Solve The Trig Equation: Sin(x)sin(7x) = Sin(5x)sin(3x)

Hey everyone! Today, we're going to dive into the world of trigonometry and tackle a fun problem: solving the equation sin(x)sin(7x) = sin(5x)sin(3x). Don't worry if it looks a little intimidating at first; we'll break it down step by step to make it super clear. This is a classic example of a trig problem that can be simplified using some clever identities. This equation, while appearing complex, yields to a systematic approach using trigonometric identities and algebraic manipulations. The core of solving this type of equation rests on transforming products of sines into sums or differences, allowing for simplification and the eventual identification of solutions. So, grab your pencils, and let's get started! Our goal is to find all the values of x that satisfy this equation. We'll utilize trigonometric identities to simplify and solve for x. The equation involves the product of sine functions, and our primary strategy will be to transform these products into sums or differences, leveraging known trigonometric identities. The solution process demands a methodical application of these identities and algebraic skills to isolate and determine the values of x.

The Strategy: Transforming Products to Sums

To begin, we'll employ the product-to-sum trigonometric identities. These are incredibly useful for converting products of trigonometric functions into sums or differences. The specific identity we'll use is: 2sin(A)sin(B) = cos(A - B) - cos(A + B). This identity is the key to unlocking the problem. The core idea is to transform the products of sines into differences of cosines. This transformation will allow us to simplify the equation and eventually solve for x. This transformation is vital for simplifying the original equation, making it easier to manage and solve. The application of this identity is a standard technique in trigonometry for solving equations involving products of sines and cosines. Understanding and effectively using the product-to-sum identities is crucial to solving problems like this. Let's apply this identity to both sides of our equation. We'll rewrite the equation with the goal of using the identity to transform the products into sums. This is a crucial step that simplifies the equation considerably. This transformation sets the stage for further simplification, bringing us closer to a solution. Specifically, we'll apply the identity to both sin(x)sin(7x) and sin(5x)sin(3x) independently. This step is designed to eliminate the products of sines, transforming the equation into a form that's easier to work with. The careful application of this identity sets up the next steps in our solution. Remember, the goal here is simplification and the eventual isolation of x.

Let's apply the product-to-sum identity to sin(x)sin(7x) and sin(5x)sin(3x). We'll rewrite the equation step by step, which will help us transform the products of sines into differences of cosines. Remember, each step here is critical to simplifying the original equation, eventually leading us to the values of x. The methodical application of the identity and algebraic manipulation allows us to move closer to isolating and solving for x. Let's transform each product of sines into differences of cosines. This step is essential in preparing the equation for further simplification and eventual solution.

Applying the Product-to-Sum Identity

Alright, let's get down to the nitty-gritty. We'll start by rewriting our original equation: sin(x)sin(7x) = sin(5x)sin(3x). Now, to use our product-to-sum identity (2sin(A)sin(B) = cos(A - B) - cos(A + B)), we need to multiply both sides of the equation by 2. This gives us: 2sin(x)sin(7x) = 2sin(5x)sin(3x). Now, apply the identity. For the left side, with A = x and B = 7x, we get: cos(x - 7x) - cos(x + 7x). This simplifies to cos(-6x) - cos(8x). For the right side, with A = 5x and B = 3x, we get: cos(5x - 3x) - cos(5x + 3x). This simplifies to cos(2x) - cos(8x). Our equation now becomes: cos(-6x) - cos(8x) = cos(2x) - cos(8x). Remember, the transformation is all about simplifying the equation using established trigonometric identities. We're getting closer to a form where we can solve for x. This process will continue with each simplification, leading us to our solution.

By carefully applying the product-to-sum identity, we've transformed the products of sines into expressions involving cosines. Now, with the equation in a new form, we can simplify further by recognizing common terms and applying additional trigonometric identities. The application of the product-to-sum identity is just the first step in solving this equation. We'll proceed by using trigonometric identities to simplify and solve for x. We are getting close to determining all values of x which satisfy the original equation. Let's simplify the equation, taking into account the properties of cosine. Specifically, we know that cos(-θ) = cos(θ), which is very useful. Let's simplify the equation by recognizing that the cosine function is even, which means cos(-6x) = cos(6x). This transforms our equation into: cos(6x) - cos(8x) = cos(2x) - cos(8x). The equation is now easier to manipulate, and we can move to the next step towards solving for x.

Further Simplification and Solving for x

Great! We've made some progress, and now it's time to simplify further and actually start solving for x. Notice that we have −cos(8x) on both sides of the equation. We can cancel these out, which leaves us with: cos(6x) = cos(2x). This is a significant simplification! It means we now need to find all the values of x where the cosine of 6x is equal to the cosine of 2x. There are a few ways to approach this. We can use the general solution for cosine equations, which states that if cos(A) = cos(B), then A = 2nπ ± B, where n is an integer. Understanding and applying the general solution for trigonometric equations is key to finding the complete set of solutions. This is where we determine the values of x that satisfy the simplified equation. Remember that the solutions are based on the periodic nature of the cosine function. Let's proceed by understanding and applying the general solution for cosine equations. Then, we will find all possible values of x.

By understanding the periodic nature of the cosine function, we can determine all possible values of x that satisfy the equation. This involves applying the general solution for cosine equations, which is a powerful tool. The general solution ensures that we capture all possible values of x for which the equation holds true. Now, let's apply the general solution. Using cos(6x) = cos(2x), we have two cases based on the general solution formula:

1. 6x = 2nπ + 2x
2. 6x = 2nπ - 2x

Let's solve these two cases.

Solving Case 1: 6x = 2nπ + 2x

Alright, let's solve the first case: 6x = 2nπ + 2x. Subtract 2x from both sides to get: 4x = 2nπ. Divide both sides by 4 to isolate x: x = nπ/2, where n is an integer. This gives us our first set of solutions. This is an important step as we are finding all possible values of x that fit the equation. These are all the values that satisfy the original equation, given the specific case. This formula provides the first set of solutions for our equation.

Solving Case 2: 6x = 2nπ - 2x

Now, let's tackle the second case: 6x = 2nπ - 2x. Add 2x to both sides to get: 8x = 2nπ. Divide both sides by 8 to isolate x: x = nπ/4, where n is an integer. This gives us our second set of solutions. These solutions, combined with the first set, give us all values of x which satisfy our original equation. Remember, these solutions are based on the periodic nature of the cosine function. So, we've found all the possible solutions by applying the general solution to cosine equations and solving for x in each case. This complete set of solutions will solve the equation. The process we used ensures that we don't miss any values of x. The values of x we've found here solve the initial equation.

The Complete Solution

So, after all the calculations, we have two sets of solutions:

1. x = nπ/2, where n is an integer
2. x = nπ/4, where n is an integer

Combining these two sets, our complete solution for the equation sin(x)sin(7x) = sin(5x)sin(3x) is x = nπ/4, where n is an integer. This is the final solution, combining the results from both cases. Remember that n can be any integer. Therefore, this formula accounts for all values of x that satisfy the equation. This is the complete solution for the original equation. Each of the steps we did helped us get to the point of solving it. Congrats on solving the equation!

I hope this explanation was clear and helpful. If you have any questions, feel free to ask! Trigonometry can be fun, and with practice, you can solve these equations too. Keep practicing and exploring, and you'll become a trigonometry whiz in no time! Remember, the key is to understand the identities, practice their application, and break down complex problems into manageable steps. Keep practicing, and you will ace these trig problems!