Solving Sin(x) + Cos(x) = 1/5: Find Sin(x)cos(x)
Hey everyone! Today, we're diving into a cool trigonometric problem. We're given that sin(x) + cos(x) = 1/5, and our mission, should we choose to accept it, is to find the value of sin(x) * cos(x). Sounds like fun, right? Let's break it down step by step and make sure we really understand what's going on. Trigonometry can seem intimidating at first, but with a bit of algebraic manipulation and some key trigonometric identities, we can totally nail this. So grab your thinking caps, and let's jump into the wonderful world of trigonometric equations!
Understanding the Problem
Before we get our hands dirty with calculations, it's super important to really understand the problem. We're given an equation that involves both sin(x) and cos(x), and we need to find a way to relate this information to the product of sin(x) and cos(x). This kind of problem often involves using trigonometric identities and algebraic manipulations to get to our desired result. Think of it like a puzzle – we have the pieces, and now we need to figure out how they fit together. Remember, math isn't just about memorizing formulas; it's about understanding the relationships between different concepts and using them creatively to solve problems.
Key Concepts and Identities
To tackle this problem effectively, we need to have a few key trigonometric identities up our sleeves. The most important one for this problem is the Pythagorean identity: sin²(x) + cos²(x) = 1. This identity is like the bread and butter of trigonometry, and it shows up in all sorts of problems. Another crucial concept is how squaring a binomial expression works. Remember that (a + b)² = a² + 2ab + b². We'll be using this algebraic identity to link the given equation to the expression we want to find. So, keep these identities in mind as we move forward – they're going to be our best friends in solving this trigonometric puzzle. It's always a good idea to refresh these basics before tackling more complex problems!
Step-by-Step Solution
Okay, guys, let's get down to business and solve this thing! Here’s the step-by-step solution, broken down to make it super clear.
1. Squaring the Given Equation
The first clever trick we're going to use is to square both sides of the given equation: sin(x) + cos(x) = 1/5. Why do we do this? Well, squaring the left side will introduce the sin²(x) and cos²(x) terms, which we know are related by the Pythagorean identity. Plus, it'll give us a term with sin(x)cos(x), which is exactly what we're trying to find! So, let's do it:
(sin(x) + cos(x))² = (1/5)²
Expanding the left side using the binomial formula (a + b)² = a² + 2ab + b², we get:
sin²(x) + 2sin(x)cos(x) + cos²(x) = 1/25
2. Applying the Pythagorean Identity
Now comes the magic! We know from the Pythagorean identity that sin²(x) + cos²(x) = 1. Let's substitute this into our equation:
1 + 2sin(x)cos(x) = 1/25
See how things are starting to simplify? We're one step closer to isolating the term we want.
3. Isolating sin(x)cos(x)
Our next step is to isolate the term 2sin(x)cos(x). To do this, we'll subtract 1 from both sides of the equation:
2sin(x)cos(x) = 1/25 - 1
To subtract the fractions, we need a common denominator. So, we'll rewrite 1 as 25/25:
2sin(x)cos(x) = 1/25 - 25/25
2sin(x)cos(x) = -24/25
4. Solving for sin(x)cos(x)
Finally, to get sin(x)cos(x) by itself, we'll divide both sides of the equation by 2:
sin(x)cos(x) = (-24/25) / 2
sin(x)cos(x) = -24/50
We can simplify this fraction by dividing both the numerator and the denominator by 2:
sin(x)cos(x) = -12/25
And there you have it! We've found the value of sin(x)cos(x). It's -12/25. High five!
Common Mistakes to Avoid
Alright, let’s chat about some common pitfalls people often stumble into when solving problems like this. Knowing these can help you dodge those mathematical bullets!
Forgetting the Binomial Expansion
A super common mistake is messing up the expansion of (sin(x) + cos(x))². Remember, it’s sin²(x) + 2sin(x)cos(x) + cos²(x), not just sin²(x) + cos²(x). That middle term, 2sin(x)cos(x), is crucial, so don’t forget it! Treat it just like any other binomial expansion; it’s all about following the rules of algebra.
Ignoring the Pythagorean Identity
The Pythagorean identity, sin²(x) + cos²(x) = 1, is like the Swiss Army knife of trigonometry. Forgetting to use it or not recognizing when to apply it can really throw you off. In this problem, it’s the key to simplifying the equation and moving towards the solution. So, always keep it in your back pocket!
Arithmetic Errors
Simple arithmetic mistakes can happen to anyone, especially when dealing with fractions and negative signs. Take your time when adding, subtracting, multiplying, and dividing. Double-check your work, especially when simplifying fractions. It's so frustrating to do all the hard trig work and then lose points on a simple math error!
Not Simplifying the Final Answer
It’s tempting to stop once you’ve found a numerical answer, but always make sure your final answer is simplified as much as possible. In our case, we needed to reduce -24/50 to -12/25. Simplifying not only gives the correct, clean answer but also shows your teacher or grader that you’ve got a good handle on the basics.
Practice Problems
Okay, now that we've conquered this problem, let's keep the momentum going! Practice makes perfect, so here are a few practice problems that are similar to what we just tackled. Work through these, and you’ll be a trig whiz in no time!
- If sin(x) - cos(x) = 1/3, find the value of sin(x)cos(x).
- Given sin(x) + cos(x) = √2/2, determine the value of sin(x)cos(x).
- If sin(x)cos(x) = 1/4 and sin(x) + cos(x) > 0, find the value of sin(x) + cos(x).
Working through these problems will help you solidify your understanding of the concepts and techniques we used. Don’t just rush through them; take your time, think carefully, and remember the steps we discussed. And hey, if you get stuck, don’t sweat it! Go back through the solution we worked on together, and see if you can spot the similarities and differences. Happy solving!
Conclusion
So, there you have it, folks! We successfully navigated through the problem where sin(x) + cos(x) = 1/5 and figured out that sin(x)cos(x) = -12/25. We walked through the step-by-step solution, highlighted some common mistakes to watch out for, and even gave you some practice problems to try on your own. Remember, tackling trigonometry (or any math, really) is all about understanding the core concepts, practicing regularly, and not being afraid to make mistakes – because that’s how we learn! Keep up the great work, and remember, math can be fun when you break it down and approach it with confidence. You’ve got this!