Finding Max & Min Values Of Trigonometric Expressions

Hey guys! Let's dive into a fun math problem today. We're going to explore the expression 3 - sin(2x + 1°) - 2 and figure out the product of its largest and smallest possible whole number values. Sounds cool, right? This isn't just about crunching numbers; it's about understanding how trigonometric functions work and how to squeeze the most (or the least!) out of an expression. So, grab your calculators (or your thinking caps!), and let's get started. We'll break down the concepts, step by step, making sure you grasp everything. Get ready to flex those math muscles!

Understanding the Core Concept: Sine Function's Behavior

First off, let's talk about the sine function. The sine function, denoted as sin(θ), is a fundamental concept in trigonometry. Its input is an angle (often measured in degrees or radians), and its output is a number that represents the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle. But here's the kicker: the sine function always oscillates between -1 and 1. Yes, sin(θ) always falls between -1 and 1, no matter what angle you throw at it. This is super important because it sets the boundaries for our expression. Understanding this is key to solving our problem.

Think of the sine function as a wave. It goes up to a peak of 1, down to a trough of -1, and then repeats. This regular, predictable behavior is what we're going to leverage to find the maximum and minimum values of our expression. The angle inside the sine function, in our case (2x + 1°), doesn't change the range of the sine function itself. It just shifts and stretches the input, but the output still stays between -1 and 1. So, regardless of the value of x, the sin(2x + 1°) part will always be between -1 and 1. This is the foundation upon which we build our solution. It's like knowing the rules of the game before you start playing; it gives you a huge advantage!

Now, let's look at how this impacts the rest of the expression. Remember, our goal is to find the greatest and smallest possible integer values that the entire expression can take. We need to work through this step by step, and the following will show us how to get it.

Decoding the Expression: Step-by-Step Analysis

Alright, let's break down the expression: 3 - sin(2x + 1°) - 2. It looks a bit intimidating at first, but trust me, it's manageable. We've already established that sin(2x + 1°) ranges from -1 to 1. Now, let's consider the rest of the terms. We have a constant '3' and another constant '-2'. These constants will shift the overall value of the expression up or down, but they won't change the inherent range determined by the sine function.

To find the maximum value, we want to maximize the contribution from the sine function. Since the sine function can be as large as 1, we want to consider how the parts of the expression would work. So, we'll try to get the smallest value of the -sin(2x + 1°). This happens when sin(2x + 1°) is equal to -1. The expression then becomes 3 - (-1) - 2 = 3 + 1 - 2 = 2. It will be the maximum value. To get the minimum value, we want to minimize the expression. The largest value of sin(2x + 1°) is 1, so -sin(2x + 1°) = -1`. The expression is then 3 - 1 - 2 = 0. This is the smallest value.

We are looking for the integer values, the whole numbers. We've basically got this, but let's formalize our approach. The easiest way to deal with this is to consider the range of the sine function and how the constants affect it. We know that -1 ≤ sin(2x + 1°) ≤ 1. When we multiply the sine function by -1, the inequality flips: -1 ≤ -sin(2x + 1°) ≤ 1. Then we add 3 to this expression, we have 3 - 1 - 2. Let's make it simpler, we know that the maximum value for sin(2x + 1°) is 1, so the minimum value for the whole expression will be 3 - 1 - 2 = 0. We know that the minimum value for sin(2x + 1°) is -1, so the maximum value for the whole expression will be 3 - (-1) - 2 = 2. These values are integers, which is what we need to solve the problem.

Calculating the Maximum and Minimum Integer Values

Okay, time to do the math and nail down those maximum and minimum values. Let's start with the maximum value. We need to find the value of x that makes the expression as large as possible. This happens when the sin(2x + 1°) has the smallest effect on the overall value. Since the expression is 3 - sin(2x + 1°) - 2, we want -sin(2x + 1°) to be as large as possible. This means sin(2x + 1°) should be -1. When sin(2x + 1°) = -1, the entire expression becomes 3 - (-1) - 2 = 3 + 1 - 2 = 2. So, the maximum value of the expression is 2.

Now, for the minimum value. We want to find the value of x that makes the expression as small as possible. This occurs when sin(2x + 1°) has the biggest positive effect on the value. This means sin(2x + 1°) should be 1. When sin(2x + 1°) = 1, the entire expression becomes 3 - 1 - 2 = 0. Therefore, the minimum value of the expression is 0. We've got our maximum and minimum values! Now we just need to calculate their product. It is pretty simple from here. The maximum is 2, and the minimum is 0, so the product is 2 * 0 = 0. We can say with confidence that the product of the largest and smallest integer values of the expression is 0. We're almost there, let's sum it all up!

The Grand Finale: Multiplication and Final Answer

We've done the hard work, guys! We've found that the maximum integer value of the expression is 2, and the minimum integer value is 0. The final step is to multiply these two values together to get our answer. So, we multiply 2 by 0, which equals 0. The product of the greatest and smallest integer values of the expression 3 - sin(2x + 1°) - 2 is 0.

And there you have it! We've successfully navigated the world of trigonometric expressions, figured out their maximum and minimum values, and multiplied them together. Pretty awesome, right? Remember, understanding the behavior of the sine function is key. It helps you grasp how these expressions behave and how to find their extreme values. Keep practicing, and you'll become a pro at these problems in no time. Math is fun, especially when you understand it! Keep exploring, keep learning, and never be afraid to tackle a challenging problem. You got this!