formulas for period calculation is not just a theoretical exercise; it's a practical necessity for accurate modeling and analysis in various scientific and engineering disciplines. By carefully considering the transformations applied to the basic trigonometric functions and utilizing the appropriate formulas, we can confidently determine the periods and effectively apply these functions in real-world contexts.
Real-World Applications of Trigonometric Functions and Period
Trigonometric functions, including sine and cosine, are not just abstract mathematical concepts; they have countless real-world applications, and the period plays a vital role in understanding these applications. Let's explore a few examples. In physics, trigonometric functions are used to model oscillatory motion, such as the motion of a pendulum, the vibration of a string, or the propagation of waves. The period of the function directly corresponds to the time it takes for one complete oscillation or cycle. For instance, in the case of a pendulum, the period is the time it takes for the pendulum to swing from one extreme to the other and back again. Similarly, in wave mechanics, the period is the time it takes for one complete wavelength to pass a given point. Understanding the period is crucial for analyzing these systems and predicting their behavior. In electrical engineering, trigonometric functions are used to describe alternating current (AC) circuits. The voltage and current in an AC circuit vary sinusoidally with time, and the period of these functions corresponds to the time it takes for one complete cycle of the AC waveform. The frequency, which is the reciprocal of the period, determines the number of cycles per second and is a crucial parameter in circuit design and analysis. In music, trigonometric functions are used to model sound waves. The frequency of a sound wave, which is related to its period, determines the pitch of the sound. Higher frequencies correspond to higher pitches, and lower frequencies correspond to lower pitches. The periodic nature of sound waves is what gives music its rhythmic and harmonic qualities. In astronomy, trigonometric functions are used to model the motion of celestial bodies. The periodic movements of planets, moons, and stars can be described using sine and cosine functions. The period of these functions corresponds to the time it takes for a celestial body to complete one orbit or rotation. For example, the Earth's rotation period determines the length of a day, and its orbital period around the Sun determines the length of a year. These are just a few examples of the many real-world applications of trigonometric functions and the importance of understanding the period. By grasping the cyclical nature of these functions and their relationship to real-world phenomena, we can gain a deeper appreciation for the power and versatility of mathematics in describing the world around us. So, the next time you encounter a sine or cosine function, remember that it's not just a mathematical abstraction; it's a powerful tool for understanding and modeling the rhythms and cycles that permeate our universe.
Conclusion
So, to wrap things up, the function y = sin 2(x - π/2) has a period of π. We arrived at this answer by understanding the horizontal compression caused by the factor of 2 and applying the formula Period = 2π / |B|. Remember, the period is a fundamental property of trigonometric functions, representing the length of one complete cycle. Grasping this concept is not only essential for solving mathematical problems but also for understanding the numerous real-world applications of these functions. I hope this breakdown has been helpful, guys! Keep exploring the fascinating world of trigonometry, and you'll continue to uncover its beauty and power. Happy calculating!