Solving 2cos(x) + √2 = 0 Trigonometric Equation For 0 ≤ X ≤ 2π

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Introduction

In this article, we will delve into the process of solving the trigonometric equation 2cos(x) + √2 = 0 within the interval 0 ≤ x ≤ 2π. Trigonometric equations are fundamental in mathematics and have wide applications in various fields such as physics, engineering, and computer science. Mastering the techniques to solve these equations is crucial for anyone pursuing studies or careers in these areas. This article aims to provide a comprehensive, step-by-step guide to solving the given equation, ensuring clarity and understanding for readers of all levels. We will begin by isolating the cosine function, then determine the reference angles and the quadrants where the solutions lie. Finally, we will express the solutions in radians, adhering to the specified interval. Understanding the unit circle and the properties of trigonometric functions is essential for this process, and we will reinforce these concepts throughout the explanation. Let’s embark on this mathematical journey to find the solutions to our equation.

Isolating the Cosine Function

To solve the trigonometric equation 2cos(x) + √2 = 0, the initial step involves isolating the cosine function. This is a crucial step as it simplifies the equation and allows us to focus solely on the trigonometric part. To achieve this, we need to manipulate the equation by performing algebraic operations. First, we subtract √2 from both sides of the equation. This gives us 2cos(x) = -√2. The goal here is to get the cosine term by itself on one side of the equation. Next, we divide both sides of the equation by 2. This isolates the cosine function, resulting in cos(x) = -√2 / 2. Now that we have isolated the cosine function, we can clearly see the value that cos(x) must equal. This simplified form allows us to determine the reference angles and the quadrants where the solutions for x will be located. Understanding this isolation process is fundamental to solving any trigonometric equation, as it sets the stage for the subsequent steps. By isolating the trigonometric function, we make it easier to identify the angles that satisfy the equation within the given interval. This step is not just a mechanical manipulation; it's a logical progression that brings us closer to the solution. In the following sections, we will use this isolated form to find the values of x that satisfy the original equation.

Determining Reference Angles and Quadrants

After isolating the cosine function and obtaining cos(x) = -√2 / 2, the next critical step is to determine the reference angle and the quadrants where the solutions lie. The reference angle is the acute angle formed between the terminal side of the angle x and the x-axis. To find the reference angle, we consider the absolute value of the cosine function, which is |cos(x)| = √2 / 2. We need to identify the angle whose cosine value is √2 / 2. From our knowledge of special triangles and the unit circle, we know that the angle whose cosine is √2 / 2 is π/4 radians or 45 degrees. This is our reference angle. Now, we need to determine the quadrants where cos(x) is negative. Recall that cosine corresponds to the x-coordinate on the unit circle. Cosine is negative in the second and third quadrants. Therefore, the solutions for x will be located in these two quadrants. Understanding the relationship between the sign of the trigonometric functions and the quadrants is essential for accurately finding the solutions. In the second quadrant, the angle x can be found by subtracting the reference angle from π (180 degrees), and in the third quadrant, the angle x can be found by adding the reference angle to π (180 degrees). This knowledge allows us to pinpoint the specific angles within the interval 0 ≤ x ≤ 2π that satisfy the given equation. In the subsequent section, we will calculate these angles and present the solutions in radians.

Finding Solutions in Radians

With the reference angle determined as π/4 and knowing that the solutions lie in the second and third quadrants, we can now find the specific values of x within the interval 0 ≤ x ≤ 2π that satisfy the equation cos(x) = -√2 / 2. In the second quadrant, the angle x is given by subtracting the reference angle from π. Therefore, x = π - π/4. To perform this subtraction, we need a common denominator, so we rewrite π as 4π/4. Thus, x = 4π/4 - π/4 = 3π/4. This is one solution within the specified interval. In the third quadrant, the angle x is given by adding the reference angle to π. Therefore, x = π + π/4. Again, we rewrite π as 4π/4, so x = 4π/4 + π/4 = 5π/4. This is our second solution within the given interval. We have now found two angles, 3π/4 and 5π/4, that satisfy the equation cos(x) = -√2 / 2 in the interval 0 ≤ x ≤ 2π. To ensure we haven't missed any solutions, we can visualize these angles on the unit circle. The angles 3π/4 and 5π/4 correspond to points in the second and third quadrants where the x-coordinate is -√2 / 2, confirming our solutions. These are the only two solutions within the interval 0 ≤ x ≤ 2π because cosine has a period of 2π, and we have covered one full period. In the next section, we will summarize our findings and present the final solutions to the equation.

Final Solutions

After a detailed step-by-step process, we have successfully solved the trigonometric equation 2cos(x) + √2 = 0 for 0 ≤ x ≤ 2π. By isolating the cosine function, determining the reference angle, and identifying the relevant quadrants, we found two solutions within the specified interval. The solutions are:

  • x = 3π/4
  • x = 5π/4

These angles, 3π/4 and 5π/4, are the only values of x within the interval 0 ≤ x ≤ 2π that satisfy the original equation. To recap, we first isolated cos(x) to get cos(x) = -√2 / 2. Then, we identified the reference angle as π/4 and determined that the solutions lie in the second and third quadrants because cosine is negative in these quadrants. Finally, we calculated the angles in these quadrants by subtracting and adding the reference angle to π, respectively. Our solutions, 3π/4 and 5π/4, can be verified by substituting them back into the original equation and confirming that they make the equation true. This methodical approach to solving trigonometric equations ensures accuracy and helps build a strong foundation in trigonometry. By understanding the unit circle, reference angles, and quadrant rules, one can confidently tackle a wide range of trigonometric problems. This concludes our solution to the equation 2cos(x) + √2 = 0 for 0 ≤ x ≤ 2π.

Check all that apply

  • 3π/4
  • 5π/4