Exploring the Relationship Between Trigonometric Functions: sin x sin y When cos x cos y 2

Introduction

In the realm of trigonometry, understanding the relationships between different trigonometric functions is crucial. This article delves into a specific problem: determining the value of (sin x sin y) when (cos x cos y 2). This problem not only tests the knowledge of trigonometric limits but also requires a careful examination of the properties of the cosine function.

Understanding the Constraint

The (cos x) and (cos y) are components of the cosine function, which has a range from -1 to 1. Therefore, the product (cos x cos y) can only fall within the interval [-1, 1]. Given that the problem states (cos x cos y 2), this condition seems impossible at first glance. However, let's explore the logic behind this apparent contradiction.

Theoretical Analysis

The Cosine Function’s Maximum Value

The maximum value of the cosine function is 1. This means that for (cos x cos y) to equal 2, both (cos x) and (cos y) would have to simultaneously equal 1. Given that the cosine function can only achieve this maximum value at specific angles, we can deduce the following:

(cos x 1) when (x 2npi) for any integer (n).

(cos y 1) when (y 2mpi) for any integer (m).

Thus, we can conclude that:

(x 2npi) and (y 2mpi) for any integers (n) and (m)

Conclusion

Now, let's proceed to the value of (sin x sin y) with the determined values of (x) and (y):

(sin (2npi) 0)

(sin (2mpi) 0)

Hence, we can deduce that:

(sin x sin y 0)

Conclusion

Therefore, the value of (sin x sin y) when (cos x cos y 2) is (boxed{0}).