Exploring Trigonometric Relationships: Finding sin x and cos x in Terms of a

Introduction

This article delves into the process of solving trigonometric equations to find the values of (sin x), (cos x), and (a), given specific conditions involving the product of sines and cosines. We will use fundamental trigonometric identities to derive a relationship between these variables. This understanding will be useful in various areas of mathematics, physics, and engineering.

Given Equations and Initial Simplifications

We start with the given equations:

The first step is to transform these expressions using the sum-to-product identities. We can rewrite the given equations as:

(sin x sin y 2 sin left(frac{x y}{2}right) cos left(frac{x - y}{2}right) a)

(cos x cos y 2 cos left(frac{x y}{2}right) cos left(frac{x - y}{2}right) a)

Dividing the Two Equations

Dividing equation (1) by equation (2), we get:

(frac{2 sin left(frac{x y}{2}right) cos left(frac{x - y}{2}right)}{2 cos left(frac{x y}{2}right) cos left(frac{x - y}{2}right)} 1)

This simplifies to:

(frac{sin left(frac{x y}{2}right)}{cos left(frac{x y}{2}right)} 1)

Therefore:

(tan left(frac{x y}{2}right) 1)

Solving for (left(frac{x y}{2}right)), we get:

(frac{x y}{2} frac{pi}{4}) or (frac{5pi}{4})

Which implies:

(x y frac{pi}{2}) or (x y frac{5pi}{2})

Substituting Back into the Original Equation

Using the first solution (left(x y frac{pi}{2}right)) and substituting (y frac{pi}{2} - x) into the original equation (sin x sin y a), we get:

(sin x sin left(frac{pi}{2} - xright) a)

Using the identity (sin left(frac{pi}{2} - xright) cos x), the equation becomes:

(sin x cos x a)

Therefore, we have successfully derived that:

(sin x cos x a)

And this relationship is boxed for clarity:

(boxed{sin x cos x a})

Conclusion and Applications

This derivation provides a straightforward method to find the value of (sin x cos x) in terms of (a), given the constraints of the original equations. Understanding this relationship can be instrumental in solving more complex trigonometric problems, especially in fields like physics and engineering.