Solving Cos X + Cos 2x + Cos 3x + Cos 4x = 0 Trigonometric Equation

Introduction

In this article, we will explore a detailed solution to a trigonometric equation problem. Specifically, we aim to find the number of real values of x that satisfy the equation cos x + cos 2x + cos 3x + cos 4x = 0 within the interval 0 ≤ x ≤ 2π. Trigonometric equations like this one often appear in various fields of mathematics and physics, making it crucial to understand the methods to solve them. This article provides a step-by-step approach, ensuring clarity and comprehension for readers of all levels. We will cover the necessary trigonometric identities, simplification techniques, and the process of finding solutions within the given interval. Let's delve into the problem and find out the real values of x that satisfy the given equation. Understanding the nuances of trigonometric equations is essential not only for academic pursuits but also for practical applications in engineering, signal processing, and many other scientific domains. Our approach will be methodical, ensuring that every step is clearly explained, and the underlying concepts are reinforced. By the end of this article, you will have a comprehensive understanding of how to tackle similar problems and a deeper appreciation for the elegance of trigonometric solutions. We will also discuss common pitfalls and strategies to avoid them, providing a robust foundation for future problem-solving endeavors. So, join us in unraveling this intriguing trigonometric puzzle and enhancing your mathematical toolkit.

Problem Statement

We are given the trigonometric equation:

cos x + cos 2x + cos 3x + cos 4x = 0

Our task is to determine the number of real values of x that satisfy this equation in the interval 0 ≤ x ≤ 2π. This problem involves the application of trigonometric identities and algebraic manipulation to simplify the equation and find its solutions within the specified range. The process begins with rearranging and grouping terms to facilitate the use of sum-to-product trigonometric identities. These identities are crucial for transforming sums of cosine functions into products, which are easier to solve. Once the equation is simplified, we can identify the values of x that make the equation true. However, we must also ensure that these values lie within the interval 0 ≤ x ≤ 2π, as this is a critical constraint of the problem. Understanding the periodic nature of trigonometric functions is essential here, as it helps in identifying all possible solutions within the given interval. The final step involves counting the number of distinct solutions that meet the criteria. This requires careful attention to detail and a thorough understanding of the unit circle and the behavior of cosine functions. By following a systematic approach, we can accurately determine the number of real values of x that satisfy the given equation, thereby reinforcing our problem-solving skills in trigonometry. This type of problem is not only a test of mathematical acumen but also an exercise in logical reasoning and precision.

Solution

Step 1: Rearranging and Grouping Terms

To begin, we rearrange the terms in the equation to group cosines of angles that have a common relationship. This rearrangement sets the stage for applying trigonometric identities effectively. Specifically, we pair cos x with cos 3x and cos 2x with cos 4x. This grouping allows us to use the sum-to-product identities, which simplify the equation significantly. The rearranged equation looks like this:

(cos 4x + cos x) + (cos 3x + cos 2x) = 0

This step is crucial because it transforms the equation from a sum of cosine functions into a form where we can apply trigonometric identities. The choice of grouping is not arbitrary; it is based on the observation that the sums of the angles within each pair are related, making the application of the sum-to-product identities more straightforward. The sum-to-product identities are fundamental tools in trigonometry, allowing us to convert sums and differences of trigonometric functions into products. This conversion is often a key step in solving trigonometric equations, as products are generally easier to handle than sums. By carefully rearranging and grouping the terms, we set ourselves up for the next step, where we will apply these identities to further simplify the equation. This strategic approach demonstrates a deep understanding of trigonometric principles and problem-solving techniques. The goal here is not just to manipulate the equation but to transform it into a more manageable form, paving the way for finding the solutions efficiently and accurately.

Step 2: Applying Sum-to-Product Identities

We will now apply the sum-to-product identity, which states:

cos A + cos B = 2 cos((A + B)/2) cos((A - B)/2)

Applying this identity to our grouped terms:

• For cos 4x + cos x: A = 4x, B = x

  cos 4x + cos x = 2 cos((4x + x)/2) cos((4x - x)/2) = 2 cos(5x/2) cos(3x/2)

• For cos 3x + cos 2x: A = 3x, B = 2x

  cos 3x + cos 2x = 2 cos((3x + 2x)/2) cos((3x - 2x)/2) = 2 cos(5x/2) cos(x/2)

Substituting these back into the equation, we get:

2 cos(5x/2) cos(3x/2) + 2 cos(5x/2) cos(x/2) = 0

This step is crucial as it transforms the sums of cosine functions into products, making the equation much easier to solve. The sum-to-product identities are powerful tools in trigonometry, and their correct application is essential for simplifying complex expressions. By converting sums into products, we create opportunities to factor out common terms and isolate the trigonometric functions. The application of these identities demonstrates a strong command of trigonometric principles and algebraic manipulation techniques. The resulting equation is now in a form that allows us to identify the values of x that make the equation true. This transformation is a testament to the elegance and efficiency of trigonometric identities in solving equations. The next step will involve further simplification by factoring out common terms, bringing us closer to the final solution. Each step in this process is carefully designed to break down the problem into manageable parts, making the solution accessible and understandable.

Step 3: Factoring and Simplifying

Now, we factor out the common term 2 cos(5x/2) from the equation:

2 cos(5x/2) [cos(3x/2) + cos(x/2)] = 0

Next, we apply the sum-to-product identity again to the terms inside the brackets:

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

Substituting this back into the equation, we get:

2 cos(5x/2) [2 cos(x) cos(x/2)] = 0

Which simplifies to:

4 cos(5x/2) cos(x) cos(x/2) = 0

This step is pivotal as it further simplifies the equation into a product of cosine functions, making it easier to identify the solutions. Factoring out common terms is a fundamental algebraic technique that is particularly useful in solving trigonometric equations. By factoring out 2 cos(5x/2), we break the equation down into manageable parts, each of which can be solved independently. The subsequent application of the sum-to-product identity transforms the sum of cosines inside the brackets into a product, further simplifying the equation. This process demonstrates the power of combining algebraic and trigonometric techniques to solve complex problems. The resulting equation, 4 cos(5x/2) cos(x) cos(x/2) = 0, is now in a form where we can easily identify the values of x that make the equation true. Each cosine term can be set to zero, leading to a set of solutions that satisfy the original equation. This step not only simplifies the equation but also provides a clear pathway to finding the solutions within the specified interval. The strategic use of factoring and trigonometric identities is a testament to the problem-solving skills required in advanced mathematics.

Step 4: Finding the Solutions

Now we set each factor to zero and solve for x:

1. cos(5x/2) = 0

   5x/2 = (2n + 1)π/2, where n is an integer.

   x = (2n + 1)π/5

   For 0 ≤ x ≤ 2π, n = 0, 1, 2, 3, 4 which gives us x = π/5, 3π/5, π, 7π/5, 9π/5 (5 solutions)

2. cos(x) = 0

   x = (2n + 1)π/2, where n is an integer.

   For 0 ≤ x ≤ 2π, n = 0, 1 which gives us x = π/2, 3π/2 (2 solutions)

3. cos(x/2) = 0

   x/2 = (2n + 1)π/2, where n is an integer.

   x = (2n + 1)π

   For 0 ≤ x ≤ 2π, n = 0 which gives us x = π (1 solution)

This step is crucial as it involves solving the simplified trigonometric equation by setting each factor to zero. Each factor corresponds to a cosine function, and finding the values of x that make these functions zero requires a solid understanding of the unit circle and the general solutions for cosine equations. For each factor, we find the general solution by considering the periodicity of the cosine function. The general solution involves an integer n, which allows us to find all possible solutions. However, we are only interested in solutions within the interval 0 ≤ x ≤ 2π, so we substitute different integer values for n to find the solutions that fall within this range. The process is methodical and requires careful attention to detail to ensure that all possible solutions are identified. For cos(5x/2) = 0, we find five solutions; for cos(x) = 0, we find two solutions; and for cos(x/2) = 0, we find one solution. This step is a direct application of trigonometric principles and algebraic techniques, demonstrating the problem-solving skills necessary to tackle such equations. The next step will involve counting the total number of distinct solutions, which will give us the final answer to the problem.

Step 5: Counting the Solutions

Combining all the solutions, we have:

x = π/5, 3π/5, π, 7π/5, 9π/5, π/2, 3π/2

Notice that π is a repeated solution, so we count it only once.

Therefore, the total number of distinct real values of x is 7.

This final step is crucial for determining the answer to the problem. After finding the solutions from each factor, we combine them into a single set. It is important to identify and eliminate any repeated solutions to ensure an accurate count. In this case, the solution x = π appears in the solutions for both cos(5x/2) = 0 and cos(x/2) = 0, so we only count it once. This careful consideration of repeated solutions is essential for avoiding errors in the final answer. The distinct solutions are π/5, 3π/5, π, 7π/5, 9π/5, π/2, and 3π/2. Counting these, we find that there are 7 distinct real values of x that satisfy the given equation in the interval 0 ≤ x ≤ 2π. This step not only provides the answer but also reinforces the importance of accuracy and attention to detail in mathematical problem-solving. The process of combining solutions and eliminating duplicates demonstrates a thorough understanding of the problem and a systematic approach to finding the solution. This final step concludes the problem-solving process, providing a clear and concise answer to the question posed.

Conclusion

The number of real values of x that satisfy the equation cos x + cos 2x + cos 3x + cos 4x = 0 in the interval 0 ≤ x ≤ 2π is 7. This problem demonstrates the application of trigonometric identities and algebraic manipulation to solve trigonometric equations. The key steps involved rearranging terms, applying sum-to-product identities, factoring out common terms, and solving for x. Understanding the periodic nature of trigonometric functions and the unit circle is crucial for finding all solutions within the given interval. The systematic approach used in this solution highlights the importance of breaking down complex problems into manageable steps. Each step, from rearranging terms to identifying and counting solutions, requires careful attention to detail and a solid understanding of trigonometric principles. This problem not only tests mathematical skills but also reinforces logical reasoning and problem-solving strategies. The ability to apply trigonometric identities effectively is a valuable skill in mathematics and has applications in various fields such as physics and engineering. The process of solving this equation also underscores the importance of verifying solutions and eliminating duplicates to ensure an accurate final answer. By mastering these techniques, one can confidently tackle a wide range of trigonometric problems, enhancing their mathematical proficiency and problem-solving capabilities. The solution presented here provides a comprehensive guide, making it accessible and understandable for learners of all levels, thereby reinforcing the beauty and power of trigonometric solutions.

Therefore, the correct answer is (c) 7.