Solving Trigonometric Equations: A Step-by-Step Guide

Hey guys! Let's dive into a cool math problem involving trigonometry. We're going to solve the equation 2 * sin(x) * cos(x) = 5 and figure out the value of sin(4x) + cos(x). I know, it might look a bit intimidating at first, but trust me, we can break it down step by step and make it super understandable. We'll go through the problem with a detailed explanation, making sure everyone gets the hang of it. Ready to get started?

Understanding the Basics of Trigonometry

Alright, before we jump into the problem, let's quickly recap some key trigonometric concepts. Trigonometry deals with the relationships between the angles and sides of triangles. Specifically, we'll be using some fundamental trigonometric identities. The core idea is that these identities help us rewrite and simplify complex trigonometric expressions. One of the most important things to remember is the relationship between sine (sin), cosine (cos), and tangent (tan) functions. These functions are based on the unit circle, where the sine represents the y-coordinate and the cosine represents the x-coordinate of a point on the circle. The equation 2 * sin(x) * cos(x) = 5 is the foundation of our problem, so let's start with this. This equation is related to the double-angle formula for sine, which is a key concept in trigonometry. The double-angle formulas are super helpful in simplifying expressions and solving equations like ours. The sine function is generally bound between -1 and 1, as is the cosine function. Their product can vary, but the equation given has an output of 5. The product of sin(x) and cos(x) has a range determined by the double-angle identity: sin(2x) = 2sin(x)cos(x). Knowing this will help us solve the problem better. Also, remember the Pythagorean identity: sin²(x) + cos²(x) = 1. This one is like the ultimate rule. We'll be using this, and several other identities to solve our problem. Another critical point is understanding the range of sine and cosine functions. Both sine and cosine functions always produce values between -1 and 1. This means the product of sin(x) and cos(x) can never be greater than 1, so the original equation 2 * sin(x) * cos(x) = 5 does not have a real solution.

Let’s try to visualize it a bit. Imagine a right triangle. The sine of an angle is the ratio of the opposite side to the hypotenuse, and the cosine is the ratio of the adjacent side to the hypotenuse. The hypotenuse is always the longest side, so the sine and cosine values are always between -1 and 1. Now, back to our equation. Since 2 * sin(x) * cos(x) = 5, if we divide both sides by 2, we get sin(x) * cos(x) = 2.5. However, since the maximum value of sin(x) * cos(x) is 0.5 (as we'll see later through the double angle identity, sin(2x) = 2sin(x)cos(x)), this means there is no real solution for x that satisfies the original equation. Nevertheless, let's explore this with the assumption that our original equation is solvable and that there is a possible real solution. Then, our goal is to find the value of sin(4x) + cos(x), given the equation 2 * sin(x) * cos(x) = 5. We need to remember some trigonometric identities. These identities are our tools for simplifying and manipulating expressions. They allow us to rewrite trigonometric functions in different forms, which is essential for solving equations and finding the solutions. You’ll be using these identities over and over again, so getting familiar with them is super important. We will also explore the double-angle formulas. These formulas are your best friends in solving problems involving trigonometric functions of multiple angles. They provide a direct relationship between the trigonometric functions of an angle and the functions of its double angle. For instance, the double-angle formula for sine is sin(2x) = 2sin(x)cos(x). Using these formulas, we can solve problems in more efficient ways.

Analyzing the Given Equation

Now, let's get down to the nitty-gritty of the problem. We are given the equation 2 * sin(x) * cos(x) = 5. Notice anything special about this? Yes, we can connect it to the double-angle formula for sine, which is sin(2x) = 2 * sin(x) * cos(x). This formula tells us that twice the product of sine and cosine is equal to the sine of twice the angle. So, our equation 2 * sin(x) * cos(x) = 5 can be rewritten as sin(2x) = 5. But wait a minute! The sine function can only have values between -1 and 1. It can never be 5. This means there's something wrong. The equation has no real solutions. This is where we need to remember the properties of the sine and cosine functions. The value of sin(2x) can't be greater than 1 or less than -1. Since the double-angle formula gives us sin(2x) = 5, we immediately know there's no real value of x that satisfies this equation. Therefore, the problem is flawed. However, assuming that our given equation is solvable, let's proceed to find the value of sin(4x) + cos(x). Given that sin(2x) = 5, we would want to find the value of sin(4x) + cos(x). Here is where the double-angle formulas strike again! The double-angle formula for sine can be used to write sin(4x) as 2 * sin(2x) * cos(2x). That’s because sin(4x) = sin(2 * 2x). Knowing this, we can move forward. Since we’ve established that the given equation is not valid for real numbers, it doesn’t have a real solution. If we assume that a real solution exists, we can move forward with our thought experiment, but it will be a bit unusual. Let's make an interesting point. In this specific case, 2 * sin(x) * cos(x) = 5 has no solution because it violates the fundamental property of sine and cosine functions. This is a super important point because it emphasizes the importance of knowing and understanding the domain and range of trigonometric functions. Now, if we were to proceed with this problem, we can use the following approach.

Step-by-Step Solution (Assuming a Valid Equation)

Okay, let's pretend for a moment that the equation 2 * sin(x) * cos(x) = 5 is actually valid and that there is a real value of x that satisfies it. We have already seen that this is not possible, but let's go with it for the sake of the exercise. So, our new goal is to calculate sin(4x) + cos(x). We know that 2 * sin(x) * cos(x) = 5, which means sin(2x) = 5. Now, we want to find sin(4x). Remember, the double-angle formula helps us again here. We can write sin(4x) as sin(2 * 2x), and therefore, sin(4x) = 2 * sin(2x) * cos(2x). We know that sin(2x) = 5, so let's plug this into the formula: sin(4x) = 2 * 5 * cos(2x), which simplifies to sin(4x) = 10 * cos(2x). Now, we need to find cos(2x). We can use another double-angle formula for cosine: cos(2x) = 1 - 2 * sin²(x) or cos(2x) = cos²(x) - sin²(x). From our initial equation, we can’t find a value for cos(2x), so we need to remember our identities once again. Let's explore the cosine double-angle formula that relates cos(2x) to sin(x) and cos(x). There are three versions of the double-angle formula for cosine: cos(2x) = cos²(x) - sin²(x), cos(2x) = 1 - 2sin²(x), and cos(2x) = 2cos²(x) - 1. We already know that sin(2x) = 5 is not possible, and it complicates the solving process. However, to solve the problem, we can use the formula cos(2x) = √(1 - sin²(2x)). We know sin(2x) = 5. Then, cos(2x) = √(1 - 25), and that’s an imaginary number. Since we can’t calculate it in real numbers, let's keep moving forward as if we can get a real number, and then we will be able to solve the equation. So, we'll continue with the solution of the problem, with the disclaimer that it is not valid in real numbers.

To continue, let's rewrite sin(4x) + cos(x). We know that sin(4x) = 10 * cos(2x), so we need to find a way to express cos(x). From the given equation, 2 * sin(x) * cos(x) = 5, divide both sides by 2, and we get sin(x) * cos(x) = 2.5. Now, let’s express cos(x) from the equation sin(4x) + cos(x). sin(4x) = 10 * cos(2x), so sin(4x) + cos(x) = 10 * cos(2x) + cos(x). Let’s express cos(2x) in terms of cos(x) or sin(x). Using the identity, cos(2x) = 1 - 2sin²(x), and then we're going to plug the cos(2x) value into the equation. So, we have sin(4x) + cos(x) = 10 * (1 - 2sin²(x)) + cos(x), or sin(4x) + cos(x) = 10 - 20sin²(x) + cos(x). Remember that sin(x) * cos(x) = 2.5, from where we can get cos(x) = 2.5 / sin(x), but it is going to make the math even harder. So, we’re going to let it stay as it is. We can replace sin(x) with cos(x) to make it easier, but the answer will be an estimate. So, we're going to assume that cos(x) is approximately equal to 0, since we know that sin(2x) = 5, which means that the original equation is not valid for real numbers. So, in our imaginary world, if we calculate it, the value would be sin(4x) + cos(x) ≈ 10.

Conclusion and Final Answer

In our problem, we found that 2 * sin(x) * cos(x) = 5 is not possible in real numbers. If we assume that it is possible, and with some calculations, we get the value of sin(4x) + cos(x) as 10. The correct answer is not found in any of the choices, because there is no real solution for x that satisfies the original equation. Thanks for sticking with me, guys! I hope this helped you understand the concepts better. Trigonometry can be fun if you break it down step-by-step. Keep practicing, and you'll become a pro in no time! Remember to always keep your trigonometric identities handy and don't be afraid to double-check your work. Keep practicing, and you'll be acing those math problems in no time. If you enjoyed this explanation, please leave a comment and let me know. Happy learning, and see you in the next one!