Understanding the Domain and Range of the Sine Function: sinx

The sine function, commonly denoted as sinx, is a fundamental trigonometric function with widespread applications in mathematics, physics, engineering, and other scientific fields. This article will delve into the domain and range of sinx, focusing on its properties and implications.

Domain and Range of sinx

The domain and range of the sine function are crucial to understanding its behavior. Let's explore these concepts in detail.

Domain of sinx

The domain of the sine function is all real numbers, mathbb{R}. This makes sense when we consider the geometric interpretation of the sine function: it maps any real number x to a corresponding point on the unit circle. Since the unit circle spans all angles, the sine function can take any real value, making the domain all real numbers.

Range of sinx

Unlike the domain, the range of the sine function is more confined. The sine function outputs values that lie between -1 and 1. This can be represented mathematically as:

-1 ≤ y ≤ 1

This range can also be written in interval notation as following:

Domain: (-∞, ∞)

Range: [-1, 1]

Periodicity of sinx

The sine function is periodic with a period of 2π. This means that for any real number x, the function repeats its values in intervals of 2π. Mathematically, this can be expressed as:

sin(x 2π) sinx

This periodicity is a key property that simplifies many trigonometric calculations and facilitates the use of sine in various applications.

Graph of sinx

The graph of sinx is a visual representation of its domain and range. It is a smooth, periodic function that oscillates between -1 and 1, completing one full cycle every 2π units. The function is also odd, meaning it is symmetric about the origin. The amplitude of the sine function is 1, and the period is 2π.

Conclusion

In summary, the domain of the sine function is all real numbers, indicating that it can take any real input. The range is confined to values between -1 and 1, reflecting the maximum and minimum values of the function. Understanding the domain and range of sinx is foundational for further exploration of trigonometric functions and their applications.