Solving Sin(α) With Trigonometric Identities: A Step-by-Step Guide

Hey guys! Today, we're diving into a trigonometric problem where we need to find the value of sin(α) given some initial conditions. This involves using trigonometric identities and a bit of algebraic manipulation. Don't worry, we'll break it down step by step so it's super easy to follow. So, let's jump right in and get this solved!

Understanding the Problem

Okay, first things first, let's make sure we fully grasp what we're dealing with. We're given that sin(π/4 - α) = -√2/10, and we know that π < α < 3π/2. Our mission, should we choose to accept it (and we do!), is to figure out what sin(α) is. This isn't as straightforward as plugging numbers into a calculator; we're going to need to use our knowledge of trigonometric identities to unravel this. Specifically, we'll be leveraging the sine subtraction formula, which is a cornerstone in solving problems of this nature. Understanding the range of α is also super important because it helps us determine the sign of sin(α) once we find its value. Trigonometry can seem daunting, but with a systematic approach, it becomes much more manageable. We’ll take each step slowly, ensuring every piece of the puzzle fits perfectly. So, stick with me, and we'll conquer this trigonometric challenge together!

The Sine Subtraction Formula

The sine subtraction formula is our main tool here, guys. It states that: sin(A - B) = sin(A)cos(B) - cos(A)sin(B). This formula is a game-changer because it allows us to expand sin(π/4 - α) into something we can actually work with. Think of it as our secret decoder ring for this problem. Now, why is this formula so crucial? Well, it lets us break down a complex trigonometric function into simpler, more manageable parts. By applying this identity, we can express the given equation in terms of sin(α) and cos(α), which are what we ultimately want to find. Remember, the beauty of trigonometry lies in these identities that connect different functions and angles. Once you've mastered these identities, a whole world of trigonometric problems opens up for you to solve! We'll apply this formula directly in the next step, so keep it in mind!

Applying the Formula

Let's plug in our values! We have A = π/4 and B = α. So, using the formula, sin(π/4 - α) becomes sin(π/4)cos(α) - cos(π/4)sin(α). Now, we know that sin(π/4) = cos(π/4) = √2/2. These are standard values that you should probably have in your trigonometric toolkit. Substituting these values into our expanded equation gives us: (√2/2)cos(α) - (√2/2)sin(α) = -√2/10. See how we've transformed the original equation into something that involves both sin(α) and cos(α)? This is a key step in solving for sin(α). We're now one step closer to our goal. By applying this formula correctly, we've set the stage for the next phase of the solution, which involves simplifying and rearranging the equation to isolate the terms we're interested in. So, keep up the great work, we're making progress!

Simplifying the Equation

Now that we've expanded the sine subtraction formula, let's simplify the equation to make it easier to handle. We have: (√2/2)cos(α) - (√2/2)sin(α) = -√2/10. The cool thing here is that we can see a common factor of √2/2 on the left side. Let's factor that out, guys! This gives us: (√2/2)[cos(α) - sin(α)] = -√2/10. Factoring simplifies things immensely because now we can divide both sides by √2/2 to isolate the term in the brackets. When we do this, we get: cos(α) - sin(α) = -1/5. This is a much cleaner equation, isn't it? We've gone from a complex trigonometric expression to a simple algebraic one. This step highlights the importance of simplification in problem-solving. By identifying common factors and performing basic algebraic operations, we can often make seemingly difficult equations much more approachable. With this simplified equation, we're well-positioned to move on to the next step, which involves using trigonometric identities to relate cos(α) and sin(α).

Isolating cos(α)

Let's isolate cos(α) in our simplified equation, cos(α) - sin(α) = -1/5. Adding sin(α) to both sides, we get: cos(α) = sin(α) - 1/5. Why are we doing this? Because it sets us up to use another super helpful trigonometric identity: the Pythagorean identity. This step is all about strategic manipulation. By isolating cos(α), we're creating an opportunity to substitute it into another equation, which will ultimately allow us to solve for sin(α). It's like setting up a chain reaction in a puzzle – each step leads us closer to the final solution. Think of it as a game of trigonometric chess, where we're carefully positioning our pieces for the win. This isolation technique is a common trick in trigonometry, and mastering it will serve you well in many problems. So, let's keep this new form of cos(α) in our back pocket as we move on to the next phase!

Using the Pythagorean Identity

Okay, guys, this is where things get really interesting! We're going to use the Pythagorean identity: sin²(α) + cos²(α) = 1. This is like the golden rule of trigonometry, and it's going to help us link sin(α) and cos(α) in a way that lets us solve for sin(α). Remember we isolated cos(α) in the last step? Now we're going to substitute that expression into this identity. This is a classic move in trigonometry – using one equation to simplify another. It's like a detective using clues to piece together a puzzle. By making this substitution, we're essentially turning our trigonometric problem into an algebraic one, which is often easier to solve. So, prepare for some algebraic fun as we transform this equation and get closer to finding the value of sin(α)!

Substituting cos(α)

Remember that we found cos(α) = sin(α) - 1/5. Now, let's substitute this into the Pythagorean identity sin²(α) + cos²(α) = 1. Replacing cos(α) gives us: sin²(α) + (sin(α) - 1/5)² = 1. See how we've turned our trigonometric equation into an algebraic equation involving only sin(α)? This is a significant step forward. By making this substitution, we've created an equation that we can actually solve. It might look a little intimidating right now, but don't worry, we're going to expand and simplify it. This substitution is a prime example of how strategic manipulation can make complex problems much more manageable. We're essentially leveraging the relationships between trigonometric functions to our advantage. So, let's roll up our sleeves and tackle this algebraic equation!

Solving the Quadratic Equation

Alright, guys, we've got a quadratic equation on our hands! After substituting and expanding, we should get something like this: sin²(α) + (sin²(α) - (2/5)sin(α) + 1/25) = 1. Let's simplify this further by combining like terms. This gives us: 2sin²(α) - (2/5)sin(α) + 1/25 = 1. Now, to make it look even more like a standard quadratic equation, let's subtract 1 from both sides: 2sin²(α) - (2/5)sin(α) - 24/25 = 0. This looks much more familiar, right? We now have a quadratic equation in terms of sin(α). To make it even cleaner, we can multiply the entire equation by 25 to get rid of the fractions: 50sin²(α) - 10sin(α) - 24 = 0. This is a classic quadratic equation that we can solve using the quadratic formula or factoring. Solving quadratic equations is a fundamental skill in math, and it's fantastic to see how it applies in trigonometry as well. We're now in the home stretch – once we solve for sin(α), we're just a step away from our final answer!

Applying the Quadratic Formula

Okay, let's bring out the big guns – the quadratic formula! For an equation in the form ax² + bx + c = 0, the solutions are given by: x = [-b ± √(b² - 4ac)] / (2a). In our case, we have 50sin²(α) - 10sin(α) - 24 = 0, so a = 50, b = -10, and c = -24. Plugging these values into the quadratic formula gives us: sin(α) = [10 ± √((-10)² - 4 * 50 * -24)] / (2 * 50). This looks a bit intimidating, but let's break it down. First, we calculate the discriminant (the part under the square root): (-10)² - 4 * 50 * -24 = 100 + 4800 = 4900. The square root of 4900 is 70. So, our equation becomes: sin(α) = [10 ± 70] / 100. This gives us two possible solutions for sin(α). Applying the quadratic formula might seem like a purely mechanical process, but it's a powerful tool for solving a wide range of problems. We're now at a critical juncture – we have two potential solutions for sin(α), and we need to determine which one is correct based on the given information. So, let's move on to the next step and filter out the extraneous solution!

Finding Possible Solutions for sin(α)

From the quadratic formula, we found two possible solutions for sin(α): sin(α) = (10 + 70) / 100 = 80 / 100 = 4/5 and sin(α) = (10 - 70) / 100 = -60 / 100 = -3/5. So, we have two candidates for the value of sin(α). But hold on, we're not done yet! We need to consider the given range for α, which is π < α < 3π/2. This is super important because it tells us which quadrant α lies in. Remember, guys, the unit circle is our friend here! Knowing the quadrant helps us determine the sign of sin(α). This step is a crucial reminder that mathematical problems often have multiple layers, and we need to consider all the given information to arrive at the correct answer. By carefully analyzing the range of α, we'll be able to eliminate one of these solutions and pinpoint the true value of sin(α). So, let's put on our detective hats and figure out which solution fits the bill!

Determining the Correct Solution

We know that π < α < 3π/2. This means α lies in the third quadrant. In the third quadrant, both sine and cosine are negative. This is a key piece of information! We have two possible values for sin(α): 4/5 and -3/5. Since sin(α) must be negative in the third quadrant, we can eliminate 4/5 as a solution. Therefore, the correct solution is sin(α) = -3/5. Woohoo! We did it! By considering the quadrant in which α lies, we were able to narrow down our solutions and find the correct answer. This step highlights the importance of connecting the algebraic solution with the geometric context of the problem. Trigonometry is all about relationships between angles and sides of triangles, and understanding these relationships is crucial for solving problems accurately. So, congratulations on making it this far – we've successfully navigated this trigonometric challenge!

Final Answer

So, guys, after all that awesome work, we've finally found the solution! Given sin(π/4 - α) = -√2/10 and π < α < 3π/2, we've determined that sin(α) = -3/5. This wasn't a simple plug-and-chug problem; we had to use trigonometric identities, algebraic manipulation, and a bit of logical deduction. We expanded the sine subtraction formula, used the Pythagorean identity, solved a quadratic equation, and considered the quadrant in which α lies. Each step was essential in leading us to the final answer. This problem is a great example of how math can be like a puzzle, where each piece fits together to form the solution. And remember, the journey is just as important as the destination. By working through this problem, we've strengthened our understanding of trigonometric concepts and problem-solving techniques. So, give yourself a pat on the back – you've earned it!