If Angle A Is Acute And Tan(a) = 3, Calculate The Value Of The Expression (sin(a) + Cos(a)) / (sin(a) - Cos(a)).
In trigonometry, solving expressions involving trigonometric functions often requires utilizing fundamental identities and relationships. One such problem involves finding the value of the expression (sin α + cos α) / (sin α - cos α), given the value of tan α. This article will provide a detailed, step-by-step solution to this problem, ensuring a clear understanding of the underlying concepts and techniques. We will leverage the relationship between tan α, sin α, and cos α, as well as algebraic manipulation, to arrive at the final answer.
Given that angle α is acute and tan α = 3, we need to find the value of the expression (sin α + cos α) / (sin α - cos α). This problem requires us to use the given information about tan α and the properties of trigonometric functions to determine the value of the expression. The key to solving this problem lies in understanding the relationships between the trigonometric functions and applying algebraic techniques to simplify the expression.
The primary trigonometric functions—sine (sin), cosine (cos), and tangent (tan)—are related in fundamental ways. The tangent of an angle is defined as the ratio of the sine to the cosine of that angle:
tan α = sin α / cos α
This relationship is crucial for solving our problem because it connects the given value of tan α to the sin α and cos α terms in the expression we want to evaluate. Additionally, the Pythagorean identity, which states that sin² α + cos² α = 1, is a cornerstone of trigonometry and can be used to relate sin α and cos α further. These relationships form the foundation upon which we will build our solution. Understanding these relationships deeply is essential for tackling a wide range of trigonometric problems.
To solve the problem, we will follow these steps:
1. Express the Target Expression in Terms of tan α
We start with the expression (sin α + cos α) / (sin α - cos α). To introduce tan α into this expression, we can divide both the numerator and the denominator by cos α. This is a valid algebraic manipulation as long as cos α ≠ 0, which is true for acute angles. Dividing both the numerator and the denominator by cos α, we get:
[(sin α + cos α) / cos α] / [(sin α - cos α) / cos α]
This simplifies to:
(sin α / cos α + cos α / cos α) / (sin α / cos α - cos α / cos α)
2. Substitute tan α = sin α / cos α
Using the definition tan α = sin α / cos α, we can substitute tan α into the expression:
(tan α + 1) / (tan α - 1)
This substitution is a key step because it transforms the original expression, which involved both sine and cosine, into an expression that only involves the tangent. Given that we know the value of tan α, this simplification brings us closer to finding the value of the entire expression.
3. Plug in the Given Value of tan α
We are given that tan α = 3. Now, we substitute this value into the simplified expression:
(3 + 1) / (3 - 1)
This step is straightforward but crucial, as it directly utilizes the information provided in the problem statement. By substituting the numerical value of tan α, we convert the problem from a symbolic manipulation to a simple arithmetic calculation.
4. Simplify the Expression
Finally, we simplify the expression by performing the arithmetic:
4 / 2 = 2
This final calculation gives us the value of the expression (sin α + cos α) / (sin α - cos α), which is 2. The simplicity of this final step underscores the importance of the earlier algebraic manipulations and substitutions.
Therefore, the value of the expression (sin α + cos α) / (sin α - cos α), given that tan α = 3, is 2. This result is obtained by strategically using the trigonometric identity tan α = sin α / cos α and performing algebraic simplifications. This solution highlights the interconnectedness of trigonometric functions and the power of algebraic manipulation in solving trigonometric problems. The final answer is a concise numerical value, demonstrating the effectiveness of the step-by-step approach we followed.
While the above method provides a direct and efficient solution, there are alternative approaches that can provide additional insights into the problem. One such approach involves using the Pythagorean identity, sin² α + cos² α = 1, to find the individual values of sin α and cos α. However, this method is generally more complex and time-consuming. Another insight is the geometric interpretation of trigonometric functions in a right-angled triangle, which can help visualize the relationships between sin α, cos α, and tan α. Exploring these alternative perspectives can deepen understanding and provide a more comprehensive grasp of trigonometric principles.
Trigonometric identities are fundamental tools in solving trigonometric problems. The identities we used in this solution, such as tan α = sin α / cos α and the implicit use of the fact that dividing by cos α is valid when cos α ≠ 0, are crucial for simplifying complex expressions and establishing relationships between trigonometric functions. Mastery of these identities is essential for anyone studying trigonometry, as they appear frequently in various contexts, from solving equations to simplifying expressions in calculus and physics. The ability to recognize and apply these identities effectively is a hallmark of a strong mathematical foundation.
Trigonometry is not merely an abstract mathematical concept; it has numerous applications in real-world scenarios. From surveying and navigation to engineering and physics, trigonometric functions are used to model periodic phenomena, calculate distances and angles, and analyze wave behavior. For instance, in engineering, trigonometric functions are used to analyze the forces acting on structures, while in physics, they are used to describe the motion of projectiles. Understanding the relationships between trigonometric functions, as demonstrated in this problem, is essential for applying these concepts to practical situations. The ability to solve problems involving trigonometric functions is a valuable skill in various technical fields.
In conclusion, finding the value of (sin α + cos α) / (sin α - cos α) given tan α = 3 is a problem that highlights the importance of understanding and applying trigonometric identities. By strategically using the definition of tan α and performing algebraic manipulations, we were able to simplify the expression and arrive at the solution, which is 2. This problem serves as a good example of how fundamental trigonometric principles can be used to solve complex expressions. The step-by-step solution provided in this article aims to enhance understanding and problem-solving skills in trigonometry. Mastering these techniques will prove beneficial in tackling more advanced mathematical problems and real-world applications. The beauty of trigonometry lies in its ability to connect abstract mathematical concepts to tangible, real-world phenomena, making it a fascinating and essential field of study.