Exploring the Domain of y sin x cos x: An In-Depth Analysis

Understanding the domain of a function is fundamental in mathematics, particularly in advanced calculus and trigonometry. This article delves into the domain of the function y sin x cos x, examining its characteristics and providing a comprehensive explanation of its properties.

Introduction to the Function

The function y sin x cos x is a product of two trigonometric functions, sine and cosine. Both sine and cosine functions are periodic and have a domain of all real numbers. Understanding the domain of this function opens up the possibilities for further analysis in calculus and related fields.

Method 1: Visual Analysis of the Graph

The domain can be clearly understood by examining the graph of y sin x cos x. From the graph, we observe that the function does not have any discontinuities or breaks. The values of the function stretch out infinitely in both the positive and negative directions, suggesting that the domain is all real numbers.

Mathematical Notation: The domain of y sin x cos x can be represented as {x: x belongs to R}, or in interval notation as (-∞, ∞).

Method 2: Analytical Approach

An alternative and more analytical approach involves considering the domains of sin x and cos x individually. Both of these functions have a domain of all real numbers, denoted as (-∞, ∞). Since y sin x cos x is a product of these two functions, the domain of the resulting function y is simply the intersection of the domains of sin x and cos x. As both functions have the same domain, the combined domain is also (-∞, ∞).

Mathematical Notation: The domain of y sin x cos x can be represented as {x: x belongs to R}, or in interval notation as (-∞, ∞).

Further Analysis: Range of the Function

While the domain of y sin x cos x is all real numbers, its range is more complex. To find the range, we can use trigonometric identities. Specifically, we can use the identity:

This identity simplifies the range calculation. The function sin 2x has a range of [-1, 1]. Therefore, the range of sin x cos x will be scaled by the factor (frac{1}{2}), resulting in a range of [-1/2, 1/2].

Mathematical Notation: The range of y sin x cos x can be represented as [ -(frac{1}{2}), (frac{1}{2}) ].

Conclusion

Through both graphical and analytical methods, we have shown that the domain of y sin x cos x is all real numbers. This encompasses the entire set of real numbers from negative infinity to positive infinity. Understanding the domain of this function is crucial for further applications in calculus and trigonometry.

For a more in-depth exploration of similar functions and their properties, consider delving into the extensive resources available online and in textbooks dedicated to trigonometric and calculus topics.