Solving Trigonometric Equations: Finding the Values of x and y in sin(x-y) cos(x/y) 1/2

In this article, we will delve into the process of solving a set of trigonometric equations, specifically sin(x-y) cos(x/y) 1/2. This type of problem requires a good understanding of trigonometric identities and algebraic manipulation. We will explore the step-by-step solution of these equations to find the values of x and y.

Simplifying the Problem

The given equations are:

[sin(x-y) cos(x/y) 1/2]

Let's break this down step by step, starting with the trigonometric identities and values that are known from standard trigonometry.

Step 1: Using Known Values

From standard trigonometric values, we know:

[cos(frac{pi}{3}) frac{1}{2}]

Thus, we can equate x/y to π/3 from the second equation:

[frac{x}{y} frac{pi}{3}]

Similarly, from the first equation, we know:

[sin(frac{pi}{6}) frac{1}{2}]

Therefore, the first equation can be written as:

[x - y frac{pi}{6}]

Step 2: Substituting and Solving

From the second equation, we can express y in terms of x:

[y frac{3x}{pi}]

Substitute this expression for y in the first equation:

[x - frac{3x}{pi} frac{pi}{6}]

Multiply through by π to clear the denominator:

[pi x - 3x frac{pi^2}{6}]

Factor out x:

[x(pi - 3) frac{pi^2}{6}]

Solve for x:

[x frac{pi^2}{6(pi - 3)} frac{pi^2}{6pi - 18}]

Thus, the value of x is:

[x frac{pi^2}{6pi - 18}]

Step 3: Finding the Value of y

Substitute the value of x back into the expression for y:

[y frac{3x}{pi} frac{3(pi^2)}{(6pi - 18)pi} frac{3pi}{6pi - 18}]

Simplify the expression:

[y frac{3pi}{6(pi - 3)} frac{pi}{2(pi - 3)} frac{pi}{2pi - 6}]

Thus, the value of y is:

[y frac{pi}{2pi - 6}]

Conclusion

The values of x and y that satisfy the given equations are:

[x frac{pi^2}{6pi - 18} text{ and } y frac{pi}{2pi - 6}]

These equations utilize fundamental trigonometric identities and algebraic manipulation to reach a solution, making them a great exercise for students and enthusiasts of mathematics.

Please Upvote if you like the answer!