Solve Sin(x) - Sin(3x) = Cos(2x): A Step-by-Step Guide

Hey everyone! Let's dive into a cool trigonometric problem today. We're going to explore the equation sin(x) - sin(3x) = cos(2x) and, more importantly, how to find all its solutions. Trigonometry can seem daunting, but breaking it down step by step makes it much more manageable. So, grab your thinking caps, and let's get started!

The Initial Spark: Using Trigonometric Identities

Our journey begins with the given equation: sin(x) - sin(3x) = cos(2x). Now, the key to cracking this is leveraging our trigonometric identities. Remember those handy formulas that help us rewrite trigonometric expressions in different forms? That's precisely what we need here. The user started off brilliantly by recognizing the sin(P) - sin(Q) pattern on the left-hand side.

Specifically, they employed the sum-to-product identity:

sin(P) - sin(Q) = -2cos((P+Q)/2)sin((P-Q)/2)

With P = x and Q = 3x, this transforms our left side into:

-2cos((x + 3x)/2)sin((x - 3x)/2) = -2cos(2x)sin(-x)

Now, recalling that sine is an odd function, meaning sin(-x) = -sin(x), we can simplify this further:

-2cos(2x)sin(-x) = 2cos(2x)sin(x)

So, our equation now looks like this:

2cos(2x)sin(x) = cos(2x)

This is progress! We've managed to consolidate some terms and introduce a common factor, which is a crucial step toward solving the equation. But remember, we're not just looking for a solution; we want all solutions. This is where a careful approach becomes vital. We will be factoring later, so this is something that we should keep in mind as it affects how we will solve the rest of the problem. The ability to manipulate trigonometric expressions using identities is a fundamental skill in trigonometry. It allows us to simplify complex equations, reveal hidden relationships, and ultimately, find solutions that might not be immediately obvious. Mastering these identities is like adding tools to your mathematical toolbox, each ready to be used to tackle different challenges. So, if you're not completely comfortable with trigonometric identities, I highly recommend spending some time reviewing them. Understanding how they work and when to apply them will make solving trigonometric equations much easier and more enjoyable. We will be using this concept later in the article, so make sure that you understand it well. Overall, by applying the sum-to-product identity and the property of sine being an odd function, we've successfully transformed the original equation into a more manageable form. This sets the stage for the next steps in finding all the solutions.

The Critical Step: Factoring and Avoiding Pitfalls

Okay, guys, we've reached a crucial juncture in solving our trigonometric puzzle. We've transformed the original equation into:

2cos(2x)sin(x) = cos(2x)

Now, the tempting thing to do here might be to divide both sides by cos(2x). But hold on! This is a major pitfall in equation solving. Why? Because if cos(2x) = 0, we're dividing by zero, which is a big no-no in the math world. More importantly, we'd be potentially throwing away solutions! Solutions where cos(2x) = 0 are perfectly valid solutions to the original equation, and we don't want to miss them. This is an important concept for factoring in math. We need to be careful in how we do it.

So, what's the correct approach? Factoring is our friend here. We want to rearrange the equation so that we have zero on one side:

2cos(2x)sin(x) - cos(2x) = 0

Now, we can factor out the common factor of cos(2x):

cos(2x)[2sin(x) - 1] = 0

This is a beautiful thing! We've transformed our equation into a product of two factors that equals zero. This means that either the first factor, cos(2x), must be zero, or the second factor, [2sin(x) - 1], must be zero, or both! This gives us two separate equations to solve, and each will lead us to a set of solutions.

This step highlights a fundamental principle in solving equations: always factor instead of dividing when you have a common factor. Dividing can lead to the loss of solutions, while factoring ensures we capture all possibilities. It's a small change in technique but makes a world of difference in the accuracy of our results. We need to make sure that we do not make this mistake. This is why we need to factor first before we move onto the next part. It is a critical step that most people would miss. We want to make sure that we are not one of them.

Factoring is not just a trick; it's a way of systematically exploring all possible scenarios. By setting each factor to zero, we create a set of simpler equations that are much easier to solve individually. This approach is applicable not only to trigonometric equations but also to algebraic equations and many other areas of mathematics. It's a powerful tool to have in your arsenal. Overall, by factoring the equation instead of dividing, we've successfully avoided a common pitfall and ensured that we'll find all possible solutions. Now, we're ready to tackle each factor separately and uncover the values of x that satisfy the original equation.

Digging Deeper: Solving cos(2x) = 0

Alright, let's tackle the first part of our factored equation: cos(2x) = 0. This equation tells us that we're looking for angles where the cosine function equals zero. Think about the unit circle – where does the x-coordinate (which represents cosine) become zero? It happens at π/2 and 3π/2. However, we need to remember that the argument of our cosine function is 2x, not just x. So, we're looking for values of 2x that make the cosine zero.

This means:

2x = π/2 + 2πk or 2x = 3π/2 + 2πk

where k is any integer. The + 2πk is crucial because it accounts for the periodic nature of the cosine function. Cosine repeats itself every 2π radians, so we need to include all possible rotations around the unit circle.

Now, to solve for x, we simply divide both sides of each equation by 2:

x = π/4 + πk or x = 3π/4 + πk

These two sets of solutions can actually be combined into a single, more elegant form. Notice that 3π/4 is simply π/2 more than π/4. Therefore, we can express all solutions for cos(2x) = 0 as:

x = π/4 + πk/2

where k is any integer. This single expression captures all the solutions we found earlier, making it a more concise and useful representation. This highlights the importance of looking for patterns and simplifying solutions whenever possible. It not only makes the answer more elegant but also deepens our understanding of the underlying relationships.

Solving cos(2x) = 0 is a great example of how understanding the unit circle and the periodicity of trigonometric functions is essential for finding all solutions to trigonometric equations. The + 2πk (and its variations) is not just a formality; it's a fundamental part of the solution, ensuring we don't miss any possible angles. This also shows the importance of solving the problem. Not just finding a solution but to find all the possible solutions for the problem.

Overall, we've successfully solved the first part of our factored equation, cos(2x) = 0, and found a general expression for all its solutions. Now, we're ready to move on to the second factor and complete our quest for all solutions to the original equation. The use of k is also important as that indicates all the possible solutions. It is not just a single solution but we are solving for a general solution.

Cracking the Code: Solving 2sin(x) - 1 = 0

Let's move on to the second equation we obtained from factoring: 2sin(x) - 1 = 0. This one looks a bit simpler, right? Our goal here is to isolate sin(x) and then find the angles x that satisfy the equation. The first step is to add 1 to both sides:

2sin(x) = 1

Then, divide both sides by 2:

sin(x) = 1/2

Now, we're looking for angles where the sine function equals 1/2. Again, let's visualize the unit circle. The y-coordinate (which represents sine) is 1/2 at two angles in the interval [0, 2π): π/6 and 5π/6.

But remember, the sine function is also periodic, repeating itself every 2π radians. So, just like with the cosine equation, we need to add multiples of 2π to our solutions to account for all possible rotations around the unit circle. This leads us to the general solutions:

x = π/6 + 2πk or x = 5π/6 + 2πk

where k is any integer. These two sets of solutions represent all the angles where the sine function equals 1/2. It's important to keep these separate, as there isn't a simple way to combine them into a single expression like we did with the cosine solutions. The solution is different so we cannot combine them like what we did with cosine. We will need to have the solutions stated separately.

Solving 2sin(x) - 1 = 0 reinforces the importance of understanding the unit circle and the periodicity of trigonometric functions. By visualizing the sine function and its values, we can easily identify the principal solutions and then generate all possible solutions by adding multiples of 2π. This systematic approach ensures that we don't miss any solutions and provides a complete answer to the equation. When we are doing trigonometric equations, we have to consider the period of the equation. We must not forget this fact when solving trigonometric equations.

Overall, we've successfully solved the second part of our factored equation, 2sin(x) - 1 = 0, and found two sets of general solutions. Now, we have all the pieces of the puzzle! We have to take each solution that we obtained to fully solve this problem. We have approached this methodically, step by step, so we should not miss any details or part of the answer.

The Grand Finale: Combining All Solutions

Okay, everyone, we've reached the finish line! We've solved both parts of our factored equation and now have all the solutions. Let's recap what we found:

From cos(2x) = 0, we obtained the solutions:

x = π/4 + πk/2

where k is any integer.

From 2sin(x) - 1 = 0, we obtained the solutions:

x = π/6 + 2πk and x = 5π/6 + 2πk

where k is any integer.

To get the complete solution set for the original equation, sin(x) - sin(3x) = cos(2x), we simply combine these solutions. There's no further simplification possible here, as these are distinct sets of solutions. So, our final answer is:

x = π/4 + πk/2, π/6 + 2πk, 5π/6 + 2πk

where k is any integer. This is the general solution to the trigonometric equations. We have to make sure that we indicate that k is any integer. This is important so that we know that we are solving for the general solution and not just a specific solution.

This comprehensive solution set captures all possible values of x that satisfy the original equation. We've successfully navigated the complexities of trigonometric identities, factoring, and periodicity to arrive at a complete and accurate answer. Solving equations like this can be challenging, but by breaking them down into smaller, manageable steps, we can conquer even the most daunting problems. This is a great example of how a combination of different types of trigonometric skills are important to solve a complex equation. We need to make sure that we are careful in each step that we do and not miss any steps. We also have to be systematic in how we are solving it.

Congratulations, you've made it through the entire solution process! You've not only found the solutions to the equation but also gained a deeper understanding of the techniques and principles involved in solving trigonometric equations. Remember, practice makes perfect, so keep exploring and challenging yourself with new problems. Keep going at it and you will eventually get it! It is important to learn from mistakes and keep on practicing. With enough practice, solving these type of problem is very doable.

Key Takeaways for Trigonometric Success

Before we wrap up, let's highlight some key takeaways from our journey:

1. Master Trigonometric Identities: They are the foundation for simplifying and manipulating trigonometric expressions.
2. Factor, Don't Divide: Factoring avoids losing solutions when dealing with common factors.
3. Visualize the Unit Circle: It's your best friend for understanding sine, cosine, and their values at different angles.
4. Embrace Periodicity: Remember to add multiples of 2π (or the appropriate period) to account for all possible solutions.
5. Break It Down: Complex equations become manageable when you break them into smaller steps.

By keeping these principles in mind, you'll be well-equipped to tackle a wide range of trigonometric problems. So, keep practicing, keep exploring, and keep unlocking those trigonometric mysteries!

Trigonometry, while sometimes tricky, is a beautiful and powerful branch of mathematics. It has applications in many fields, from physics and engineering to computer graphics and music. By mastering the fundamentals and developing a problem-solving mindset, you can unlock its potential and apply it to solve real-world problems. Keep the faith in yourself and continue to work hard to achieve it. I hope that this article has helped you solve a tough problem. Hopefully, we will be able to explore more trigonometric problems together!