Unveiling The Distance: Lili's Stick And Trigonometry
Hey guys! Let's dive into a fun geometry problem involving Lili and her trusty stick. This isn't just about math; it's about seeing how the world works around us, using simple tools and a bit of clever thinking. We'll be using some basic trigonometry to find out the distance between the bottom of Lili's stick and the wall. Ready to crack the code? Let's get started!
Understanding the Problem: Lili's Stick Scenario
Alright, so here's the deal: Lili has a stick that's 1.2√2 meters long. She leans it against a wall, and the bottom of the stick forms a 45-degree angle with the floor. The question is: How far is the bottom of the stick from the wall? This problem is a classic example of how trigonometry can be used in everyday situations. Think about it: construction workers, architects, and even carpenters use these principles all the time to make sure things are perfectly aligned. We're essentially doing the same thing here, but on a smaller scale. We are trying to find the horizontal distance, or the adjacent side of the triangle formed by the stick, the wall, and the floor. The stick itself is the hypotenuse (the longest side, opposite the right angle), and the angle is 45 degrees. Our goal? To find the length of the base. This scenario sets the stage for a straightforward trigonometry problem. We know the length of the stick (the hypotenuse) and the angle it makes with the ground, and we want to find the distance from the wall (the adjacent side). This calls for a little bit of trigonometric magic. This problem uses the principles of right-angled triangles, angles, and trigonometric functions to find the missing side, it is a fun puzzle.
Before we jump into the solution, it's worth understanding the core concepts at play. We're dealing with a right-angled triangle, meaning one of the angles is 90 degrees. This is key because it allows us to use trigonometric ratios like sine, cosine, and tangent. These ratios relate the angles of a triangle to the lengths of its sides. In our case, the 45-degree angle is crucial, as it simplifies things significantly. A 45-degree angle in a right triangle tells us that the other angle (besides the 90-degree angle) is also 45 degrees, making it an isosceles right triangle. This means the two sides opposite the 45-degree angles are equal. So, when we use the cosine function, we're essentially looking at the ratio of the adjacent side (the distance we want to find) to the hypotenuse (the length of the stick).
Applying Trigonometry: Cosine to the Rescue!
Alright, let's get down to the nitty-gritty and use some trigonometry! Since we know the hypotenuse (the stick's length) and we're trying to find the adjacent side (the distance from the wall), we'll use the cosine function. Cosine is defined as:
Cos(θ) = Adjacent / Hypotenuse
Here's how we break it down:
- Identify the knowns:
- Angle (θ) = 45 degrees
- Hypotenuse = 1.2√2 meters
- Set up the equation:
- Cos(45°) = Adjacent / 1.2√2
- Find the cosine of 45 degrees:
- Cos(45°) = 1/√2 (which is approximately 0.707)
- Rearrange and solve for the Adjacent:
- Adjacent = Cos(45°) * Hypotenuse
- Adjacent = (1/√2) * 1.2√2
- Adjacent = 1.2 meters
See? It's not as scary as it looks, right? The key is to understand the relationships between the sides and angles of the triangle, and then use the correct trigonometric function. The beauty of this is that it gives us a simple, direct way to calculate the unknown distance. By using the cosine function, we translate the angle and the known side into a value that directly tells us the length of the side we're trying to find.
It is important to remember that trigonometric functions like cosine have specific values for specific angles. These values are derived from the ratios within right-angled triangles. When you input cosine 45, you're essentially telling the calculator to apply the ratio that defines the relationship between the adjacent side and the hypotenuse in a 45-degree triangle. This calculation then gives us a numerical value that allows us to find the actual length of the adjacent side. This method is incredibly useful not only for solving this particular problem but for a wide range of geometrical and engineering calculations. Being familiar with these functions is like having a secret weapon for tackling various problems involving angles and distances!
The Answer and What It Means
So, after all that, the distance between the bottom of the stick and the wall is 1.2 meters. This means option (a) is correct! But more than just getting the right answer, think about what this means in the real world. This calculation allows us to understand spatial relationships and measure distances accurately, helping us better understand the world around us. In our case, we used the cosine function because we had the angle and the hypotenuse (the stick's length) and wanted to find the adjacent side (the distance from the wall). We applied the formula Cosine = Adjacent / Hypotenuse, and by knowing the cosine value for 45 degrees (which is approximately 0.707 or 1/√2), we could calculate the distance directly. The result provides us with a precise measurement of the distance from the wall. This is a practical example of how mathematics provides the tools to solve real-world problems. Whether it's setting up a ladder, building a house, or even just figuring out how far away something is, trigonometry helps. This ability to make these calculations is a testament to the power of mathematics! The ability to calculate distances and angles accurately is crucial in many fields, from architecture and engineering to navigation and surveying. Being able to solve this problem demonstrates the practical application of trigonometry.
Understanding and correctly applying trigonometric functions is essential. It highlights the importance of using formulas to accurately measure distances and angles. This ability translates into a deeper understanding of spatial relationships and how things fit together in the world around us. In addition to solving the problem, the process also gives us a clear understanding of the principles that underpin the solution. It's a great example of applying theoretical concepts to a practical scenario, making math more relatable and useful. Remember, practice makes perfect! The more you work through these types of problems, the easier it will become to recognize the patterns and apply the appropriate trigonometric functions. This will not only improve your problem-solving skills but will also deepen your appreciation for the practical applications of mathematics.
Why This Matters: Beyond the Classroom
Now, you might be thinking,