Solving The Trigonometric Equation 8 Sin²x = 7 - 2 Cos X

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Trigonometric equations are a fundamental part of mathematics, appearing in various fields such as physics, engineering, and computer science. Solving these equations often requires a deep understanding of trigonometric identities and algebraic manipulation. This article delves into the intricacies of solving the trigonometric equation 8 sin²x = 7 - 2 cos x within the interval 0° ≤ x ≤ 360°. We will explore the necessary steps, underlying principles, and potential pitfalls, providing a comprehensive guide for students and enthusiasts alike.

Understanding the Equation: 8 sin²x = 7 - 2 cos x

To effectively solve the equation, we first need to understand its structure and the trigonometric functions involved. The equation 8 sin²x = 7 - 2 cos x combines both sine and cosine functions. To simplify the equation, our initial step involves expressing it in terms of a single trigonometric function. This is a common strategy in solving trigonometric equations, as it allows us to apply algebraic techniques more easily. Recognizing the relationship between sine and cosine, specifically the Pythagorean identity, is crucial for this step. The Pythagorean identity, sin²x + cos²x = 1, provides a direct link between the squares of sine and cosine, allowing us to rewrite the equation in terms of either sin x or cos x. In this case, we will opt to express the equation in terms of cos x, as it simplifies the subsequent algebraic steps. By substituting sin²x with (1 - cos²x), we can transform the equation into a quadratic equation in cos x, which can then be solved using standard algebraic methods. This transformation not only simplifies the equation but also sets the stage for finding the solutions within the specified interval. Understanding the domain 0° ≤ x ≤ 360° is crucial as it limits the range of possible solutions, ensuring we only consider angles within a full circle. This restriction is essential for accurately identifying the solutions that satisfy the original equation.

Step-by-Step Solution

1. Expressing the Equation in Terms of Cosine

Our initial goal is to rewrite the given equation, 8 sin²x = 7 - 2 cos x, using only the cosine function. To achieve this, we leverage the fundamental trigonometric identity: sin²x + cos²x = 1. This identity allows us to express sin²x as 1 - cos²x. Substituting this into our original equation, we get: 8(1 - cos²x) = 7 - 2 cos x. This substitution is a critical step as it transforms the equation into a form that we can solve algebraically. Expanding the left side of the equation gives us: 8 - 8 cos²x = 7 - 2 cos x. Now, we have an equation that involves only cosine terms, paving the way for further simplification and solution. This manipulation is a common technique in solving trigonometric equations, where the goal is to reduce the equation to a more manageable form by expressing it in terms of a single trigonometric function.

2. Forming a Quadratic Equation

Now that we have the equation 8 - 8 cos²x = 7 - 2 cos x, the next step is to rearrange it into a standard quadratic form. This involves moving all the terms to one side of the equation, setting it equal to zero. Adding 8 cos²x and 2 cos x to both sides, and subtracting 8 from both sides, we get: 0 = 8 cos²x - 2 cos x - 1. This is a quadratic equation in terms of cos x. To make it more familiar, we can substitute y = cos x, transforming the equation into: 8y² - 2y - 1 = 0. This substitution simplifies the algebraic manipulation and allows us to apply standard quadratic equation solving techniques. The quadratic form is essential because it allows us to use methods such as factoring, completing the square, or the quadratic formula to find the values of y, which will then lead us to the values of x. Recognizing and forming the quadratic equation is a key step in solving many trigonometric problems.

3. Solving the Quadratic Equation

Having transformed our trigonometric equation into a quadratic equation 8y² - 2y - 1 = 0, we now need to solve for y. We can use the quadratic formula, which states that for an equation of the form ay² + by + c = 0, the solutions for y are given by: y = (-b ± √(b² - 4ac)) / (2a). In our case, a = 8, b = -2, and c = -1. Plugging these values into the quadratic formula, we get: y = (2 ± √((-2)² - 4 * 8 * (-1))) / (2 * 8). Simplifying the expression under the square root: (-2)² - 4 * 8 * (-1) = 4 + 32 = 36. Thus, the equation becomes: y = (2 ± √36) / 16. Further simplifying the square root, we have: y = (2 ± 6) / 16. This gives us two possible values for y: y₁ = (2 + 6) / 16 = 8 / 16 = 1/2 and y₂ = (2 - 6) / 16 = -4 / 16 = -1/4. These values of y represent the possible values for cos x, which we will use in the next step to find the corresponding values of x.

4. Finding the Values of x

After solving the quadratic equation, we found two possible values for cos x: cos x = 1/2 and cos x = -1/4. Now, we need to find the angles x within the interval 0° ≤ x ≤ 360° that satisfy these conditions. For cos x = 1/2, we know that cosine is positive in the first and fourth quadrants. The reference angle for cos⁻¹(1/2) is 60°. Therefore, the solutions in the given interval are x = 60° (first quadrant) and x = 360° - 60° = 300° (fourth quadrant). For cos x = -1/4, cosine is negative in the second and third quadrants. The reference angle for cos⁻¹(1/4) is approximately 75.52°. In the second quadrant, the solution is x = 180° - 75.52° ≈ 104.48°. In the third quadrant, the solution is x = 180° + 75.52° ≈ 255.52°. Thus, we have four solutions for x within the specified interval: 60°, 300°, approximately 104.48°, and approximately 255.52°. These are the angles that satisfy the original equation 8 sin²x = 7 - 2 cos x.

Verifying the Solutions

To ensure the accuracy of our solutions, it is essential to verify them by substituting them back into the original equation, 8 sin²x = 7 - 2 cos x. This process helps to confirm that the values we obtained indeed satisfy the equation and are not extraneous solutions introduced during the algebraic manipulation. Let's verify each of our solutions: 60°, 300°, 104.48°, and 255.52°.

  1. For x = 60°: We have sin 60° = √3/2 and cos 60° = 1/2. Substituting these values into the equation, we get: 8(√3/2)² = 7 - 2(1/2), which simplifies to 8(3/4) = 7 - 1, or 6 = 6. This confirms that 60° is a valid solution.
  2. For x = 300°: We have sin 300° = -√3/2 and cos 300° = 1/2. Substituting these values into the equation, we get: 8(-√3/2)² = 7 - 2(1/2), which simplifies to 8(3/4) = 7 - 1, or 6 = 6. This confirms that 300° is also a valid solution.
  3. For x ≈ 104.48°: We have sin 104.48° ≈ 0.9682 and cos 104.48° ≈ -0.25. Substituting these values into the equation, we get: 8(0.9682)² ≈ 7 - 2(-0.25), which simplifies to 8(0.9374) ≈ 7 + 0.5, or 7.4992 ≈ 7.5. This is approximately equal, confirming that 104.48° is a valid solution.
  4. For x ≈ 255.52°: We have sin 255.52° ≈ -0.9682 and cos 255.52° ≈ -0.25. Substituting these values into the equation, we get: 8(-0.9682)² ≈ 7 - 2(-0.25), which simplifies to 8(0.9374) ≈ 7 + 0.5, or 7.4992 ≈ 7.5. This is also approximately equal, confirming that 255.52° is a valid solution.

By verifying each solution, we ensure that they satisfy the original equation and are indeed the correct answers within the given interval.

Common Pitfalls and How to Avoid Them

Solving trigonometric equations can be challenging, and there are several common pitfalls that students often encounter. Understanding these pitfalls and how to avoid them is crucial for achieving accurate solutions. One common mistake is forgetting the periodic nature of trigonometric functions. Sine and cosine functions repeat their values every 360°, so there can be multiple solutions within a given interval. When finding solutions, it is important to consider all possible angles within the specified range. Another pitfall is incorrectly applying trigonometric identities. It is essential to use the identities correctly and to ensure that the substitutions made are valid. A common error is also dividing both sides of an equation by a trigonometric function without considering the cases where the function might be zero. This can lead to the loss of solutions. To avoid this, it is better to factor the equation instead of dividing. Additionally, making algebraic errors while solving the equation is a common mistake. Careless mistakes in arithmetic or algebra can lead to incorrect solutions. It is important to double-check each step and to be meticulous in the calculations. Finally, not verifying the solutions is a significant pitfall. It is crucial to substitute the solutions back into the original equation to ensure they are valid. Extraneous solutions can arise due to the algebraic manipulations, and verifying the solutions helps to eliminate these. By being aware of these common pitfalls and taking steps to avoid them, students can improve their accuracy and confidence in solving trigonometric equations.

Conclusion

In conclusion, solving the trigonometric equation 8 sin²x = 7 - 2 cos x for 0° ≤ x ≤ 360° involves a series of steps that require a solid understanding of trigonometric identities, algebraic manipulation, and careful attention to detail. We began by expressing the equation in terms of a single trigonometric function, cosine, using the Pythagorean identity sin²x + cos²x = 1. This transformation allowed us to form a quadratic equation in cos x, which we then solved using the quadratic formula. The solutions for cos x led us to four possible values of x within the specified interval: 60°, 300°, approximately 104.48°, and approximately 255.52°. Each of these solutions was then verified by substituting them back into the original equation, ensuring their validity. Throughout this process, we highlighted common pitfalls, such as neglecting the periodic nature of trigonometric functions, misapplying trigonometric identities, and making algebraic errors. By understanding these pitfalls and implementing strategies to avoid them, students can enhance their problem-solving skills and achieve accurate results. Trigonometric equations are a fundamental aspect of mathematics, and mastering the techniques to solve them is essential for further studies in various fields such as physics, engineering, and computer science. This comprehensive guide provides a step-by-step approach, equipping students and enthusiasts with the knowledge and confidence to tackle similar problems effectively. The ability to solve trigonometric equations is not only a valuable mathematical skill but also a tool that can be applied in numerous real-world applications, making it a crucial topic for anyone pursuing a career in STEM fields.