Which of the following lists all the exact solutions in radians to the equation $\sin x=0$?

• A. $x=0+k\pi$, where $k$ is any integer
• B. $x=\frac{\pi }{2}+k\pi$, where $k$ is any integer
• C. $x=k\pi$, where $k$ is any real number
• D. $x=0+k\frac{\pi }{2}$, where $k$ is any integer

Answer: (A)

Verified step by step guidance

1. Recall the general solutions for the equation \( \sin x = 0 \). The sine function equals zero at integer multiples of \( \pi \), that is, \( x = k\pi \) where \( k \) is any integer.
2. Understand that the sine function has zeros at \( 0, \pi, 2\pi, 3\pi, \ldots \) and also at their negative counterparts \( -\pi, -2\pi, -3\pi, \ldots \). This pattern repeats every \( \pi \) radians.
3. Express the solution set as \( x = k\pi \), where \( k \in \mathbb{Z} \) (the set of all integers), to include all these points.
4. Note that solutions like \( x = 0 + k\pi \) and \( x = k\pi \) are equivalent ways to write the same solution set, emphasizing the periodicity of sine zeros.
5. Avoid solutions involving fractions like \( \frac{\pi}{2} + k\pi \) because sine equals \( \pm 1 \) at those points, not zero.