1. The problem is to understand and analyze the function $\sin x$. 2. The sine function is a fundamental trigonometric function defined for all real numbers $x$ and is periodic with period $2\pi$. 3. The formula for the sine function is $y = \sin x$. 4. Important properties include: - Range: $[-1,1]$ - Periodicity: $\sin(x + 2\pi) = \sin x$ - Zeros at multiples of $\pi$: $x = k\pi$, where $k$ is an integer. 5. The function oscillates smoothly between $-1$ and $1$. 6. To graph or analyze, consider key points such as $0$, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, and $2\pi$. 7. For example, $\sin 0 = 0$, $\sin \frac{\pi}{2} = 1$, $\sin \pi = 0$, $\sin \frac{3\pi}{2} = -1$, and $\sin 2\pi = 0$. 8. This function is widely used in physics, engineering, and mathematics to model periodic phenomena.