Solving Trigonometric Equations A Step-by-Step Guide To Sin(4x-10) = Cos(40-x)

In this detailed guide, we'll delve into the realm of trigonometry, specifically focusing on solving equations involving sine and cosine functions. Trigonometric equations, while seemingly complex, can be unraveled with a systematic approach and a solid understanding of trigonometric identities. Our primary focus will be on deciphering the equation sin (4x - 10)° = cos (40 - x)°. This type of problem frequently appears in mathematics curricula, covering trigonometry, and serves as an excellent exercise for honing problem-solving skills. This comprehensive guide aims to provide you with a step-by-step method, ensuring you grasp the core concepts and techniques involved.

In tackling this equation, we will employ fundamental trigonometric identities, primarily the cofunction identity, which establishes a relationship between sine and cosine functions. This identity is the cornerstone of our approach, allowing us to transform the equation into a more manageable form. By leveraging this identity, we will rewrite the equation in terms of a single trigonometric function, paving the way for isolating the variable x. This process will not only lead us to the solution but also deepen our understanding of trigonometric relationships and their applications. Understanding trigonometric equations is not just about finding the correct answer; it's about building a strong foundation in mathematics that can be applied to more advanced concepts and real-world problems. This guide will serve as a valuable resource for students, educators, and anyone seeking to enhance their grasp of trigonometry.

Our journey through this equation will also highlight the importance of careful algebraic manipulation. Once we've applied the cofunction identity, we'll need to solve a linear equation. This involves combining like terms, isolating the variable, and performing arithmetic operations with precision. Each step is crucial, and even a minor error can lead to an incorrect solution. Therefore, we'll emphasize the importance of double-checking your work and maintaining a clear, organized approach. Moreover, we will discuss the significance of verifying the solution. In trigonometry, it's essential to ensure that the solution we obtain satisfies the original equation. This verification step acts as a safeguard against potential errors and reinforces our confidence in the correctness of the answer. By the end of this guide, you'll not only know how to solve this specific equation but also gain valuable insights into the broader principles of trigonometric problem-solving.

The Core Trigonometric Identity: Unveiling the Cofunction Relationship

The cornerstone of solving the equation sin(4x - 10)° = cos(40 - x)° lies in understanding and applying a fundamental trigonometric identity: the cofunction identity. This identity reveals a crucial relationship between sine and cosine functions, stating that the sine of an angle is equal to the cosine of its complement. Mathematically, this is expressed as sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ), where θ represents any angle. This seemingly simple relationship is a powerful tool in simplifying and solving trigonometric equations. The cofunction identity stems from the geometric properties of right-angled triangles. In a right-angled triangle, the two acute angles are complementary, meaning they add up to 90°. The sine of one acute angle is defined as the ratio of the opposite side to the hypotenuse, while the cosine of the other acute angle is defined as the ratio of the adjacent side to the hypotenuse. These ratios are, in fact, the same, leading to the cofunction identity.

Understanding the cofunction identity goes beyond memorizing the formula; it's about grasping the underlying geometric principles. This understanding allows you to apply the identity flexibly and confidently in various contexts. For instance, you can use the cofunction identity to rewrite a sine function as a cosine function, or vice versa, which can be particularly useful in simplifying complex expressions or solving equations. In our case, the equation sin(4x - 10)° = cos(40 - x)° presents a situation where the cofunction identity can be directly applied. By recognizing that the sine of an angle is equal to the cosine of its complement, we can transform the equation into a more manageable form. This transformation is the key to unlocking the solution and demonstrating the power of trigonometric identities in problem-solving. Furthermore, the cofunction identity is not limited to sine and cosine; it extends to other trigonometric functions as well. For example, the tangent and cotangent functions are cofunctions, as are the secant and cosecant functions. This broader understanding of cofunction relationships enhances your overall grasp of trigonometry and its applications.

In essence, the cofunction identity is a fundamental building block in trigonometry. It provides a bridge between different trigonometric functions and allows us to manipulate equations and expressions with greater ease. By mastering this identity, you'll be well-equipped to tackle a wide range of trigonometric problems, from simple equations to more complex applications in physics, engineering, and other fields. The ability to recognize and apply the cofunction identity is a hallmark of a strong trigonometric foundation, and it's a skill that will serve you well in your mathematical journey. As we proceed with solving the equation sin(4x - 10)° = cos(40 - x)°, remember that the cofunction identity is our guiding principle, transforming the problem into a solvable form.

Step-by-Step Solution: Cracking the Code of the Trigonometric Equation

Let's embark on a step-by-step journey to solve the trigonometric equation sin(4x - 10)° = cos(40 - x)°. Our guiding principle, as discussed earlier, is the cofunction identity. This identity allows us to relate sine and cosine functions, paving the way for a straightforward solution. The first crucial step is to apply the cofunction identity to either the sine or cosine term. In this case, let's rewrite the sine term using the identity sin(θ) = cos(90° - θ). Applying this to our equation, we get:

cos(90° - (4x - 10)°) = cos(40 - x)°

This transformation is the key to simplifying the equation. By expressing both sides in terms of the cosine function, we've eliminated one layer of complexity. Now, we can focus on the arguments of the cosine functions. The next step involves equating the angles. Since the cosine functions are equal, their arguments must also be equal. This gives us the equation:

90° - (4x - 10)° = 40° - x°

This is a linear equation in x, which is much easier to solve than the original trigonometric equation. The next step is to simplify the equation by removing the parentheses and combining like terms. Distribute the negative sign on the left side:

90° - 4x° + 10° = 40° - x°

Now, combine the constant terms on the left side:

100° - 4x° = 40° - x°

To isolate x, we need to move all the x terms to one side and the constant terms to the other. Add 4x° to both sides:

100° = 40° + 3x°

Subtract 40° from both sides:

60° = 3x°

Finally, divide both sides by 3 to solve for x:

x = 20°

And there we have it! The value of x that satisfies the equation sin(4x - 10)° = cos(40 - x)° is 20°. This step-by-step solution demonstrates the power of the cofunction identity in simplifying trigonometric equations and the importance of careful algebraic manipulation.

Verification: Ensuring the Accuracy of Our Solution

While we've diligently followed the steps to solve the equation sin(4x - 10)° = cos(40 - x)°, it's crucial to verify our solution. Verification is a fundamental aspect of problem-solving in mathematics, ensuring that our answer is not only plausible but also accurate. In this case, we found that x = 20°. To verify this, we'll substitute this value back into the original equation and check if both sides are equal.

First, let's substitute x = 20° into the left side of the equation:

sin(4x - 10)° = sin(4(20) - 10)° = sin(80 - 10)° = sin(70)°

Now, let's substitute x = 20° into the right side of the equation:

cos(40 - x)° = cos(40 - 20)° = cos(20)°

So, our equation now looks like this:

sin(70)° = cos(20)°

To confirm if this is true, we can again use the cofunction identity. We know that sin(θ) = cos(90° - θ). Therefore:

sin(70)° = cos(90 - 70)° = cos(20)°

This confirms that our solution, x = 20°, satisfies the original equation. The left side equals the right side, and we can confidently conclude that our answer is correct. Verification is not just a formality; it's an essential step that helps prevent errors and builds confidence in your problem-solving abilities. By taking the time to verify your solutions, you develop a more thorough understanding of the concepts and processes involved.

In this case, the verification process reinforced our understanding of the cofunction identity and its application. It also highlighted the importance of careful substitution and evaluation. By verifying our solution, we've not only ensured the accuracy of our answer but also strengthened our grasp of the underlying trigonometric principles. This practice of verification is a valuable habit to cultivate in all areas of mathematics, as it promotes accuracy, critical thinking, and a deeper understanding of the subject matter.

Conclusion: Mastering Trigonometric Equations and Beyond

In this comprehensive guide, we've successfully navigated the trigonometric equation sin(4x - 10)° = cos(40 - x)°, demonstrating a systematic approach to solving such problems. Our journey began with understanding the core concept: the cofunction identity. This identity, which reveals the relationship between sine and cosine functions, served as the cornerstone of our solution. We then meticulously applied this identity to transform the equation, making it easier to solve. The step-by-step solution involved careful algebraic manipulation, including simplifying expressions, combining like terms, and isolating the variable x. Each step was executed with precision, highlighting the importance of attention to detail in mathematical problem-solving. Furthermore, we emphasized the crucial role of verification in ensuring the accuracy of our solution. By substituting our answer back into the original equation, we confirmed its validity and reinforced our understanding of the concepts involved.

The process of solving this equation extends far beyond the specific problem itself. It provides valuable insights into the broader realm of trigonometric equations and their applications. Trigonometry is a fundamental branch of mathematics with widespread applications in various fields, including physics, engineering, computer science, and navigation. Mastering trigonometric concepts and techniques is essential for success in these areas. The skills we've developed in this guide, such as applying trigonometric identities, manipulating equations, and verifying solutions, are transferable to a wide range of mathematical problems. Moreover, the systematic approach we've employed can be applied to other areas of problem-solving, fostering critical thinking and analytical skills.

As you continue your mathematical journey, remember that practice is key. The more you engage with trigonometric problems, the more comfortable and confident you'll become. Don't be afraid to tackle challenging problems, and always strive to understand the underlying principles. By building a strong foundation in trigonometry, you'll unlock a world of possibilities and enhance your problem-solving abilities in mathematics and beyond. The equation sin(4x - 10)° = cos(40 - x)° served as a valuable learning experience, demonstrating the power of trigonometric identities, the importance of careful algebraic manipulation, and the significance of verification. Embrace these principles, and you'll be well-equipped to conquer any trigonometric challenge that comes your way.

Answer

The correct answer is C. x = 20