Solving 2sin^2(x)-5sin(x)-3=0 A Step-by-Step Guide

Trigonometric equations can often seem daunting, but with the right approach, they become manageable and even enjoyable to solve. In this comprehensive guide, we will tackle the equation 2sin²(x) - 5sin(x) - 3 = 0. We will break down the problem step by step, making it easy to understand and implement the solution. Whether you are a student grappling with trigonometry or someone looking to brush up on your skills, this article is designed to help you master the art of solving trigonometric equations. Our main keywords are trigonometric equations, solving equations, sine function, quadratic form, and solutions for x, which we will weave into our discussion to keep the focus sharp and clear.

Understanding the Problem

Before diving into the solution, it is crucial to understand the nature of the equation. The given equation, 2sin²(x) - 5sin(x) - 3 = 0, is a trigonometric equation involving the sine function. Recognizing this is the first step towards finding the solution. To make the equation easier to handle, we can use a substitution. Let's set u = sin(x). This substitution transforms the trigonometric equation into a quadratic equation, which is a more familiar form to work with. This technique of substitution is a cornerstone in solving equations, allowing us to simplify complex expressions into manageable forms. By recognizing the underlying structure, we pave the way for a straightforward solution process. This approach of breaking down complex problems into simpler steps is a valuable skill in mathematics and beyond, enhancing our ability to tackle challenges with confidence and clarity.

Transforming the Equation

By substituting u = sin(x), the original equation 2sin²(x) - 5sin(x) - 3 = 0 transforms into a quadratic equation in terms of u. This transformation is a pivotal step in simplifying the problem. The new equation is 2u² - 5u - 3 = 0. Now, we have a standard quadratic equation that we can solve using various methods, such as factoring, completing the square, or using the quadratic formula. Factoring is often the quickest and most efficient method if the quadratic equation is factorable. In this case, the equation is indeed factorable, making it an ideal approach. This method highlights the importance of recognizing patterns and structures in mathematical problems, allowing for the application of appropriate techniques. Understanding these transformations is essential for solving equations effectively and efficiently. By converting the trigonometric equation into a quadratic form, we have essentially translated the problem into a domain where we have well-established methods for finding solutions. This process of transformation not only simplifies the problem but also deepens our understanding of the underlying mathematical principles at play.

Factoring the Quadratic Equation

Now that we have the quadratic equation 2u² - 5u - 3 = 0, we need to factor it. Factoring involves expressing the quadratic expression as a product of two binomials. In this case, we are looking for two binomials that, when multiplied, give us the original quadratic expression. To factor 2u² - 5u - 3, we look for two numbers that multiply to (2)(-3) = -6 and add up to -5. These numbers are -6 and 1. We can then rewrite the middle term, -5u, as -6u + u, and proceed with factoring by grouping. This method is a standard technique in algebra and is crucial for solving equations of this type. The ability to factor quadratic equations efficiently is a valuable skill in mathematics, enabling us to find solutions quickly and accurately. By breaking down the quadratic expression into its factors, we reveal the roots of the equation, which are the values of u that make the equation true. This process not only provides the solutions but also enhances our understanding of the relationship between factors and roots in algebraic expressions. Factoring is a fundamental tool in the mathematician's toolkit, essential for solving a wide range of problems.

Solving for u

After factoring the quadratic equation 2u² - 5u - 3 = 0, we obtain the factored form (2u + 1)(u - 3) = 0. This factored form is a significant step towards finding the solutions for u. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This principle allows us to set each factor equal to zero and solve for u individually. Setting 2u + 1 = 0 gives us u = -1/2, and setting u - 3 = 0 gives us u = 3. These values are potential solutions for u, but we need to check if they are valid in the context of our original problem. This step highlights the importance of verifying solutions, especially when dealing with substitutions or transformations. Understanding the zero-product property is fundamental in solving equations, as it provides a direct link between factored forms and solutions. By systematically applying this property, we can efficiently find the values that satisfy the equation. The solutions for u are crucial stepping stones towards finding the solutions for x in the original trigonometric equation.

Finding the values of sin(x)

Recall that we made the substitution u = sin(x). Now that we have the values of u, we can find the values of sin(x). We found that u = -1/2 and u = 3. Therefore, we have two equations to consider: sin(x) = -1/2 and sin(x) = 3. However, we know that the sine function has a range of -1 ≤ sin(x) ≤ 1. This means that the equation sin(x) = 3 has no solution, as 3 is outside the range of the sine function. The equation sin(x) = -1/2, on the other hand, does have solutions within the specified range. Understanding the properties and range of trigonometric functions is crucial in solving trigonometric equations. By recognizing the limitations of the sine function, we can eliminate extraneous solutions and focus on the valid ones. This step underscores the importance of a thorough understanding of trigonometric concepts when tackling such problems. The range restriction of trigonometric functions is a fundamental aspect that must be considered to ensure the accuracy of the solutions. By applying this knowledge, we avoid the pitfall of including solutions that do not exist.

Solving for x

Now that we have sin(x) = -1/2, we need to find the values of x that satisfy this equation. To do this, we consider the unit circle and the angles where the sine function is equal to -1/2. The sine function is negative in the third and fourth quadrants. The reference angle for sin(x) = 1/2 is π/6. Therefore, the angles in the third and fourth quadrants where sin(x) = -1/2 are 7π/6 and 11π/6. However, we need to consider all possible solutions, which means adding integer multiples of 2π to these angles. The general solutions for x are x = 7π/6 + 2πk and x = 11π/6 + 2πk, where k is an integer. These general solutions represent all possible angles that satisfy the equation sin(x) = -1/2. Finding the general solutions is a crucial step in solving trigonometric equations, as it ensures that we capture all possible answers. The use of the unit circle and reference angles is a standard technique in trigonometry, providing a visual and intuitive way to find solutions. By understanding the periodic nature of trigonometric functions, we can express the solutions in a general form that accounts for all possible angles.

General Solutions and Periodicity

As we discussed, the general solutions for x are x = 7π/6 + 2πk and x = 11π/6 + 2πk, where k is an integer. These solutions reflect the periodic nature of the sine function. The sine function repeats its values every 2π radians, which means that there are infinitely many solutions to the equation sin(x) = -1/2. The general solutions capture all these possibilities by including the term 2πk, where k can be any integer. This periodicity is a fundamental characteristic of trigonometric functions and must be considered when solving trigonometric equations. Understanding periodicity allows us to express the complete set of solutions in a concise and accurate manner. The parameter k in the general solutions is crucial for capturing all possible angles that satisfy the equation. By varying the value of k, we can generate an infinite number of solutions that lie on the unit circle. This concept of periodicity is essential for a comprehensive understanding of trigonometric functions and their behavior.

Verifying the Solutions

To ensure the accuracy of our solutions, it is always a good practice to verify them. We can plug the values of x back into the original equation 2sin²(x) - 5sin(x) - 3 = 0 to check if they satisfy the equation. For example, let's consider x = 7π/6. We have sin(7π/6) = -1/2. Plugging this into the equation, we get 2(-1/2)² - 5(-1/2) - 3 = 2(1/4) + 5/2 - 3 = 1/2 + 5/2 - 3 = 6/2 - 3 = 3 - 3 = 0. Thus, x = 7π/6 is indeed a solution. Similarly, we can verify that x = 11π/6 is also a solution. This verification step is crucial in solving equations, as it helps us catch any potential errors in our calculations or reasoning. By plugging the solutions back into the original equation, we can confirm their validity and gain confidence in our results. This practice reinforces the importance of accuracy and attention to detail in mathematical problem-solving.

Conclusion

In this article, we have provided a detailed, step-by-step guide to solving the trigonometric equation 2sin²(x) - 5sin(x) - 3 = 0. We started by recognizing the quadratic form of the equation and using a substitution to simplify it. We then factored the quadratic equation, found the values of sin(x), and determined the general solutions for x. Throughout the process, we emphasized the importance of understanding the properties of the sine function, the range restriction, and the periodicity of trigonometric functions. By following these steps, you can confidently tackle similar trigonometric equations. We hope this guide has been helpful and has enhanced your understanding of solving trigonometric equations. The ability to solve these types of problems is a valuable skill in mathematics and has applications in various fields, including physics, engineering, and computer science. Remember to practice regularly and apply the techniques learned here to a variety of problems to further solidify your understanding.

Equivalent Equation

Based on our solution, the equivalent equation to the given equation 2sin²(x) - 5sin(x) - 3 = 0 after the substitution u = sin(x) and factoring is (2u + 1)(u - 3) = 0. Therefore, the correct option is A. This final step confirms our step-by-step approach and highlights the accuracy of our solution. Understanding how to transform and solve equations is a fundamental skill in mathematics, and this example provides a clear illustration of the process. The ability to recognize and apply the appropriate techniques is key to success in solving mathematical problems.