Solving Cos(2x)/(cos(x) + Sin(x)) = 0 For 0° ≤ X ≤ 180°

Trigonometric equations are fundamental in mathematics, with applications spanning various fields such as physics, engineering, and computer science. These equations involve trigonometric functions like sine, cosine, and tangent, and solving them requires a deep understanding of trigonometric identities and principles. In this comprehensive guide, we will delve into the process of solving a specific trigonometric equation: cos(2x) / (cos(x) + sin(x)) = 0, where 0° ≤ x ≤ 180°. We will break down the problem step by step, providing clear explanations and insightful strategies to help you master the art of solving trigonometric equations.

Understanding Trigonometric Identities

Before we dive into solving the equation, it is crucial to have a solid grasp of trigonometric identities. These identities are equations that hold true for all values of the variables involved, and they are essential tools for simplifying and manipulating trigonometric expressions. In this case, we will primarily utilize the double-angle formula for cosine: cos(2x) = cos²(x) - sin²(x). This identity allows us to express cos(2x) in terms of cos(x) and sin(x), which will be crucial for simplifying the equation and finding solutions. Moreover, familiarity with the Pythagorean identity, sin²(x) + cos²(x) = 1, and other related identities can further aid in simplifying expressions and solving trigonometric equations effectively.

Setting up the Equation

The given equation is cos(2x) / (cos(x) + sin(x)) = 0, with the constraint 0° ≤ x ≤ 180°. The first step in solving this equation is to recognize that a fraction can only be equal to zero if its numerator is equal to zero, and its denominator is not equal to zero. This is a fundamental principle in algebra and applies directly to our trigonometric equation. Therefore, we can rewrite the equation as cos(2x) = 0, provided that cos(x) + sin(x) ≠ 0. This condition ensures that we are not dividing by zero, which would make the equation undefined. By focusing on the numerator, we simplify the equation and narrow down our search for solutions.

Applying the Double-Angle Formula

To further simplify the equation, we apply the double-angle formula for cosine: cos(2x) = cos²(x) - sin²(x). Substituting this into our equation, we get cos²(x) - sin²(x) = 0. This step is crucial because it expresses the equation in terms of cos(x) and sin(x), which are the basic trigonometric functions. Now, we can use algebraic techniques to factor or simplify the expression. Recognizing the difference of squares pattern, we can factor the left-hand side of the equation as (cos(x) - sin(x))(cos(x) + sin(x)) = 0. This factorization is a key step in isolating the possible solutions for x.

Factoring and Solving

Having factored the equation as (cos(x) - sin(x))(cos(x) + sin(x)) = 0, we can apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This leads us to two separate equations: cos(x) - sin(x) = 0 and cos(x) + sin(x) = 0. Solving each of these equations will give us potential solutions for x. For the first equation, cos(x) - sin(x) = 0, we can rearrange it to get cos(x) = sin(x). Similarly, for the second equation, cos(x) + sin(x) = 0, we can rearrange it to get cos(x) = -sin(x). These two equations represent different relationships between cos(x) and sin(x) and will yield different sets of solutions.

Analyzing the Solutions

Now, let's analyze the solutions for each equation within the given range of 0° ≤ x ≤ 180°. For the equation cos(x) = sin(x), we know that this equality holds true when x = 45°. This is because at 45°, both cosine and sine have the same value, which is √2/2. For the equation cos(x) = -sin(x), we need to find angles where the cosine and sine have opposite signs. This occurs in the second quadrant, where cosine is negative and sine is positive. The angle that satisfies this condition within the given range is x = 135°. At 135°, cos(x) = -√2/2 and sin(x) = √2/2, confirming that cos(x) = -sin(x). Therefore, we have identified two potential solutions: x = 45° and x = 135°.

Checking for Extraneous Solutions

It is crucial to check for extraneous solutions, which are solutions that satisfy the simplified equations but not the original equation. Recall that we had the condition cos(x) + sin(x) ≠ 0. We need to ensure that our solutions do not violate this condition. For x = 45°, cos(45°) + sin(45°) = √2/2 + √2/2 = √2, which is not equal to zero. Therefore, x = 45° is a valid solution. For x = 135°, cos(135°) + sin(135°) = -√2/2 + √2/2 = 0. This violates our condition, so x = 135° is an extraneous solution and must be discarded. By checking for extraneous solutions, we ensure the accuracy and validity of our final answer.

Final Solution

After analyzing the solutions and checking for extraneous roots, we conclude that the only possible value for x that satisfies the equation cos(2x) / (cos(x) + sin(x)) = 0 within the range 0° ≤ x ≤ 180° is x = 45°. This comprehensive guide has walked you through the process of solving the equation step by step, from understanding trigonometric identities to checking for extraneous solutions. By mastering these techniques, you will be well-equipped to tackle a wide range of trigonometric equations and excel in mathematics.

In summary, solving trigonometric equations requires a combination of algebraic manipulation, trigonometric identities, and careful analysis. By understanding the principles and techniques outlined in this guide, you can confidently approach and solve a variety of trigonometric equations. Remember to always check for extraneous solutions and ensure that your answers are valid within the given constraints. With practice and perseverance, you can develop a strong foundation in trigonometry and its applications.

The possible value for x is 45° only.