Trigonometric functions, particularly sine and cosine, play a critical role in various fields including calculus, engineering, and physics. In this article, we'll delve into the maximum and minimum values of the function sin(xy)cos(x) - y to provide a comprehensive understanding of its behavior. We will explore the mathematical principles behind these values and provide a step-by-step analysis to solidify the concepts.

1. Introduction to Trigonometric Functions

Trigonometric functions, such as sine (sin) and cosine (cos), are periodic and oscillate between fixed ranges. The sine function, for example, has a range of [-1, 1], and the same applies to the cosine function. Understanding these bases is crucial for analyzing more complex functions like sin(xy)cos(x) - y.

2. Analyzing the Function sin(xy)cos(x) - y

Consider the function sin(xy)cos(x) - y. This function combines several trigonometric elements, making it a challenging but interesting subject for analysis. We can break down the function into more manageable parts to identify its maximum and minimum values.

2.1 Simplifying the Expression

Starting with the expression sin(xy)cos(x) - y, let's break it down step-by-step. We can express it as:

(sin(xy) cos(x) - y sin(x) sin(y) cos(x) - y)

By taking cos(x) and sin(x) in common, we get:

(-cos(y) sin(x) cos(x) - y)

2.2 Maximum and Minimum Values of sin(x)cos(x)

The maximum value of sin(x)cos(x) is derived from the fact that the product of sine and cosine has a maximum value of (frac{1}{2}). This is derived from the identity:

(sin(x) cos(x) frac{1}{2} sin(2x))

Therefore, the maximum value of sin(2x) is 1, making the maximum of sin(x)cos(x) equal to (frac{1}{2}).

When we substitute this into the function, we get:

(-cos(y) cdot frac{1}{2} - y)

The maximum value of this expression occurs when cos(y) 1, leading to:

(-frac{1}{2} - y)

2.3 Evaluating the Specific Cases

Given the insight that (sin(xy) cos(x) frac{1}{2} sin(2x) cos(y)), we can evaluate the function at specific points to find its maximum and minimum values. The bounding values can be attained as follows:

-2 is the minimum value when xy -frac{pi}{2} and x - y pi.

2 is the maximum value when xy frac{pi}{2} and x - y 0.

3. Conclusion

In this article, we've explored the function sin(xy)cos(x) - y and determined its maximum and minimum values. By breaking down the function and analyzing specific cases, we can better understand its behavior.

Understanding such functions is crucial in various applications, including solving differential equations, analyzing periodic phenomena, and optimizing complex systems. Whether you're a student, an engineer, or a mathematician, the insights gained from this analysis can be applied to a wide range of problems.