Solving 2sin(x) + 3cos(y) = 1 And 4sin(x) - Cos(y) = 2 A Step-by-Step Guide

Introduction

In this article, we will delve into the process of solving a system of trigonometric equations. Specifically, we will tackle the equations 2sin(x) + 3cos(y) = 1 and 4sin(x) - cos(y) = 2. These types of problems often appear in mathematics, physics, and engineering contexts, requiring a blend of algebraic manipulation and trigonometric identities to arrive at a solution. Understanding how to solve such systems is crucial for anyone looking to deepen their grasp of mathematical problem-solving techniques. This article aims to provide a step-by-step guide, ensuring clarity and comprehension for readers of all backgrounds. We will explore the underlying concepts and strategies, making this a valuable resource for students, educators, and enthusiasts alike. Our focus will be on breaking down the problem into manageable steps, making use of substitutions and eliminations to isolate variables and ultimately find the solutions for x and y. By the end of this exploration, you should feel confident in your ability to approach and solve similar trigonometric systems. Remember, the key to mastering mathematics lies in practice and understanding the fundamental principles. Let's embark on this journey together and unlock the solutions to these intriguing equations.

Setting up the Equations

The initial step in solving any system of equations is to clearly define and understand the given equations. In our case, we have two equations involving trigonometric functions: 2sin(x) + 3cos(y) = 1 and 4sin(x) - cos(y) = 2. To make the solution process more manageable, we can label these equations for easy reference. Let's denote the first equation as Equation (1) and the second as Equation (2). This simple step helps in organizing our work and referring back to specific equations during the solution process. Before diving into complex manipulations, it's important to recognize the variables we are trying to solve for: x and y. These variables are embedded within the sine and cosine functions, which adds a layer of complexity to the problem. However, by using techniques such as substitution and elimination, we can simplify the system and isolate these variables. The goal is to find values for x and y that satisfy both equations simultaneously. This often involves strategic algebraic manipulations and the application of trigonometric identities. As we move forward, we will explore these techniques in detail, providing a clear roadmap to solving the system. The ability to set up and understand the equations is a foundational skill in mathematics, and it's the cornerstone upon which the entire solution rests. By paying close attention to this initial step, we pave the way for a more efficient and accurate solution.

Elimination Method

The elimination method is a powerful technique for solving systems of equations. It involves manipulating the equations in such a way that one of the variables is eliminated when the equations are added or subtracted. In our system, we have 2sin(x) + 3cos(y) = 1 (Equation 1) and 4sin(x) - cos(y) = 2 (Equation 2). Our goal is to eliminate either sin(x) or cos(y). A close look reveals that cos(y) can be easily eliminated. To achieve this, we can multiply Equation (2) by 3. This will give us 12sin(x) - 3cos(y) = 6. Let's call this new equation Equation (3). Now, we have two equations with terms that are additive inverses of each other: Equation (1): 2sin(x) + 3cos(y) = 1 and Equation (3): 12sin(x) - 3cos(y) = 6. By adding these two equations, the cos(y) terms will cancel out, leaving us with an equation involving only sin(x). This is a crucial step as it simplifies the system significantly. The resulting equation will be 14sin(x) = 7. This equation is much easier to solve for sin(x). Once we find the value of sin(x), we can substitute it back into one of the original equations to solve for cos(y). The elimination method is a cornerstone of solving systems of equations, and its application here demonstrates its effectiveness in simplifying complex problems. By strategically manipulating the equations, we can isolate variables and move closer to the solution. This method is not only useful in trigonometry but also in various other branches of mathematics and science. Understanding and mastering the elimination method is a valuable asset in any problem-solver's toolkit.

Solving for sin(x)

Having successfully eliminated cos(y) from our system of equations, we now have a simplified equation: 14sin(x) = 7. This equation is straightforward to solve for sin(x). To isolate sin(x), we simply divide both sides of the equation by 14. This gives us sin(x) = 7/14, which simplifies to sin(x) = 1/2. This is a significant milestone in our solution process. We have found a specific value for the sine of x. Now, we need to determine the values of x that satisfy this condition. Recall that the sine function represents the y-coordinate on the unit circle. The values of x for which sin(x) = 1/2 correspond to the angles where the y-coordinate on the unit circle is 1/2. These angles are typically found in the first and second quadrants. The reference angle for sin(x) = 1/2 is π/6 (or 30 degrees). Therefore, the two solutions in the interval [0, 2π) are x = π/6 and x = 5π/6. However, since the sine function is periodic with a period of 2π, there are infinitely many solutions. The general solutions for x can be expressed as x = π/6 + 2πn and x = 5π/6 + 2πn, where n is an integer. These solutions represent all possible angles for which sin(x) = 1/2. Understanding how to solve for trigonometric functions like sine is fundamental in mathematics and its applications. This step demonstrates the power of algebraic manipulation in conjunction with trigonometric principles. With the value of sin(x) in hand, we can now move on to finding the corresponding values of cos(y).

Solving for cos(y)

With the value of sin(x) determined as 1/2, we can now proceed to solve for cos(y). To do this, we can substitute the value of sin(x) back into one of the original equations. Let's use Equation (2): 4sin(x) - cos(y) = 2. Substituting sin(x) = 1/2 into this equation, we get 4(1/2) - cos(y) = 2. This simplifies to 2 - cos(y) = 2. Now, we need to isolate cos(y). Subtracting 2 from both sides of the equation gives us -cos(y) = 0. Multiplying both sides by -1, we find that cos(y) = 0. This result is a crucial piece of the puzzle. We have found a specific value for the cosine of y. To find the values of y that satisfy cos(y) = 0, we need to consider the unit circle. The cosine function represents the x-coordinate on the unit circle. The values of y for which cos(y) = 0 correspond to the angles where the x-coordinate is 0. These angles are typically π/2 and 3π/2. Since the cosine function is periodic with a period of 2π, there are infinitely many solutions. The general solutions for y can be expressed as y = π/2 + πm, where m is an integer. This concise expression captures all possible angles for which cos(y) = 0. The ability to solve for trigonometric functions like cosine is essential in various mathematical and scientific contexts. This step highlights the importance of substitution in solving systems of equations. By substituting the value of sin(x) into one of the original equations, we were able to isolate cos(y) and find its value. With the values of both sin(x) and cos(y) determined, we are now in a position to find the general solutions for x and y.

General Solutions for x and y

Now that we have found sin(x) = 1/2 and cos(y) = 0, we can express the general solutions for x and y. As we discussed earlier, the general solutions for x are given by x = π/6 + 2πn and x = 5π/6 + 2πn, where n is an integer. These solutions account for all angles that have a sine of 1/2, considering the periodic nature of the sine function. Similarly, the general solutions for y are given by y = π/2 + πm, where m is an integer. This expression captures all angles that have a cosine of 0, taking into account the periodicity of the cosine function. To fully understand these solutions, it's helpful to visualize them on the unit circle. The solutions for x represent two sets of angles that are 2π radians (or 360 degrees) apart, while the solutions for y represent angles that are π radians (or 180 degrees) apart. These general solutions provide a complete picture of all possible values of x and y that satisfy the original system of equations. However, in some contexts, we may be interested in specific solutions within a certain interval, such as [0, 2π). In such cases, we can substitute different integer values for n and m to find the solutions within that interval. For example, if we want to find the solutions for x in the interval [0, 2π), we can substitute n = 0 into the general solutions, giving us x = π/6 and x = 5π/6. The same approach can be used to find specific solutions for y. Understanding how to express and interpret general solutions is a crucial skill in mathematics. It allows us to capture the complete set of solutions to an equation or system of equations, taking into account the periodic nature of trigonometric functions. With the general solutions for x and y in hand, we have successfully solved the given system of equations.

Verification of Solutions

After obtaining the general solutions for x and y, it's crucial to verify that these solutions indeed satisfy the original equations. This step ensures that our calculations are correct and that we haven't introduced any extraneous solutions along the way. To verify our solutions, we can substitute the general expressions for x and y back into the original equations: 2sin(x) + 3cos(y) = 1 and 4sin(x) - cos(y) = 2. Let's start by substituting x = π/6 + 2πn and y = π/2 + πm into the first equation: 2sin(π/6 + 2πn) + 3cos(π/2 + πm) = 1. Since sin(π/6) = 1/2 and cos(π/2) = 0, the equation becomes 2(1/2) + 3(0) = 1, which simplifies to 1 = 1. This confirms that the first equation is satisfied. Now, let's substitute the same expressions into the second equation: 4sin(π/6 + 2πn) - cos(π/2 + πm) = 2. Again, using sin(π/6) = 1/2 and cos(π/2) = 0, the equation becomes 4(1/2) - 0 = 2, which simplifies to 2 = 2. This confirms that the second equation is also satisfied. We can repeat this process for x = 5π/6 + 2πn and y = π/2 + πm to ensure that these solutions also satisfy the original equations. The verification step is a vital part of the problem-solving process. It provides a check on our work and helps us catch any errors that may have occurred. By verifying our solutions, we can have confidence in our results and know that we have truly solved the system of equations. This practice is not only important in mathematics but also in various other fields where problem-solving is a key skill. Always remember to verify your solutions to ensure accuracy and completeness.

Conclusion

In this comprehensive exploration, we have successfully navigated the process of solving the system of trigonometric equations 2sin(x) + 3cos(y) = 1 and 4sin(x) - cos(y) = 2. We began by setting up the equations and understanding the problem, then strategically employed the elimination method to simplify the system. By eliminating cos(y), we were able to isolate sin(x) and solve for its value. Subsequently, we substituted this value back into one of the original equations to solve for cos(y). With the values of sin(x) and cos(y) determined, we derived the general solutions for x and y, taking into account the periodic nature of the trigonometric functions. Finally, we emphasized the importance of verifying our solutions to ensure accuracy and completeness. This journey has highlighted several key mathematical concepts and techniques, including algebraic manipulation, trigonometric identities, and the application of the unit circle. We have demonstrated the power of the elimination method in simplifying complex systems of equations and the importance of understanding the periodicity of trigonometric functions. Furthermore, we have underscored the value of verification in ensuring the correctness of our solutions. By mastering these skills, you will be well-equipped to tackle a wide range of mathematical problems. Remember, the key to success in mathematics is practice and a deep understanding of the fundamental principles. We hope this article has provided you with valuable insights and a clear roadmap for solving trigonometric systems of equations. Keep practicing, and you will continue to grow your mathematical abilities.