To find the zeros of the function [tex]\( y = \sin(x) \)[/tex], we need to determine where the sine function equals zero, i.e., where [tex]\( \sin(x) = 0 \)[/tex].
The sine function is known to have zeros at specific points on the x-axis. These points are the values where the function intersects the x-axis. The sine function [tex]\( \sin(x) \)[/tex] equals zero whenever [tex]\( x \)[/tex] is a multiple of [tex]\( \pi \)[/tex]. This is because the sine of any integer multiple of [tex]\( \pi \)[/tex] is zero. Mathematically, this can be expressed as:
[tex]\[ x = k \pi \][/tex]
where [tex]\( k \)[/tex] is an integer (not just positive integers, but any integer – positive, negative, or zero).
Thus, the zeros of [tex]\( y = \sin(x) \)[/tex] occur at:
[tex]\[ x = k \pi \][/tex]
for any integer [tex]\( k \)[/tex].
Among the given options, the correct formula that represents these zeros is:
[tex]\[ k \pi \text{ for any integer } k \][/tex]
Therefore, the correct answer is not among the provided options. If there had been an accurate representation, it would have been:
[tex]\[ k \pi \text{ for any integer } k \][/tex]
