Finding Cos(2x) When Sin(x) Equals 4/5 And Cos(x) Is Negative

Hey guys! Let's dive into a trigonometric problem where we need to find the value of cos(2x) given some information about sin(x) and cos(x). It might sound a bit complex at first, but don't worry, we'll break it down step by step.

The Problem: A Quick Overview

We're given that sin(x) = 4/5 and cos(x) < 0. Our mission, should we choose to accept it, is to determine the value of cos(2x). We have a few options presented to us:

• A. -14/25
• B. -7/25
• C. 7/25
• D. 14/25

To crack this, we'll need to dust off our trigonometric identities and do a little detective work to figure out which quadrant our angle x is in. Let's get started!

Unpacking the Givens

The first crucial piece of information is that sin(x) = 4/5. Remember, in the unit circle context, sine corresponds to the y-coordinate. Since sin(x) is positive, we know that our angle x must lie in either the first or second quadrant, because those are the quadrants where the y-coordinate is positive. So far so good!

Now, here's where it gets interesting. We're also told that cos(x) < 0. Cosine corresponds to the x-coordinate in the unit circle. Thus, cos(x) being negative tells us that x must lie in either the second or third quadrant, where the x-coordinate is negative. That's key information for us to proceed in solving this trigonometric puzzle.

Combining these two clues, we can deduce that x must reside in the second quadrant. This is because the second quadrant is the only one where both sin(x) is positive and cos(x) is negative. Knowing the specific quadrant helps us nail down the sign of trigonometric functions, which is super important for getting the right answer.

Why the Quadrant Matters so much

You might be wondering, why do we make such a fuss about the quadrant? Well, the quadrant determines the signs (+ or -) of the trigonometric functions. In the first quadrant, all trigonometric functions are positive. In the second quadrant, only sine is positive (cosine and tangent are negative). In the third quadrant, only tangent is positive (sine and cosine are negative), and in the fourth quadrant, only cosine is positive (sine and tangent are negative). This is often remembered using the mnemonic "All Students Take Calculus," which indicates which function is positive in each quadrant (starting from the first quadrant and going counterclockwise).

In our case, knowing that x is in the second quadrant confirms that sin(x) is positive (as given) and cos(x) is negative, which will be crucial when we calculate cos(x) directly.

Finding cos(x) Using the Pythagorean Identity

Okay, so we know sin(x) = 4/5, and we need to find cos(x). How do we do it? This is where the Pythagorean identity comes to our rescue. This fundamental identity states that:

sin²(x) + cos²(x) = 1

This identity is like the Swiss Army knife of trigonometry; it's incredibly versatile and pops up everywhere.

We already know sin(x), so we can plug that into the identity and solve for cos(x). Let's do it:

(4/5)² + cos²(x) = 1

16/25 + cos²(x) = 1

Now, we isolate cos²(x):

cos²(x) = 1 - 16/25

cos²(x) = 25/25 - 16/25

cos²(x) = 9/25

To find cos(x), we take the square root of both sides:

cos(x) = ±√(9/25)

cos(x) = ±3/5

But hold on! We're not done yet. Remember, we already determined that x is in the second quadrant, where cosine is negative. Therefore, we choose the negative root:

cos(x) = -3/5

So, we've successfully found cos(x). Pat yourself on the back – that was a crucial step!

Unleashing the Double-Angle Formula for cos(2x)

Now that we've found cos(x), we can finally tackle the main question: finding cos(2x). For this, we'll need the double-angle formula for cosine. There are a few versions of this formula, but the one that's most convenient for us, given the information we have, is:

cos(2x) = cos²(x) - sin²(x)

This formula elegantly expresses cos(2x) in terms of sin(x) and cos(x), both of which we know. It's like having the perfect key to unlock the problem.

Let's plug in the values we have: sin(x) = 4/5 and cos(x) = -3/5:

cos(2x) = (-3/5)² - (4/5)²

cos(2x) = 9/25 - 16/25

cos(2x) = -7/25

And there we have it! The value of cos(2x) is -7/25. We've successfully navigated through the trigonometric terrain and emerged victorious.

The Final Answer and Why It Matters

So, the correct answer is B. -7/25. Woohoo! We solved it. This problem is a classic example of how trigonometric identities and quadrant analysis work together. It's not just about memorizing formulas; it's about understanding how they connect and how the properties of the unit circle influence the signs and values of trigonometric functions.

The reason this type of problem is important is that it reinforces your understanding of the fundamental concepts of trigonometry. It tests your ability to:

• Apply trigonometric identities.
• Determine the sign of trigonometric functions based on the quadrant.
• Combine information to deduce specific values.

These skills are not only essential for further studies in mathematics but also for various applications in physics, engineering, and computer science, where trigonometric functions are used to model periodic phenomena, waves, and oscillations. So, mastering these concepts is a great investment in your academic and professional journey.

Wrapping Up and Key Takeaways

Okay, guys, let's recap what we've learned. We started with sin(x) = 4/5 and cos(x) < 0 and were asked to find cos(2x). Here’s the roadmap we followed:

1. Identified the Quadrant: We used the given information to determine that x lies in the second quadrant.
2. Found cos(x): We used the Pythagorean identity (sin²(x) + cos²(x) = 1) to find cos(x), remembering to choose the negative value since we were in the second quadrant.
3. Applied the Double-Angle Formula: We used the double-angle formula for cosine (cos(2x) = cos²(x) - sin²(x)) to calculate cos(2x).

The key takeaway here is the power of combining different pieces of information and using trigonometric identities as tools. Don't just memorize the identities; understand how they're derived and when to apply them. Also, always pay attention to the quadrant, as it's a crucial detail for determining the correct sign.

Trigonometry might seem daunting at first, but with practice and a solid understanding of the fundamentals, you'll be solving these problems like a pro in no time. Keep up the great work, and remember, every problem solved is a step closer to mastering the math! You've got this!