Hey guys! Let's dive deep into the fascinating world of trigonometric functions, specifically focusing on sin(3x) and cos(3x), and get a solid grasp on when they're increasing and decreasing. This isn't just about memorizing formulas; it's about understanding the behavior of these waves. When we talk about a function increasing, we mean that as the input (in this case, 'x') gets larger, the output of the function also gets larger. Conversely, a function is decreasing when its output gets smaller as the input increases. This concept is fundamental in calculus and many other areas of math and science. We'll break down how the '3x' inside the sine and cosine functions actually affects their typical wave patterns, making them oscillate faster. Understanding this shift is key to predicting where these functions will reach their peaks and troughs, and importantly, where they change direction. So, buckle up, and let's make trigonometry make sense!

The Basics of Sine and Cosine Waves

Before we tackle sin(3x) and cos(3x), let's quickly refresh our memory about the standard sin(x) and cos(x). You know, the ones you probably learned about first. sin(x) starts at 0, goes up to a maximum value of 1 at x = π/2, comes back down to 0 at x = π, hits a minimum of -1 at x = 3π/2, and returns to 0 at x = 2π. This completes one full cycle. It's increasing from 0 to π/2 and again from 3π/2 to 2π (and so on, adding multiples of 2π). It's decreasing from π/2 to 3π/2. Now, cos(x) is a bit different. It starts at its maximum value of 1 at x = 0, goes down to 0 at x = π/2, hits its minimum of -1 at x = π, comes back up to 0 at x = 3π/2, and reaches 1 again at x = 2π. So, cos(x) is decreasing from 0 to π and increasing from π to 2π. The key takeaway here is that these functions have a period of 2π, meaning they repeat their pattern every 2π units on the x-axis. The amplitude (the height from the center line to the peak) is 1 for both. Understanding these basic shapes and behaviors is crucial because sin(3x) and cos(3x) are essentially scaled versions of these fundamental waves. Think of it like stretching or compressing the graph horizontally. The peaks and valleys are still there, but they happen more frequently. We'll explore precisely how that '3' factor changes the game in the following sections, making sure you can confidently identify where these modified functions are on the rise and where they're on the fall.

How the '3x' Changes Everything

Alright, guys, let's get down to the nitty-gritty: what happens when we introduce that 3 inside our sine and cosine functions, turning sin(x) into sin(3x) and cos(x) into cos(3x)? This little number has a huge impact on the period of the function. Remember how sin(x) and cos(x) have a period of 2π? Well, for a function of the form sin(bx) or cos(bx), the new period is calculated as 2π / |b|. In our case, b = 3, so the period for both sin(3x) and cos(3x) becomes 2π / 3. This means these functions complete a full cycle three times faster than their original sin(x) and cos(x) counterparts. Imagine the standard waves being squished horizontally. This compression means they will increase and decrease much more rapidly, and over shorter intervals. For sin(3x), it will start at 0, increase to its maximum, decrease through zero to its minimum, and return to zero, all within an interval of 2π/3. This cycle repeats three times within the original 2π interval. Similarly, cos(3x) will go through its entire pattern of reaching maximum, decreasing, reaching minimum, increasing, and returning to maximum within that same 2π/3 period. This change in period is the fundamental reason why the intervals of increasing and decreasing behavior are different from the standard sine and cosine functions. We're essentially looking at three waves packed into the space where one used to be. It's like speed dating for waves – they go through their whole routine much quicker!

Finding Intervals of Increase for sin(3x)

So, how do we pinpoint exactly when sin(3x) is increasing? The general rule for sin(θ) is that it increases when θ is in the intervals (-π/2 + 2kπ, π/2 + 2kπ), where 'k' is any integer. Since our angle here is 3x, we'll substitute 3x for θ:

• -π/2 + 2kπ < 3x < π/2 + 2kπ

Now, to find the intervals for 'x', we need to isolate 'x' by dividing all parts of the inequality by 3:

• (-π/2 + 2kπ) / 3 < x < (π/2 + 2kπ) / 3

Simplifying this gives us:

• -π/6 + 2kπ/3 < x < π/6 + 2kπ/3

This means that sin(3x) is increasing in these specific intervals. For example, when k=0, the interval is (-π/6, π/6). When k=1, it's (-π/6 + 2π/3, π/6 + 2π/3), which simplifies to (π/2, 5π/6). And when k=-1, it's (-π/6 - 2π/3, π/6 - 2π/3), which simplifies to (-5π/6, -π/2). See how the 2kπ/3 part reflects the new, shorter period of 2π/3? This is the magic of transformation! These intervals are where the 'upward slope' of the sin(3x) wave occurs. It's where the function's value is climbing towards its peak. Understanding these intervals is super useful for graphing and analyzing the function's behavior. We’re essentially identifying all the 'uphill' sections of the compressed sine wave. It's like finding all the moments the wave is catching a ride to the top!

Identifying Intervals of Decrease for sin(3x)

Now, let's flip the coin and talk about when sin(3x) is decreasing. For the standard sin(θ), it decreases when θ is in the intervals (π/2 + 2kπ, 3π/2 + 2kπ), where 'k' is any integer. Again, we replace θ with our angle, 3x:

• π/2 + 2kπ < 3x < 3π/2 + 2kπ

To find the intervals for 'x', we divide every part by 3:

• (π/2 + 2kπ) / 3 < x < (3π/2 + 2kπ) / 3

Simplifying this yields:

• π/6 + 2kπ/3 < x < π/2 + 2kπ/3

These are the intervals where sin(3x) is decreasing. Let's look at some examples. For k=0, the interval is (π/6, π/2). For k=1, we get (π/6 + 2π/3, π/2 + 2π/3), which simplifies to (5π/6, 7π/6). For k=-1, it's (π/6 - 2π/3, π/2 - 2π/3), simplifying to (-π/2, -π/6). Notice how these 'downhill' intervals perfectly follow the 'uphill' intervals we found earlier, fitting within the 2π/3 period? This confirms our understanding of the function's behavior. These are the segments where the wave is heading downwards, reaching its lowest points before starting to climb again. It’s where the function is losing value. It's like watching the wave take a dive before it prepares for its next ascent. The pattern is consistent, just happening more rapidly.

Finding Intervals of Increase for cos(3x)

Let's shift our focus to cos(3x) and when it's increasing. Remember, the standard cos(θ) function increases when θ is in the intervals (π + 2kπ, 2π + 2kπ), where 'k' is any integer. Substituting 3x for θ, we get:

• π + 2kπ < 3x < 2π + 2kπ

To find the 'x' intervals, we divide all parts by 3:

• (π + 2kπ) / 3 < x < (2π + 2kπ) / 3

This simplifies to:

• π/3 + 2kπ/3 < x < 2π/3 + 2kπ/3

These are the intervals where cos(3x) is increasing. Let's check a few. For k=0, the interval is (π/3, 2π/3). For k=1, it's (π/3 + 2π/3, 2π/3 + 2π/3), which equals (π, 4π/3). For k=-1, we get (π/3 - 2π/3, 2π/3 - 2π/3), which simplifies to (-π/3, 0). Notice how these 'uphill' segments for cosine are shifted compared to sine, and they also occur with the compressed period of 2π/3? The cosine function starts its cycle at a peak, so its first increasing interval happens after it has already passed its maximum and started to decrease. These are the sections where the value of cos(3x) is climbing towards its next peak. It's the 'comeback' phase of the cosine wave. It's where the function is regaining value after hitting a low point.

Identifying Intervals of Decrease for cos(3x)

Finally, let's determine when cos(3x) is decreasing. The standard cos(θ) function decreases when θ is in the intervals (0 + 2kπ, π + 2kπ), which is the same as (2kπ, π + 2kπ), where 'k' is any integer. Replacing θ with 3x:

• 2kπ < 3x < π + 2kπ

Dividing by 3 to solve for 'x':

• (2kπ) / 3 < x < (π + 2kπ) / 3

This simplifies to:

• 2kπ/3 < x < π/3 + 2kπ/3

These are the intervals where cos(3x) is decreasing. Let's test them. For k=0, the interval is (0, π/3). For k=1, we have (2π/3, π/3 + 2π/3), which simplifies to (2π/3, π). For k=-1, it's (-2π/3, π/3 - 2π/3), simplifying to (-2π/3, -π/3). These 'downhill' intervals for cosine are where the function's value is dropping from its peak towards its trough. This aligns perfectly with the 'uphill' intervals we just discussed, completing the 2π/3 cycle. It’s where the function is losing value rapidly, heading towards its minimum. It's the initial plunge of the cosine wave after reaching its summit. It’s crucial to see how these decreasing intervals logically follow the increasing ones, creating the continuous wave pattern.

Putting It All Together: The Visual Story

Understanding these intervals of increasing and decreasing sin(3x) and cos(3x) helps us paint a clearer picture of their graphs. Remember, the 3 in 3x compresses the waves, making them oscillate three times as fast within the standard 2π interval. This means there are three full cycles of sin(3x) and cos(3x) packed into the space where you'd normally see just one cycle of sin(x) or cos(x). The key is that the shape of the wave remains the same, but its frequency increases. The intervals we found are just the segments where the graph is going