Solving For Tanx: A Trigonometry Problem Explained
Hey guys! Let's dive into a classic trigonometry problem. We're given an equation involving sine and cosine, and our goal is to find the value of tanx. Specifically, we have the equation (4sinx + cosx) / (3cosx - sinx) = 2/3. This might seem a little intimidating at first, but trust me, it's totally manageable! We'll break it down step-by-step to make sure everything clicks. This kind of problem is super common in math exams, so understanding the process is key. We'll be using some fundamental trigonometric identities and algebraic manipulations to get to the answer. The core concept here is to isolate tanx which is the ratio of sinx to cosx. Let's get started!
First things first, to tackle this problem effectively, we should first remember the fundamental trigonometric relationship, tanx = sinx / cosx. This is the key to solving the problem and helps to get the final result. Now, going back to the original equation: (4sinx + cosx) / (3cosx - sinx) = 2/3. We need to manipulate this equation to get the tanx value. The best way is to simplify the given equation through cross-multiplication. Let’s multiply both sides by 3 to get rid of the fraction.
Now, here's how we're going to approach this problem. We'll start by cross-multiplying to eliminate the fractions. This will give us a more straightforward equation to work with. Then, we'll aim to group the sine and cosine terms together. Finally, we'll divide both sides of the equation by a carefully chosen term to isolate tanx. Sounds good? Alright, let’s begin!
Step-by-Step Solution
Cross-Multiplication
Alright, let’s start by cross-multiplying the given equation: (4sinx + cosx) / (3cosx - sinx) = 2/3. Multiplying both sides by (3cosx - sinx) and by 3, we get:
3 * (4sinx + cosx) = 2 * (3cosx - sinx).
This simplifies to: 12sinx + 3cosx = 6cosx - 2sinx. This step is crucial because it transforms the equation into a linear form in terms of sine and cosine, making it easier to manipulate. See, not so bad, right? We've already simplified the equation and now it's easier to find the solution. Now, let’s continue to the next step which will make it even simpler.
Grouping Sine and Cosine Terms
Next, we're going to group the sine terms on one side and the cosine terms on the other. This helps us to consolidate the terms. Add 2sinx to both sides and subtract 3cosx from both sides. This gives us:
12sinx + 2sinx = 6cosx - 3cosx.
Which simplifies to: 14sinx = 3cosx. The purpose of this arrangement is to prepare the equation for the final step. Now, we are ready to calculate tanx. See how it's all coming together? We are moving closer to the final answer. We're making progress every step of the way, and we're just about to crack it. Let's move on to the final step to determine the value of tanx.
Isolating tanx
Finally, to find tanx, we recall that tanx = sinx / cosx. To get this form, we need to divide both sides of the equation 14sinx = 3cosx by cosx. This gives us:
(14sinx) / cosx = (3cosx) / cosx.
Which simplifies to: 14 * (sinx / cosx) = 3.
Since sinx / cosx = tanx, we can rewrite this as:
14 * tanx = 3.
Now, to solve for tanx, we divide both sides by 14: tanx = 3/14. Boom! We have our answer. And that's it! We've successfully solved for tanx. It really is that straightforward once you break it down into manageable steps. Pretty cool, right? With a bit of practice, you'll be acing these trigonometry problems in no time. This final step is the most crucial, because by dividing both sides by cosx and isolating tanx. We directly arrive at the final solution. The answer makes sense, and is mathematically correct.
Why This Works: Understanding the Logic
So, why does this method work? The magic lies in leveraging the fundamental definition of tanx. By manipulating the original equation to get sinx and cosx on opposite sides, we can then create the ratio that defines tanx. It's all about strategic algebraic manipulation and recognizing the key trigonometric identity. This approach is not only effective but also provides a clear path to the solution. The core of this method is the ability to transform the original equation. Each step we took was based on valid mathematical principles. It’s all interconnected. This method is applicable to similar trigonometric problems. It is really important to grasp these fundamentals to solve similar problems. If you can understand this process, you will easily be able to solve similar problems. Moreover, understanding this approach will increase your confidence in solving similar mathematical problems. The ability to break down the problem into smaller parts and solve it systematically helps you to apply the same strategy in other math problems.
Conclusion: Your Trigonometry Toolkit
Alright, guys, we’ve successfully solved for tanx. We found that tanx = 3/14. We started with a seemingly complex equation and, through a series of logical steps, arrived at a clear solution. Remember, trigonometry is all about understanding the relationships between angles and sides of triangles. By mastering the basic definitions and algebraic manipulation, you can conquer any trigonometry problem. Keep practicing, and you'll find that these problems become easier and more intuitive. Now, you’ve got a solid framework for approaching similar problems. Remember the key steps: cross-multiplication, grouping terms, and isolating tanx. Now go out there and tackle those math problems with confidence! Keep practicing and you’ll get better with each problem you solve. Have fun and keep learning! That's all for today. See you next time, and keep exploring the amazing world of math!