Analyzing Sin(3x)cos(3x): Increasing Or Decreasing?

Hey there, math enthusiasts! Ever wondered about the ups and downs of trigonometric functions? Today, we're diving deep into the world of trigonometry to explore whether the function sin(3x)cos(3x) is increasing or decreasing. This might sound a bit complex, but don't worry, we'll break it down into easy-to-understand chunks. We'll use concepts from calculus, like derivatives, to understand how this function behaves. This is important because understanding whether a function is increasing or decreasing helps us understand its behavior, find its maximum and minimum values, and even graph it accurately. Are you ready to unravel the secrets of sin(3x)cos(3x)? Let's get started!

Unveiling the Function: sin(3x)cos(3x)

First off, what exactly is sin(3x)cos(3x)? This is a trigonometric function, which means it involves sine and cosine, those familiar friends from your geometry and trigonometry classes. However, it's not just a simple sine or cosine function. Here, we've got 3x inside the sine and cosine, which means the function's behavior will be a bit more complicated. It also contains the product of sine and cosine functions. sin(3x)cos(3x) combines the sine and cosine functions. Understanding this is key to what comes next. Before we dive into its increasing and decreasing properties, let's refresh our memory on some fundamental trigonometric concepts. Recall that sine and cosine functions oscillate between -1 and 1. The '3x' inside the functions means that the oscillation will be faster. The function sin(3x)cos(3x) itself is also a periodic function, meaning its values repeat over certain intervals. Because this function is a product of trigonometric functions, its behavior is somewhat different. It's not a simple sine or cosine wave; instead, its amplitude and frequency are influenced by both the sine and cosine components. These nuances are what make the analysis of this function interesting and important. Think of it like this: the sine function is like a wave, and the cosine function is another wave, and when you multiply them together, you get a new kind of wave. So, let’s begin our journey to figuring out whether this wave is going up or down.

The Importance of Derivatives

To understand if our function is increasing or decreasing, we need to bring in calculus, specifically the derivative. The derivative of a function tells us its rate of change. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing. It is important to know about derivatives. The derivative is like a speedometer for the function. It tells you the function’s speed and direction at any given point. To find the derivative of sin(3x)cos(3x), we'll need to use the product rule from calculus. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function. Remember the product rule: If you have a function like f(x) * g(x), then its derivative is f'(x) * g(x) + f(x) * g'(x). This is a really important tool for our analysis. We'll use this rule to break down our function and see how it behaves.

Applying the Product Rule: Step-by-Step

Alright, time to get our hands dirty with some calculus! Our function is sin(3x)cos(3x). Let’s consider sin(3x) as our first function and cos(3x) as our second function. Now, we need to find the derivatives of sin(3x) and cos(3x). The derivative of sin(3x) is 3cos(3x), and the derivative of cos(3x) is -3sin(3x). The '3' comes from the chain rule. Remember the chain rule? It’s another key concept in calculus. The chain rule helps us find the derivative of a composite function. Now, we apply the product rule: the derivative of sin(3x)cos(3x) is [3cos(3x)]cos(3x) + sin(3x)[-3sin(3x)]. This simplifies to 3cos²(3x) - 3sin²(3x). This new function, 3cos²(3x) - 3sin²(3x), tells us the rate of change of our original function, sin(3x)cos(3x). Next, we’re going to simplify this even further, using a trigonometric identity, to make our analysis easier. So buckle up, because the ride is getting a little bit trickier.

Simplifying the Derivative: A Trigonometric Identity

We've got the derivative, 3cos²(3x) - 3sin²(3x), but it can be simplified further using a trigonometric identity. This identity is cos(2θ) = cos²(θ) - sin²(θ). In our case, θ is 3x. So, we can rewrite our derivative as 3[cos²(3x) - sin²(3x)]. Applying the identity, this simplifies to 3cos(6x). Now, we have a much simpler form of the derivative, 3cos(6x). This is much easier to analyze to determine where our original function is increasing or decreasing. The function 3cos(6x) is also a periodic function. However, the period is shorter than the original function. The argument of the cosine function is now 6x, which means it oscillates faster. It has the same amplitude but the frequency is different. Knowing this allows us to understand how sin(3x)cos(3x) changes more efficiently.

Analyzing the Derivative: When is it Increasing or Decreasing?

Now, let's analyze our simplified derivative, 3cos(6x), to determine when the original function, sin(3x)cos(3x), is increasing or decreasing. Remember, the function is increasing when its derivative is positive, and decreasing when its derivative is negative. The cosine function, cos(6x), varies between -1 and 1. Therefore, 3cos(6x) will also vary between -3 and 3. The function 3cos(6x) is positive when cos(6x) > 0, and it's negative when cos(6x) < 0. Let's find out the intervals. The cosine function is positive in the first and fourth quadrants. In terms of 6x, this means 6x is between 0 and π/2, and between 3π/2 and 2π. Hence, 3cos(6x) is positive (and sin(3x)cos(3x) is increasing) when 0 < 6x < π/2 and 3π/2 < 6x < 2π. This simplifies to 0 < x < π/12 and π/4 < x < π/3. Conversely, the cosine function is negative in the second and third quadrants. So, 6x is between π/2 and 3π/2. This means 3cos(6x) is negative (and sin(3x)cos(3x) is decreasing) when π/2 < 6x < 3π/2. This simplifies to π/12 < x < π/4. That's our increasing and decreasing intervals. We can now visualize how our function sin(3x)cos(3x) changes across different values of x. Understanding these intervals will also help us in graphing the function accurately.

Visualizing the Behavior

To better understand the behavior of sin(3x)cos(3x), let's think about its graph. The function oscillates, but its amplitude is not constant. The graph's behavior changes, going up and down in a regular pattern because of the periodic nature of the trigonometric functions. When 3cos(6x) is positive, the function sin(3x)cos(3x) is increasing, meaning the graph goes upwards. When 3cos(6x) is negative, the function sin(3x)cos(3x) is decreasing, meaning the graph goes downwards. The points where 3cos(6x) = 0 are where the function changes direction. Those points are the critical points of our function, where it transitions from increasing to decreasing, or vice versa. These points are also where we might find the maximum and minimum values of the function. If you were to graph this function, you'd see it oscillating with a changing rate. The graph will rise and fall, mirroring the behavior of the derivative, 3cos(6x). To get a perfect graph, we could use graphing tools, but understanding the increasing and decreasing intervals gives us a good picture of its behavior. Graphing is a great way to double-check our work and visualize our findings.

Conclusion: The Increasing and Decreasing Nature of sin(3x)cos(3x)

So, what's the final verdict? We have explored whether the function sin(3x)cos(3x) is increasing or decreasing by using calculus, particularly the derivative. We found that the derivative of sin(3x)cos(3x) is 3cos(6x). We discovered that sin(3x)cos(3x) is increasing in the intervals (0, π/12) and (π/4, π/3), and it's decreasing in the interval (π/12, π/4). Also, we’ve covered the trigonometric identities. We've simplified the derivative using the cosine double-angle identity. Analyzing the derivative gives us a clear picture of the function’s behavior. This analysis shows the function’s varying behavior, which is a key characteristic of trigonometric functions. This is a journey through calculus. We hope this exploration has made the concept clearer. Remember that understanding derivatives is key to understanding how functions change. If you're interested in more advanced topics, you could explore the second derivative, which would tell you about the concavity of the function. Keep exploring, keep learning, and don't hesitate to ask questions. Math can be fun!