Which formula gives all zeros (x-intercepts) of the function $y=\sin \left(x\right)$?

• A. $x=\frac{\pi }{2}+2n\pi$, where $n$ is any integer
• B. $x=n\pi$, where $n$ is any integer
• C. $x=\frac{\pi }{4}+n\pi$, where $n$ is any integer
• D. $x=\pi +2n\pi$, where $n$ is any integer

Answer: (B)

Verified step by step guidance

1. Recall that the zeros (x-intercepts) of the function \( y = \sin(x) \) occur where \( \sin(x) = 0 \).
2. Identify the values of \( x \) for which \( \sin(x) = 0 \). From the unit circle, \( \sin(x) = 0 \) at integer multiples of \( \pi \), i.e., \( x = n\pi \) where \( n \) is any integer.
3. Understand that the general formula for all zeros of \( \sin(x) \) is therefore \( x = n\pi \), capturing all points where the sine wave crosses the x-axis.
4. Compare this with the given options and recognize that the correct formula matches \( x = n\pi \), where \( n \) is any integer.
5. Note that other options represent shifts or different periodic points that do not correspond to all zeros of \( \sin(x) \).