To determine the domain of the function [tex]\( y = \sin(x) \)[/tex], we must understand what inputs [tex]\( x \)[/tex] we are allowed to use in the function such that [tex]\( y = \sin(x) \)[/tex] is defined.
The sine function [tex]\( \sin(x) \)[/tex] is a trigonometric function that relates the angle [tex]\( x \)[/tex] to a ratio of sides in a right-angled triangle. Importantly, the sine function is defined for all real numbers [tex]\( x \)[/tex]. This means that no matter what real number you choose for [tex]\( x \)[/tex], you will always get a valid output [tex]\( \sin(x) \)[/tex].
Let's analyze the options provided to identify the correct domain:
A. [tex]\( (-\infty, \infty) \)[/tex]: This notation represents all real numbers, meaning [tex]\( x \)[/tex] can be any real number. Since we know that the sine function is defined for all real numbers, this option could be correct.
B. [tex]\( 2\pi \)[/tex]: This notation [tex]\( 2\pi \)[/tex] alone does not represent a range or set of values but rather a single number. By specifying [tex]\( 2\pi \)[/tex], it implies that the domain would only include the number [tex]\( 2\pi \)[/tex], which is incorrect, as [tex]\( \sin(x) \)[/tex] is defined for more than just this single value.
C. [tex]\( [0, \infty) \)[/tex]: This represents all non-negative real numbers, including [tex]\( 0 \)[/tex]. Although the sine function is defined for those values, the domain of [tex]\( \sin(x) \)[/tex] should include negative values as well.
D. [tex]\(\{1, 1\}\)[/tex]: This notation is not correctly formatted as a domain representation. Also, it seems to represent the set containing just the number 1. This is incorrect because the function [tex]\( y = \sin(x) \)[/tex] is not limited to just [tex]\( x = 1 \)[/tex]; it is defined for all real numbers.
Having examined all options, we can conclude that the correct domain of the function [tex]\( y = \sin(x) \)[/tex] is:
A. [tex]\( (-\infty, \infty) \)[/tex].
Therefore, the correct answer is [tex]\( A \)[/tex].
