Solving Trigonometric Equations: A Step-by-Step Guide

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Hey math enthusiasts! Today, we're diving into a fascinating trigonometric equation: 2 * (cos(5x) / sin(5x)) = sin(10x). Don't worry if it looks a little intimidating at first; we'll break it down step-by-step to make it crystal clear. This equation involves trigonometric functions like cosine, sine, and their relationships. Our goal is to find the values of x that satisfy this equation. Ready? Let's get started!

Understanding the Basics: Trigonometric Identities are Key

Before we start solving, let's refresh our memory on some fundamental trigonometric identities. These are the tools that will help us simplify the equation and find our solutions. Specifically, we'll be leaning on a couple of crucial identities:

  1. Cotangent Identity: cot(θ) = cos(θ) / sin(θ). This identity tells us that the cotangent of an angle is equal to the cosine of that angle divided by its sine. This will be super helpful because our equation contains cos(5x) / sin(5x), which is basically the cotangent of 5x. This will make our lives much easier.
  2. Double-Angle Identity for Sine: sin(2θ) = 2sin(θ)cos(θ). This identity shows us how to express the sine of twice an angle in terms of the sine and cosine of the original angle. We will use this to express sin(10x) using the sine and cosine of 5x. It will let us rewrite the right side of the equation.

Knowing these identities is like having the right tools for a construction project. Without them, it's really hard to make any progress. Now that we've got our tools ready, we're prepared to make some progress on solving the equation! We will take the trigonometric equation, and manipulate it. This will involve applying the trigonometric identities to make the equation solvable. We will need to be careful with the algebra, as well as the trigonometric identities.

Step-by-Step Solution: Unraveling the Equation

Alright, guys, let's get down to the nitty-gritty and solve this equation. We'll proceed systematically, making sure each step is clear.

  1. Rewrite Using Cotangent: We start with our original equation: 2 * (cos(5x) / sin(5x)) = sin(10x). Notice that cos(5x) / sin(5x) is the cotangent of 5x. So, let's rewrite the equation using the cotangent identity: 2 * cot(5x) = sin(10x). This simplifies things right off the bat, making the equation a little less cluttered.
  2. Express in Terms of Sine and Cosine: From the double-angle identity, we know that sin(10x) = 2sin(5x)cos(5x). Let's substitute that into our equation: 2 * cot(5x) = 2sin(5x)cos(5x). Now we have only the sine and cosine of 5x in the equation! Notice that we are getting closer to solving the equation.
  3. Use the cot(x) formula: We know that cot(x) = cos(x) / sin(x). Substituting this into the equation, we get: 2 * (cos(5x) / sin(5x)) = 2sin(5x)cos(5x). Now we can see what to do next to simplify our equation!
  4. Simplify and Rearrange: We can cancel the 2 from both sides of the equation. This yields cos(5x) / sin(5x) = sin(5x)cos(5x). Multiply both sides by sin(5x) to get rid of the fraction: cos(5x) = sin²(5x)cos(5x).
  5. Further Simplification and Factoring: Subtract sin²(5x)cos(5x) from both sides to get cos(5x) - sin²(5x)cos(5x) = 0. We can factor out cos(5x) from the left side, giving us cos(5x) * (1 - sin²(5x)) = 0. Recall that 1 - sin²(θ) = cos²(θ). Now we have cos(5x) * cos²(5x) = 0, which further simplifies to cos³(5x) = 0. We're almost there!
  6. Find the Solutions: For cos³(5x) = 0 to be true, we need cos(5x) = 0. The cosine function equals zero at odd multiples of π/2. So, we can write 5x = (2n + 1) * π/2, where n is an integer. Dividing both sides by 5, we get x = ((2n + 1) * π) / 10. These are our potential solutions! But we are not done yet, since we need to account for certain values, as described below.

Important Considerations: Domain Restrictions and Valid Solutions

When solving trigonometric equations, we must always consider the domain of the functions involved. This is very important. In our original equation, we had sin(5x) in the denominator. That means sin(5x) cannot be equal to zero, because division by zero is undefined. This is a very critical detail. So, before we can say we have the final solutions, we need to check this condition.

  1. Checking for Domain Restrictions: We need to make sure that our solutions do not make sin(5x) = 0. If sin(5x) = 0, then 5x = kπ, where k is an integer. Thus, x = kπ/5. So, any of our solutions where x = kπ/5 are not valid, and we will have to remove them.
  2. Excluding Invalid Solutions: Now, compare our general solution, x = ((2n + 1) * π) / 10, with x = kπ/5. We need to find which values of n give us the same solutions as x = kπ/5. We can rewrite x = kπ/5 as x = (2kπ) / 10. If we equate this with our solution, ((2n + 1) * π) / 10 = (2kπ) / 10, we get 2n + 1 = 2k. But the left side is always odd, and the right side is always even, so there is no value of n and k that can satisfy the equation.
  3. Final Solution: This means that none of our solutions will violate the domain restrictions. Therefore, our final solution is x = ((2n + 1) * π) / 10, where n is any integer. So, we've solved the trigonometric equation, and the result is quite elegant, don't you think?

Conclusion: Mastering Trigonometric Equations

Well, guys, we made it! We successfully solved the trigonometric equation 2 * (cos(5x) / sin(5x)) = sin(10x). We used our knowledge of trigonometric identities, simplified the equation step by step, and carefully considered the domain restrictions. This process highlights the importance of understanding the relationships between trigonometric functions and being meticulous in your calculations. By following these steps, you can tackle similar trigonometric equations with confidence. Remember to always double-check your work and consider the domain. With practice, you'll become a pro at solving these types of problems. Keep practicing and keep exploring the wonderful world of mathematics! You've got this!

I hope you found this guide helpful. If you have any questions or want to try another problem, feel free to ask. Happy solving! This is the end of the explanation, and hopefully it was an enlightening experience. Keep in mind that solving these types of equations takes practice. So keep at it and you will be a math whiz in no time.