Answer :
To determine the domain of the function [tex]\( y = \sin x \)[/tex]:
1. Understanding the Sine Function: The sine function, denoted as [tex]\( \sin(x) \)[/tex], is a periodic function that oscillates between -1 and 1. It is a fundamental trigonometric function commonly used in various mathematical applications.
2. Analyzing the Function's Definition: The sine function is defined for all real numbers. There is no restriction on the values that [tex]\( x \)[/tex] can take because the sine function can compute the sine of any real number.
3. Evaluating Potential Constraints: Let's evaluate each option given in the question.
- Option A: [tex]\( -1 \leq x \leq 1 \)[/tex] is incorrect. This describes the range of [tex]\( \sin(x) \)[/tex], i.e., the output values of [tex]\( \sin(x) \)[/tex], rather than the domain.
- Option B: [tex]\( x \neq n \pi \)[/tex] (where [tex]\( n \)[/tex] is any integer) is also incorrect. This would imply the function is undefined at integer multiples of [tex]\( \pi \)[/tex], which is not true.
- Option C: All real numbers. This is the correct option because [tex]\( \sin(x) \)[/tex] is defined for every real number x.
- Option D: [tex]\( -1 \leq y \leq 1 \)[/tex] is incorrect. This depicts the range of the function rather than the domain.
4. Conclusion: The correct option for the domain of [tex]\( y = \sin x \)[/tex] is:
[tex]\[ \boxed{\text{C. All real numbers}} \][/tex]
1. Understanding the Sine Function: The sine function, denoted as [tex]\( \sin(x) \)[/tex], is a periodic function that oscillates between -1 and 1. It is a fundamental trigonometric function commonly used in various mathematical applications.
2. Analyzing the Function's Definition: The sine function is defined for all real numbers. There is no restriction on the values that [tex]\( x \)[/tex] can take because the sine function can compute the sine of any real number.
3. Evaluating Potential Constraints: Let's evaluate each option given in the question.
- Option A: [tex]\( -1 \leq x \leq 1 \)[/tex] is incorrect. This describes the range of [tex]\( \sin(x) \)[/tex], i.e., the output values of [tex]\( \sin(x) \)[/tex], rather than the domain.
- Option B: [tex]\( x \neq n \pi \)[/tex] (where [tex]\( n \)[/tex] is any integer) is also incorrect. This would imply the function is undefined at integer multiples of [tex]\( \pi \)[/tex], which is not true.
- Option C: All real numbers. This is the correct option because [tex]\( \sin(x) \)[/tex] is defined for every real number x.
- Option D: [tex]\( -1 \leq y \leq 1 \)[/tex] is incorrect. This depicts the range of the function rather than the domain.
4. Conclusion: The correct option for the domain of [tex]\( y = \sin x \)[/tex] is:
[tex]\[ \boxed{\text{C. All real numbers}} \][/tex]
