To find the zeros of the function [tex]\( y = \sin(x) \)[/tex], we need to determine the values of [tex]\( x \)[/tex] at which [tex]\( \sin(x) = 0 \)[/tex].
1. Recall that the sine function equals zero at integer multiples of [tex]\(\pi\)[/tex]. That is to say:
[tex]\[
\sin(k\pi) = 0
\][/tex]
for any integer [tex]\( k \)[/tex].
2. Since [tex]\( k \)[/tex] can take any integer value, including negative values, positive values, and zero, this means:
[tex]\[
x = k\pi
\][/tex]
for any integer [tex]\( k \)[/tex].
3. Now, let us review the provided answer choices:
- [tex]\( k\pi \)[/tex] for any positive integer [tex]\( k \)[/tex]
- [tex]\( k\pi \)[/tex] for any integer [tex]\( k \)[/tex]
- [tex]\( \frac{k\pi}{2} \)[/tex] for any positive integer [tex]\( k \)[/tex]
- [tex]\( \frac{k\pi}{2} \)[/tex] for any integer [tex]\( k \)[/tex]
4. We know that the zeros of the sine function specifically occur at integer multiples of [tex]\(\pi\)[/tex]. Therefore, the correct formula should represent all integers [tex]\( k \)[/tex].
The correct formula that gives the zeros of [tex]\( y = \sin(x) \)[/tex] is:
[tex]\[
k\pi \text{ for any integer } k
\][/tex]
