Step-by-Step Simplification Of Sin(x-y)/cos(x-y) Trigonometric Transformation

In the realm of mathematics, particularly within trigonometry, transformations and manipulations of trigonometric identities are fundamental skills. These skills not only aid in simplifying complex expressions but also pave the way for solving equations and understanding the relationships between angles and their respective trigonometric functions. This article will dissect the transformation of the expression sin(x-y)/cos(x-y), a common trigonometric ratio representing the tangent of the difference of two angles. We'll delve into the step-by-step process, providing a detailed explanation of each stage, and highlighting the underlying trigonometric identities that make this transformation possible. Our focus will be on ensuring that every step is clear, concise, and logically sound, allowing readers to grasp the intricacies of the process and apply it to other similar problems.

This exploration is crucial for students and enthusiasts alike, as it bridges the gap between theoretical knowledge and practical application. Understanding how trigonometric expressions can be manipulated and simplified is not just an academic exercise; it’s a vital tool in various fields, including physics, engineering, and computer graphics. By the end of this article, readers should be well-equipped to tackle similar trigonometric problems with confidence and a solid understanding of the underlying principles. The journey through trigonometric transformations is not just about memorizing formulas, it's about developing a deeper appreciation for the elegance and interconnectedness of mathematical concepts.

In the initial step, we are presented with the expression: sin(x-y)/cos(x-y). This expression is a fundamental trigonometric ratio, representing the tangent of the difference between two angles, x and y. The tangent function, often abbreviated as tan, is defined as the ratio of the sine of an angle to the cosine of the same angle. Therefore, the expression sin(x-y)/cos(x-y) is equivalent to tan(x-y). This is a crucial starting point because it sets the stage for further manipulation and simplification using trigonometric identities. Understanding this foundational relationship is paramount to navigating the subsequent steps of the transformation.

The tangent function's representation as a ratio of sine and cosine is not merely a definition; it's a gateway to a plethora of trigonometric identities. By expressing tan(x-y) in this form, we open the door to utilizing the sine and cosine angle subtraction identities. These identities are essential tools in trigonometry, allowing us to break down complex trigonometric functions into simpler, more manageable components. The ability to recognize and apply these identities is a hallmark of proficiency in trigonometry. The initial step, therefore, is not just a simple restatement; it's a strategic move that unlocks the potential for further simplification and manipulation. It's a reminder that in mathematics, the way we represent a problem can significantly impact our ability to solve it. The power of trigonometric identities lies in their ability to transform expressions, making them more amenable to analysis and computation.

The next crucial step involves applying the angle subtraction identities for sine and cosine. These identities are fundamental tools in trigonometry that allow us to expand trigonometric functions of the difference of two angles into expressions involving the individual angles. The angle subtraction identity for sine is given by: sin(x - y) = sin(x)cos(y) - cos(x)sin(y). Similarly, the angle subtraction identity for cosine is given by: cos(x - y) = cos(x)cos(y) + sin(x)sin(y). These identities are not arbitrary formulas; they are derived from geometric considerations and the unit circle definition of trigonometric functions. Understanding their derivation provides a deeper appreciation for their validity and applicability.

Applying these identities to our expression, sin(x-y)/cos(x-y), we substitute the expanded forms of sin(x-y) and cos(x-y) into the numerator and denominator, respectively. This yields the expression: (sin(x)cos(y) - cos(x)sin(y)) / (cos(x)cos(y) + sin(x)sin(y)) . This step is a significant advancement in the simplification process. By expanding the sine and cosine of the difference of angles, we have transformed the original expression into one that is more amenable to further manipulation. The expanded form reveals the individual contributions of the angles x and y to the overall trigonometric ratio. This decomposition is often a key step in solving trigonometric equations or simplifying complex expressions. The mastery of angle subtraction identities is, therefore, a cornerstone of trigonometric proficiency.

The expression presented in the original step 2, (sin(x)cos(y) + cos(x)sin(y))/((A)cos(y) - sin(x)sin(y)), contains an error. The numerator should be sin(x)cos(y) - cos(x)sin(y) (note the subtraction), and the denominator should be cos(x)cos(y) + sin(x)sin(y) (note the addition). Therefore, to correct the expression, (A) should be cos(x) and the operation in the numerator should be a subtraction. This correction is essential to maintain the mathematical integrity of the transformation.

In the third step, we aim to further simplify the expression obtained in the previous step. The strategy here is to divide both the numerator and the denominator by a common factor. This technique is a standard practice in algebraic manipulation and is particularly useful in trigonometry for transforming expressions into more recognizable forms. The common factor chosen in this case is cos(x)cos(y). Dividing both the numerator and the denominator by this factor allows us to express the terms in terms of tangent functions, which can lead to further simplification or a more insightful representation of the expression.

Starting with the corrected expression from Step 2, (sin(x)cos(y) - cos(x)sin(y)) / (cos(x)cos(y) + sin(x)sin(y)), we divide each term in the numerator and the denominator by cos(x)cos(y). This gives us: [(sin(x)cos(y) / cos(x)cos(y)) - (cos(x)sin(y) / cos(x)cos(y))] / [(cos(x)cos(y) / cos(x)cos(y)) + (sin(x)sin(y) / cos(x)cos(y))]. Now, we can simplify each term by canceling out common factors. This leads to: [(sin(x)/cos(x)) - (sin(y)/cos(y))] / [1 + (sin(x)/cos(x))(sin(y)/cos(y))]. Recognizing that sin(x)/cos(x) = tan(x) and sin(y)/cos(y) = tan(y), we can rewrite the expression as: (tan(x) - tan(y)) / (1 + tan(x)tan(y)). This final form is the tangent subtraction identity, which provides a concise and elegant representation of the original expression.

Therefore, in Step 3, (B) should be cos(y). The division by the common factor cos(x)cos(y) has successfully transformed the expression into a form that involves tangent functions, revealing the underlying trigonometric relationship in a more transparent way. This step highlights the power of algebraic manipulation in simplifying trigonometric expressions and connecting them to fundamental identities. The ability to identify and apply appropriate common factors is a crucial skill in trigonometric problem-solving.

In conclusion, the journey from the initial expression sin(x-y)/cos(x-y) to the final form (tan(x) - tan(y)) / (1 + tan(x)tan(y)) demonstrates the power and elegance of trigonometric transformations. By systematically applying trigonometric identities and algebraic manipulations, we have successfully simplified a complex expression into a more manageable and insightful form. This process has highlighted the importance of understanding and utilizing fundamental trigonometric identities, such as the angle subtraction identities for sine and cosine, as well as the definition of the tangent function.

Each step in this transformation played a crucial role in reaching the final result. The initial expression, representing the tangent of the difference of two angles, set the stage for applying the angle subtraction identities. These identities allowed us to expand the sine and cosine of the difference of angles into expressions involving the individual angles. Then, by strategically dividing by a common factor, cos(x)cos(y), we were able to transform the expression into a form that involved tangent functions. This final form, (tan(x) - tan(y)) / (1 + tan(x)tan(y)), is the tangent subtraction identity, providing a concise representation of the original expression.

This transformation is not just a mathematical exercise; it's a demonstration of the interconnectedness of trigonometric concepts. It showcases how different trigonometric functions and identities relate to each other and how they can be manipulated to simplify complex expressions. The ability to perform such transformations is essential for anyone working in fields that rely on trigonometry, such as physics, engineering, and computer graphics. By mastering these techniques, students and professionals alike can gain a deeper understanding of the mathematical principles that underpin these fields.

• Step 2: The expression in Step 2 is incorrect. The numerator should use subtraction instead of addition, and the denominator should use addition. Also, what is the value of A? Correct the expression and determine the value of A.
• Step 3: What should B be in step 3?

Trigonometric Transformation Step-by-Step Simplification of sin(x-y)/cos(x-y)