Determining the period of a function that's a product of two trigonometric functions, like cos(x)sin(πx/2), involves a bit of mathematical exploration. Let's break down how to find it, making sure we understand each step along the way. It's not always as straightforward as finding the period of individual trig functions, but with a methodical approach, we can crack the code.

Understanding the Basics of Trigonometric Periods

Before diving into the specific function cos(x)sin(πx/2), let's refresh our understanding of trigonometric periods in general. The period of a trigonometric function is the interval over which the function's graph repeats itself. For instance, the cosine function, cos(x), has a period of 2π. This means that the graph of cos(x) repeats every 2π units along the x-axis. Similarly, the sine function, sin(x), also has a period of 2π. When we deal with functions like sin(kx) or cos(kx), where k is a constant, the period changes. The new period is given by 2π/k. So, sin(2x) would have a period of π, and cos(x/2) would have a period of 4π. Understanding these basics is crucial because when we have a product (or sum, difference, or quotient) of trigonometric functions, the resulting period isn't always obvious. It often involves finding the least common multiple (LCM) of the individual periods or employing trigonometric identities to simplify the expression. In our case, cos(x)sin(πx/2), we have two trigonometric functions multiplied together, each with potentially different periods. This is what makes the problem interesting and requires a careful, step-by-step analysis to determine the overall period of the combined function. Remember, the goal is to find the smallest interval over which the entire function cos(x)sin(πx/2) repeats its values. This involves not just looking at the individual periods but also how these functions interact when multiplied together. So, keep these fundamental concepts in mind as we proceed to solve for the period of our function.

Analyzing cos(x) and sin(πx/2) Separately

Okay, let's start by dissecting our function, cos(x)sin(πx/2). First, we'll look at cos(x). As we know, the period of cos(x) is 2π. This means that the cosine function completes one full cycle from 0 to 2π and then starts repeating itself. Simple enough, right? Now, let's move on to the second part, sin(πx/2). This is where it gets a tad more interesting because we have a constant, π/2, multiplied by x inside the sine function. To find the period of sin(πx/2), we use the formula: Period = 2π / k, where k is the coefficient of x. In this case, k = π/2. Plugging this into our formula, we get: Period = 2π / (π/2) = 2π * (2/π) = 4. So, the period of sin(πx/2) is 4. This means that the sine function sin(πx/2) completes one full cycle from 0 to 4 and then starts repeating. Now, we have the individual periods: cos(x) has a period of 2π, and sin(πx/2) has a period of 4. The next step is to figure out how these two periods interact when the functions are multiplied together. It's not as simple as just adding them or multiplying them. Instead, we need to find a common multiple or use trigonometric identities to simplify the expression and determine the overall period of the combined function, cos(x)sin(πx/2). Remember, the goal is to find the smallest value T such that f(x + T) = f(x) for all x, where f(x) = cos(x)sin(πx/2). So, let's move on to the next step and see how we can combine these periods to find the overall period of the function.

Finding the Period of the Product

Alright, guys, we've established that cos(x) has a period of 2π and sin(πx/2) has a period of 4. Now, the million-dollar question: what's the period of their product, cos(x)sin(πx/2)? This isn't as straightforward as just finding the least common multiple (LCM) because one period involves π, which is irrational. In such cases, a direct LCM approach won't work. Instead, we should consider if there’s a T such that cos(x + T)sin(π(x + T)/2) = cos(x)sin(πx/2) for all x. This is tough to solve directly. However, sometimes trigonometric identities can come to our rescue! Let’s use the product-to-sum identity:

cos(A)sin(B) = 1/2 [sin(A + B) - sin(A - B)]

Applying this to our function, we get:

cos(x)sin(πx/2) = 1/2 [sin(x + πx/2) - sin(x - πx/2)] = 1/2 [sin((2 + π)x/2) - sin((2 - π)x/2)]

Now we have a difference of two sine functions. The periods of these sine functions are:

Period_1 = 2π / ((2 + π)/2) = 4π / (2 + π) Period_2 = 2π / ((2 - π)/2) = 4π / (2 - π)

Since we have a difference of sine functions, we are looking for a common multiple of Period_1 and Period_2. Still, because of the π in the denominator, it's unlikely that a simple common multiple exists that will give us a clean period. This indicates that the original function, cos(x)sin(πx/2), might not have a well-defined period in the traditional sense. The interaction between cos(x) and sin(πx/2) creates a function that doesn't repeat in a regular, predictable manner. In practical terms, this means that if you were to graph the function cos(x)sin(πx/2), you wouldn't see a repeating pattern that defines a clear period. The function might exhibit some quasi-periodic behavior, but it won't have a T such that f(x + T) = f(x) for all x. Therefore, after careful analysis using trigonometric identities and considering the nature of the individual periods, we can conclude that the function cos(x)sin(πx/2) does not have a simple, easily expressible period.

Conclusion

So, folks, after a thorough investigation, we've determined that the function cos(x)sin(πx/2) doesn't have a straightforward period. While cos(x) has a period of 2π and sin(πx/2) has a period of 4, their product doesn't result in a periodic function in the traditional sense. The use of trigonometric identities helped us rewrite the function as a difference of sine functions, but the resulting periods were not easily combined to find a common multiple. This highlights an important point: not all functions, even those built from periodic trigonometric functions, are themselves periodic. The interaction between the cosine and sine functions in cos(x)sin(πx/2) leads to a complex behavior that lacks a clear, repeating pattern. Therefore, when faced with such problems, it's crucial to analyze the individual components, explore trigonometric identities, and consider whether a common multiple of the periods exists. In this case, the absence of a simple common multiple suggests that the function does not have a well-defined period. Understanding this nuanced behavior of trigonometric functions is key to tackling more advanced problems in mathematics and physics. Keep exploring, and don't be afraid to dive deep into the world of functions and their properties!