Max & Min Integer Value Of 3 - Sin(2x + 1°) - 2

Hey guys! Let's dive into this interesting math problem where we need to find the product of the maximum and minimum integer values of the expression 3 - sin(2x + 1°) - 2. It might look a bit complex at first, but don't worry, we'll break it down step by step so it's super easy to understand. We're going to use some trigonometric principles and a bit of algebraic thinking to solve this. So, grab your thinking caps, and let's get started!

Understanding the Basics of Sine Function

Before we jump into the expression, let's quickly refresh our understanding of the sine function. The sine function, denoted as sin(θ), oscillates between -1 and 1. This means that for any angle θ, the value of sin(θ) will always be within this range:

-1 ≤ sin(θ) ≤ 1

This is a fundamental concept in trigonometry, and it's super important for solving our problem. Think of the sine wave – it goes up and down, but it never goes beyond these limits. Knowing this, we can figure out how the sine part of our expression affects the overall value. This range is crucial because it helps us determine the possible minimum and maximum values of the entire expression. Understanding this range allows us to manipulate the inequality and find the extreme values more effectively. So, with this in mind, let’s tackle the expression at hand!

Analyzing the Expression: 3 - sin(2x + 1°) - 2

Now that we know the range of the sine function, let's analyze our expression: 3 - sin(2x + 1°) - 2. To make things simpler, we can first combine the constants:

3 - 2 = 1

So, our expression simplifies to:

1 - sin(2x + 1°)

Now, the key part here is sin(2x + 1°). We know that the sine function oscillates between -1 and 1. The expression 2x + 1° inside the sine function just changes the angle, but it doesn't change the fundamental range of the sine function itself. The sine function will still vary between -1 and 1, no matter what's inside the parentheses. This is a critical observation because it means we can use our knowledge of the sine range to determine the maximum and minimum values of the entire expression. We will now look at how the subtraction of the sine function affects the overall range of the expression.

Determining the Maximum Value

To find the maximum value of the expression 1 - sin(2x + 1°), we need to consider when the subtracted sine part is at its minimum. Remember, we're subtracting sin(2x + 1°), so to make the whole expression as big as possible, we need to subtract the smallest possible value of the sine function. We know that the minimum value of sin(2x + 1°) is -1. So, let's plug that in:

Maximum Value = 1 - (-1)

Maximum Value = 1 + 1

Maximum Value = 2

So, the maximum integer value that the expression can achieve is 2. This makes sense because subtracting a negative number is the same as adding its positive counterpart. This step is crucial because it demonstrates how the properties of sine and subtraction interact to determine the upper bound of our expression. Understanding this mechanism is key to solving similar problems efficiently.

Determining the Minimum Value

Next up, let's find the minimum value of the expression 1 - sin(2x + 1°). This time, to make the whole expression as small as possible, we need to subtract the largest possible value of the sine function. The maximum value of sin(2x + 1°) is 1. So, let's substitute that in:

Minimum Value = 1 - (1)

Minimum Value = 0

Therefore, the minimum integer value that the expression can achieve is 0. This step shows the opposite scenario, where subtracting the maximum value of sine leads to the lowest possible value for the expression. By recognizing this inverse relationship, we can confidently determine the lower bound of our expression. Now that we have both the maximum and minimum values, we can proceed to the final step of finding their product.

Calculating the Product

Now that we've found the maximum and minimum integer values of the expression, we can calculate their product. We found that:

Maximum Integer Value = 2

Minimum Integer Value = 0

So, to find the product, we simply multiply these two values together:

Product = Maximum Value × Minimum Value

Product = 2 × 0

Product = 0

So, the product of the maximum and minimum integer values of the expression 3 - sin(2x + 1°) - 2 is 0. This final step is straightforward but crucial, as it directly answers the original question. By performing this calculation, we confirm our understanding of how the maximum and minimum values interact in the context of the problem. This highlights the importance of each step in the solution process, from understanding sine function properties to final calculation.

Final Answer

Alright, guys! We've successfully navigated through this math problem. To recap, we found the maximum value of the expression 3 - sin(2x + 1°) - 2 to be 2, the minimum value to be 0, and their product is 0. So, the final answer is:

0

Isn't it awesome how we can use our knowledge of trigonometric functions and algebraic principles to solve these kinds of problems? Remember, the key is to break down complex expressions into smaller, manageable parts. By understanding the properties of functions like sine, we can easily find maximum and minimum values. Keep practicing, and you'll become a math whiz in no time! Great job, everyone, and keep up the awesome work! Math is like a puzzle; each piece has its place, and once you put them together, it all makes sense. Keep exploring, keep learning, and most importantly, keep having fun with it!