Finding Integer Values In The Range Of Y = 3 - 2sin(x)

Hey guys! Today, we're diving into a super interesting algebra problem that involves trigonometric functions and their ranges. We're going to explore the function y = 3 - 2sin(x) and figure out the sum of all the integer values that fall within its range. This might sound a bit complex at first, but trust me, we'll break it down step by step so it's easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Sine Function and Its Range

To really grasp what's going on with our function, we first need to have a solid understanding of the sine function, sin(x). The sine function is a fundamental trigonometric function that describes the relationship between an angle and the ratio of the opposite side to the hypotenuse in a right triangle. But for our purposes, we're more interested in its graphical representation and, most importantly, its range. When we graph the sine function, we see a smooth, wave-like curve that oscillates between two values. The sine function, sin(x), oscillates between -1 and 1. This means that for any value of x, the output of sin(x) will always be within this range. Mathematically, we can express this as: -1 ≤ sin(x) ≤ 1. This is a crucial piece of information because it forms the foundation for understanding the range of our given function, y = 3 - 2sin(x).

Now, why is understanding this range so important? Well, the range of a function tells us all the possible output values that the function can produce. In other words, it's the set of all possible y-values. Knowing the range of sin(x) allows us to manipulate the inequality and ultimately determine the range of y. Think of it like building blocks – we know the range of sin(x), and we're going to use that to figure out the range of our more complex function. This is a common strategy in mathematics: breaking down complex problems into smaller, more manageable parts. We've got the range of sin(x) down, so we're well on our way to solving the problem.

Determining the Range of y = 3 - 2sin(x)

Okay, so we know that -1 ≤ sin(x) ≤ 1. How do we use this information to find the range of y = 3 - 2sin(x)? This is where we start manipulating the inequality. Our goal is to transform the inequality representing the range of sin(x) into an inequality that represents the range of y. First, let's focus on the term -2sin(x). To get this, we need to multiply the entire inequality by -2. Remember, when we multiply an inequality by a negative number, we need to flip the inequality signs. So, multiplying -1 ≤ sin(x) ≤ 1 by -2 gives us: 2 ≥ -2sin(x) ≥ -2. Notice how the inequality signs flipped! It's super important to remember this rule when working with inequalities. Now, let's rewrite this inequality in the standard order, from least to greatest: -2 ≤ -2sin(x) ≤ 2. We're getting closer to our target range for y!

Next, we need to add 3 to all parts of the inequality because our function is y = 3 - 2sin(x). Adding 3 to the inequality -2 ≤ -2sin(x) ≤ 2, we get: -2 + 3 ≤ 3 - 2sin(x) ≤ 2 + 3. Simplifying this, we have: 1 ≤ 3 - 2sin(x) ≤ 5. And there you have it! We've found the range of y. Since y = 3 - 2sin(x), we can say that the range of y is 1 ≤ y ≤ 5. This means that the output values of the function y will always be between 1 and 5, inclusive. This is a fantastic result because it gives us a clear boundary for the possible values of y. Now that we know the range, we can move on to identifying the integer values within that range and finding their sum.

Identifying Integer Values in the Range and Calculating Their Sum

Now that we've nailed down the range of our function, which is 1 ≤ y ≤ 5, we can easily identify the integer values that fall within this range. Remember, integers are whole numbers (no fractions or decimals), so we're looking for the whole numbers between 1 and 5, including 1 and 5 themselves. Listing them out, we have: 1, 2, 3, 4, and 5. These are the only whole number values that y can take within its range. This is a pretty straightforward step, but it's crucial because these are the numbers we'll be summing up.

The final step is to calculate the sum of these integer values. This is a simple addition problem: 1 + 2 + 3 + 4 + 5. Adding these numbers together, we get a total of 15. So, the sum of all the integer values within the range of y = 3 - 2sin(x) is 15. And that's it! We've successfully solved the problem. We started by understanding the range of the sine function, then we manipulated inequalities to find the range of our function, and finally, we identified the integer values within that range and calculated their sum. This is a great example of how different concepts in algebra and trigonometry can come together to solve a single problem. You guys did awesome! Let's recap what we've learned.

Conclusion: Summing Up Our Findings

Wow, we've covered a lot in this problem! We started with the function y = 3 - 2sin(x) and set out to find the sum of all the integer values within its range. To do this, we first had to understand the fundamental concept of the sine function and its range, which oscillates between -1 and 1. This understanding was crucial because it allowed us to manipulate inequalities and determine the range of our given function.

We then skillfully manipulated the inequality representing the range of sin(x) to find the range of y = 3 - 2sin(x). Remember how we multiplied by -2 and flipped the inequality signs? That's a key technique to remember when working with inequalities. After a few steps, we successfully found that the range of y is 1 ≤ y ≤ 5. This was a major step forward because it gave us the boundaries within which we needed to find our integer values.

Next, we easily identified the integer values within the range 1 ≤ y ≤ 5 as 1, 2, 3, 4, and 5. Finally, we added these integers together to find the sum, which turned out to be 15. So, the sum of all the integer values in the range of y = 3 - 2sin(x) is 15. This problem beautifully illustrates how a solid understanding of basic trigonometric functions and algebraic manipulations can lead us to the solution of more complex problems. You guys rocked it! Keep practicing, and you'll become masters of algebra and trigonometry in no time. This journey of problem-solving is what makes math so engaging and rewarding. Until next time, keep those brains buzzing and those pencils moving!