Finding Tan(α): Solving Trigonometric Equations
Hey guys! Let's dive into a cool math problem today that involves finding the tangent of an acute angle. This is a classic trigonometry question that combines trigonometric identities and a bit of algebraic manipulation. We're given that the angle α is acute, meaning it lies between 0 and 90 degrees. We also have the equation 5tan(α) - 2sin²(α) = 2cos²(α). Our mission, should we choose to accept it, is to find the value of tan(α). Sounds like fun, right? Buckle up, and let's get started!
Understanding the Problem
Before we jump into solving, let's break down what we know and what we need to find. We are dealing with trigonometric functions, specifically tangent (tan), sine (sin), and cosine (cos). The relationship between these functions, especially the Pythagorean identity, will be crucial. Our main keyword here is tangent (tan(α)), as that’s what we're ultimately trying to find. The equation we have involves both sin²(α) and cos²(α), which hints at using the fundamental trigonometric identity sin²(α) + cos²(α) = 1. This identity is a cornerstone in solving trigonometric problems, guys, so make sure you're super familiar with it. The fact that α is an acute angle is also important because it tells us that all trigonometric functions for α will be positive. This helps us narrow down our solutions later on. Remember, acute angles are our friends in these scenarios, making the math a bit cleaner and more straightforward. So, keep in mind the basics: what are the trigonometric functions, and how are they related? And most importantly, what does it mean for an angle to be acute? Got it? Great, let's move on to the next step.
Leveraging Trigonometric Identities
Okay, so we have our equation: 5tan(α) - 2sin²(α) = 2cos²(α). The first thing that probably jumps out is the sin²(α) and cos²(α) terms. These scream for us to use the Pythagorean identity, sin²(α) + cos²(α) = 1. But how do we make that substitution? Well, notice that the right side of our equation has 2cos²(α). If we can somehow get the left side to look like something involving sin²(α) + cos²(α), we're in business. Let's rearrange the given equation a bit. We can rewrite 2cos²(α) as 2(1 - sin²(α)), using our Pythagorean identity. This gives us: 5tan(α) - 2sin²(α) = 2 - 2sin²(α). Now, something cool happens! We have -2sin²(α) on both sides of the equation. We can add 2sin²(α) to both sides, and these terms cancel out, simplifying our equation significantly. This leaves us with 5tan(α) = 2. See how much cleaner that is? This step is crucial because it eliminates the squared trigonometric functions, bringing us closer to isolating tan(α). Remember, the goal here is to manipulate the equation using known identities to make it simpler and solvable. It's like a puzzle, guys, where each identity is a piece that fits just right. So, with this simplification, we're on the home stretch to finding our value for tan(α).
Solving for tan(α)
Alright, after simplifying our equation using the Pythagorean identity, we've arrived at a much more manageable form: 5tan(α) = 2. Now, this looks like something we can easily solve! Our main keyword, tan(α), is almost isolated. To find the value of tan(α), all we need to do is divide both sides of the equation by 5. This gives us tan(α) = 2/5. And there you have it! We've found the value of the tangent of the angle α. This step highlights the importance of algebraic manipulation in trigonometry. Once we've used trigonometric identities to simplify the equation, the final solution often comes down to basic algebra. It's like the grand finale of our mathematical journey! Now, let's take a moment to think about what this means in the context of the problem. We were given that α is an acute angle, and we've found that tan(α) = 2/5. Since the tangent of an acute angle is positive, this result makes sense. If we had gotten a negative value, we'd know something went wrong somewhere. So, always double-check that your answer is reasonable within the given context. We're not just solving for a number, guys; we're finding a value that fits the geometric conditions of the problem.
Checking the Answer Options
Now that we've calculated tan(α) = 2/5, let's look back at the answer options provided in the original question. This is a crucial step in any problem-solving process, especially in multiple-choice questions. We need to make sure that our answer matches one of the given choices. The options were:
A. 5 B. 1/5 C. 5/2 D. 2/5
Comparing our result with these options, we can see that option D, 2/5, matches our calculated value perfectly. This confirms that we've solved the problem correctly! It's always satisfying when your hard work pays off and your answer aligns with the given choices. This step also serves as a final check against any potential errors. If our answer hadn't matched any of the options, we'd know to go back and review our steps. So, never skip this verification process, guys. It's the last line of defense against mistakes. Plus, it gives you that extra confidence boost when you know you've nailed it.
Conclusion
So, there you have it! We've successfully found the value of tan(α) in this trigonometric problem. We started with a given equation, used the Pythagorean identity to simplify it, and then solved for tan(α) using basic algebra. The key takeaway here is the power of trigonometric identities in simplifying complex equations. Remember, identities like sin²(α) + cos²(α) = 1 are your best friends in these scenarios. They allow you to transform equations into more manageable forms. And, of course, don't forget the importance of algebraic manipulation in isolating the variable you're trying to find. It's a combination of trigonometric knowledge and algebraic skills that leads to the solution. This problem also highlights the significance of understanding the context of the problem. The fact that α was an acute angle helped us ensure that our answer was reasonable. Math isn't just about formulas and equations, guys; it's about logical reasoning and problem-solving. So, keep practicing, keep exploring, and keep having fun with it! You'll be surprised at how much you can achieve. Until next time, happy solving! Remember, the answer is D. 2/5.