Math Problem: Solving Trigonometric Equations

Hey math enthusiasts! Today, we're diving into a neat little trigonometry problem. We're given that 5x = π/2, and we're tasked with finding the value of a rather interesting expression. Let's break it down step by step, shall we? This is a great example of how we can use trigonometric identities to simplify complex expressions and find a numerical answer. So, grab your pencils and let's get started! Remember, the key to solving these problems is to stay organized and to apply the right formulas at the right time. We will go through the formula step by step, and explain the logic behind each step. This method is designed to help you learn and understand how to solve complex trigonometric problems. Are you ready to see how it's done?

First off, let's clarify what we're dealing with. We're looking at the expression: (cos 3x) / (sin 2x) - (tan x) / (cot 4x). Our ultimate goal is to simplify this down, using the given condition of 5x = π/2 which means that x = π/10. This will help us eliminate the variables and find an actual numerical value. But first, it looks like we need to get creative with our trig identities to simplify this before we start plugging in numbers. Trigonometry is all about relationships between angles and sides of triangles. We will focus on rewriting and adjusting the expression in order to get a result. The process may seem a bit complex, but it is the application of basic concepts that will help us solve it.

Breaking Down the Problem with Trigonometric Identities

Alright, guys, let's get our hands dirty with some trig identities! The expression we're working with has several trigonometric functions, including cosine, sine, tangent, and cotangent. Our goal is to transform this expression into something we can evaluate using the given condition 5x = π/2. This means that x = π/10. Trigonometric identities are your best friends in these situations. They allow you to rewrite expressions in a way that makes them easier to handle. Let's start by looking at the different parts of the equation individually and then combine them back together. Understanding these identities will make the whole process much more accessible. Always keep in mind that practice is key. The more you practice, the better you'll get at recognizing the right identities and applying them effectively. Now, let’s move on to the next stage of solving this problem.

Let's start with the first term, (cos 3x) / (sin 2x). We could try using the triple angle formula for cosine, but let's hold off on that for a second. Instead, let's focus on the second term, (tan x) / (cot 4x). Remember that cotangent is the reciprocal of tangent, so cot(4x) = 1/tan(4x). Thus, (tan x) / (cot 4x) becomes (tan x) * tan(4x). Our expression now looks like this: (cos 3x) / (sin 2x) - tan(x)tan(4x). Now the question becomes: How do we simplify this? It's all about finding the right path through the maze of trigonometric identities. There are often multiple ways to approach a problem like this, but the key is to keep trying different identities until you find one that leads to a simplification. This is where your knowledge of trig identities becomes crucial. And it’s also where a little bit of cleverness comes into play. The more problems you work through, the better you'll become at spotting the patterns and choosing the most effective approach. So, stay with me, and we'll get through this together. Now, let's try to use the condition 5x = π/2 more directly.

Applying the Condition 5x = π/2

Okay, time to bring our secret weapon into play: 5x = π/2. This means x = π/10. Notice that 5x is in the equation, which could potentially help us to reduce the variables. We can rewrite some of the terms using this condition, especially the ones that have a coefficient to 'x' that could be made into 5x. Because we have the value of x, we can start substituting this into the expression, and slowly reduce the expression into a single value. Now we know that x = π/10, we can also find out what 2x, 3x, and 4x are: 2x = π/5, 3x = 3π/10, and 4x = 2π/5. We can replace the variables in the equation (cos 3x) / (sin 2x) - tan(x)tan(4x). Let’s substitute all the variables and check what we get from them. With a bit of patience and some well-chosen substitutions, we should be able to find the answer. It’s like a puzzle, where each piece of information helps us to complete the bigger picture. This step will help us reduce the complexity of our equation by substituting the value of x which we have previously calculated. Remember that simplification is the name of the game.

Let's substitute the values. We will start with (cos 3x) / (sin 2x). We know that 3x = 3π/10 and 2x = π/5. Therefore, the first part of the equation can be written as: cos(3π/10) / sin(π/5). And the second part of the equation is tan(x)tan(4x). Since x = π/10 and 4x = 2π/5, the second part of the equation can be written as: tan(π/10)tan(2π/5). Putting them back into our original equation, we get: cos(3π/10) / sin(π/5) - tan(π/10)tan(2π/5). This looks more manageable, don’t you think? We've turned a complex expression into something much simpler by substituting values. Now we can try to find the final result, by continuing with the trigonometric identities.

Further Simplification and Calculation

We’ve simplified the equation to cos(3π/10) / sin(π/5) - tan(π/10)tan(2π/5). Now, let's dig deeper into some trigonometric identities to see if we can simplify this further. One useful identity is the cofunction identity: cos(π/2 - θ) = sin(θ). Let’s rewrite cos(3π/10) to see if we can take advantage of it. Since 3π/10 = π/2 - π/5, we can write: cos(3π/10) = cos(π/2 - π/5) = sin(π/5). This simplifies the first part of the equation quite nicely. Thus, the cos(3π/10) / sin(π/5) transforms into sin(π/5) / sin(π/5) = 1. We are now in the last part of solving the problem. With a few more steps, we will have our final answer. We've got the first part done. It’s time to move on to the second part of the equation, which will involve the same procedure as the first one, and finish it off. Now, let’s simplify the second term, tan(π/10)tan(2π/5). Remember that 2π/5 = π/2 - π/10. So, we can rewrite tan(2π/5) as tan(π/2 - π/10). We also know that tan(π/2 - θ) = cot(θ). Therefore, tan(π/2 - π/10) = cot(π/10). Now the second term becomes tan(π/10) * cot(π/10), and since cotangent is the reciprocal of tangent, tan(π/10) * cot(π/10) = 1. So, the second term simplifies to 1 as well. Our original equation cos(3π/10) / sin(π/5) - tan(π/10)tan(2π/5) simplifies to 1 - 1.

Therefore, the final result is 1 - 1 = 0. So, the answer to the equation is 0! We have solved it! Congrats, guys! This shows how powerful trigonometric identities can be when you know how to use them. By systematically applying the right formulas and being patient, you can tackle even the most complex-looking problems. Remember, practice makes perfect. The more you work through these types of problems, the more comfortable and confident you'll become in your ability to solve them. Keep up the great work, and keep exploring the fascinating world of trigonometry!