Finding Xy + Yz + Zx Given Trigonometric Equations - A Detailed Solution
In mathematics, especially in trigonometry and algebra, we often encounter problems that require us to find specific expressions given a set of equations. One such common problem involves finding the value of the expression xy + yz + zx when we are given trigonometric equations relating x, y, and z. This article delves into a detailed example demonstrating the techniques and steps involved in solving such problems. We will explore how trigonometric identities, algebraic manipulations, and logical reasoning come together to provide an elegant solution. Whether you're a student tackling homework, a teacher preparing lessons, or simply a math enthusiast, this guide aims to enhance your problem-solving skills and deepen your understanding of mathematical concepts.
Before diving into the solution, it's essential to fully understand the problem statement and the underlying concepts. In this specific example, we are given trigonometric equations involving variables x, y, and z. Our objective is to find the value of the expression xy + yz + zx. This type of problem often appears in various mathematical contexts, including algebra, trigonometry, and coordinate geometry. The key to solving these problems lies in identifying the relationships between the given equations and the expression we need to find. Often, this involves using trigonometric identities, algebraic manipulations, and clever substitutions to transform the given equations into a form that allows us to compute xy + yz + zx directly or indirectly. The challenge is not just about applying formulas but also about developing a strategic approach to problem-solving.
Let's consider a specific example to illustrate the process. Suppose we are given the following trigonometric equations:
- x = A * cos(θ)
- y = A * cos(θ + 2π/3)
- z = A * cos(θ + 4π/3)
where A is a constant and θ is an angle. Our goal is to find the value of xy + yz + zx in terms of A. This problem combines trigonometric functions with algebraic expressions, making it a rich exercise in mathematical problem-solving. We will need to utilize trigonometric identities, such as the cosine addition formula, and algebraic techniques to simplify and solve for the desired expression. The problem not only tests our knowledge of these concepts but also our ability to apply them in a creative and logical manner.
The approach to solving this problem involves several key steps, each requiring careful attention and application of relevant mathematical principles. First, we will compute the individual terms xy, yz, and zx using the given expressions for x, y, and z. This will involve multiplying the trigonometric functions and applying the cosine addition formula to simplify the resulting expressions. Second, we will sum these terms to obtain the expression xy + yz + zx. This step will require further simplification using trigonometric identities and algebraic manipulations. Finally, we will analyze the simplified expression to find its value in terms of A. This may involve recognizing patterns, canceling out terms, or applying additional trigonometric identities. The entire process is a testament to the power of combining different mathematical techniques to solve complex problems.
Step 1: Compute xy, yz, and zx
Let's start by computing xy. Using the given expressions for x and y, we have:
xy = (A * cos(θ)) * (A * cos(θ + 2π/3))
Expanding this, we get:
xy = A² * cos(θ) * cos(θ + 2π/3)
Now, we apply the cosine addition formula, which states that cos(a + b) = cos(a)cos(b) - sin(a)sin(b). Thus,
cos(θ + 2π/3) = cos(θ)cos(2π/3) - sin(θ)sin(2π/3)
We know that cos(2π/3) = -1/2 and sin(2π/3) = √3/2, so we can substitute these values:
cos(θ + 2π/3) = cos(θ) * (-1/2) - sin(θ) * (√3/2)
Substituting this back into the expression for xy, we get:
xy = A² * cos(θ) * [(-1/2)cos(θ) - (√3/2)sin(θ)]
xy = A² * [(-1/2)cos²(θ) - (√3/2)cos(θ)sin(θ)]
Similarly, we can compute yz and zx:
yz = (A * cos(θ + 2π/3)) * (A * cos(θ + 4π/3))
yz = A² * cos(θ + 2π/3) * cos(θ + 4π/3)
Applying the cosine addition formula to both terms, we have:
cos(θ + 4π/3) = cos(θ)cos(4π/3) - sin(θ)sin(4π/3)
We know that cos(4π/3) = -1/2 and sin(4π/3) = -√3/2, so:
cos(θ + 4π/3) = cos(θ) * (-1/2) - sin(θ) * (-√3/2)
cos(θ + 4π/3) = (-1/2)cos(θ) + (√3/2)sin(θ)
Substituting the expressions for cos(θ + 2π/3) and cos(θ + 4π/3) into yz, we get:
yz = A² * [(-1/2)cos(θ) - (√3/2)sin(θ)] * [(-1/2)cos(θ) + (√3/2)sin(θ)]
yz = A² * [(1/4)cos²(θ) - (3/4)sin²(θ)]
Finally, let's compute zx:
zx = (A * cos(θ + 4π/3)) * (A * cos(θ))
zx = A² * cos(θ + 4π/3) * cos(θ)
Substituting the expression for cos(θ + 4π/3), we have:
zx = A² * [(-1/2)cos(θ) + (√3/2)sin(θ)] * cos(θ)
zx = A² * [(-1/2)cos²(θ) + (√3/2)cos(θ)sin(θ)]
This step involves meticulous application of trigonometric identities and algebraic manipulation to express xy, yz, and zx in terms of trigonometric functions of θ. The attention to detail in this step is crucial for the subsequent simplification and solution.
Step 2: Sum xy + yz + zx
Now that we have expressions for xy, yz, and zx, we can sum them to find the value of xy + yz + zx:
xy + yz + zx = A² * [(-1/2)cos²(θ) - (√3/2)cos(θ)sin(θ)] + A² * [(1/4)cos²(θ) - (3/4)sin²(θ)] + A² * [(-1/2)cos²(θ) + (√3/2)cos(θ)sin(θ)]
We can factor out A² and combine like terms:
xy + yz + zx = A² * [(-1/2)cos²(θ) - (√3/2)cos(θ)sin(θ) + (1/4)cos²(θ) - (3/4)sin²(θ) + (-1/2)cos²(θ) + (√3/2)cos(θ)sin(θ)]
Notice that the terms involving cos(θ)sin(θ) cancel out:
xy + yz + zx = A² * [(-1/2)cos²(θ) + (1/4)cos²(θ) + (-1/2)cos²(θ) - (3/4)sin²(θ)]
Combining the cos²(θ) terms, we get:
xy + yz + zx = A² * [(-3/4)cos²(θ) - (3/4)sin²(θ)]
Now, we can factor out (-3/4):
xy + yz + zx = A² * (-3/4) * [cos²(θ) + sin²(θ)]
Step 3: Simplify the Expression
Using the Pythagorean trigonometric identity, we know that cos²(θ) + sin²(θ) = 1. Therefore,
xy + yz + zx = A² * (-3/4) * 1
xy + yz + zx = (-3/4)A²
Thus, the value of xy + yz + zx is (-3/4)A². This result demonstrates the power of algebraic manipulation and trigonometric identities in simplifying complex expressions. The systematic approach of computing individual terms, summing them, and then simplifying using known identities allowed us to arrive at a concise and elegant solution.
In this article, we have demonstrated a step-by-step approach to finding the value of xy + yz + zx given trigonometric equations. By applying trigonometric identities and algebraic manipulations, we were able to simplify the expression and arrive at the solution (-3/4)A². This example highlights the importance of understanding fundamental mathematical concepts and developing problem-solving strategies. The ability to break down a complex problem into smaller, manageable steps is a crucial skill in mathematics and beyond. Through careful computation, simplification, and logical reasoning, we can tackle challenging problems and gain a deeper appreciation for the beauty and power of mathematics.
This comprehensive example provides valuable insights into solving trigonometric problems and showcases the interplay between algebra and trigonometry. By mastering these techniques, students and enthusiasts can enhance their mathematical skills and confidently approach similar problems in various contexts. The key takeaway is the importance of a systematic approach, meticulous application of identities, and the ability to recognize and utilize patterns to simplify complex expressions.
Trigonometric equations, xy + yz + zx, trigonometric identities, cosine addition formula, algebraic manipulation, problem-solving strategies, mathematical concepts, simplification, Pythagorean identity, complex expressions
