Answer :
To determine the domain of the function [tex]\(y = \cos(x)\)[/tex], let's analyze the properties of the cosine function:
1. Definition of Cosine Function:
- The cosine function, [tex]\(\cos(x)\)[/tex], is one of the basic trigonometric functions. It is defined by the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle or as the x-coordinate on the unit circle for a given angle [tex]\(x\)[/tex].
2. Periodicity:
- The cosine function is periodic with a period of [tex]\(2\pi\)[/tex], meaning that [tex]\(\cos(x) = \cos(x + 2\pi k)\)[/tex] for any integer [tex]\(k\)[/tex].
3. Continuity and Real Values:
- The cosine function is continuous and defined for all real numbers. There are no restrictions on the values that [tex]\(x\)[/tex] can take.
4. Range of the Function:
- Although the range of the cosine function is limited to [tex]\([-1, 1]\)[/tex], the domain is not constrained by this fact.
Given these properties, we can conclude that the cosine function is defined for all real numbers. Therefore, its domain includes all real numbers.
So, the correct answer is:
D. [tex]\( (-\infty, \infty) \)[/tex]
1. Definition of Cosine Function:
- The cosine function, [tex]\(\cos(x)\)[/tex], is one of the basic trigonometric functions. It is defined by the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle or as the x-coordinate on the unit circle for a given angle [tex]\(x\)[/tex].
2. Periodicity:
- The cosine function is periodic with a period of [tex]\(2\pi\)[/tex], meaning that [tex]\(\cos(x) = \cos(x + 2\pi k)\)[/tex] for any integer [tex]\(k\)[/tex].
3. Continuity and Real Values:
- The cosine function is continuous and defined for all real numbers. There are no restrictions on the values that [tex]\(x\)[/tex] can take.
4. Range of the Function:
- Although the range of the cosine function is limited to [tex]\([-1, 1]\)[/tex], the domain is not constrained by this fact.
Given these properties, we can conclude that the cosine function is defined for all real numbers. Therefore, its domain includes all real numbers.
So, the correct answer is:
D. [tex]\( (-\infty, \infty) \)[/tex]
