1. The problem is to understand the trigonometric function $\sin \theta$. 2. $\sin \theta$ represents the ratio of the length of the opposite side to the hypotenuse in a right triangle for an angle $\theta$. 3. It is a periodic function with period $2\pi$, meaning $\sin(\theta + 2\pi) = \sin \theta$. 4. The values of $\sin \theta$ range from $-1$ to $1$. 5. The sine function can be defined on the unit circle as the y-coordinate of the point corresponding to an angle $\theta$ measured from the positive x-axis. 6. Its graph is a smooth wave oscillating between $-1$ and $1$, crossing zero at multiples of $\pi$. 7. Important properties include $\sin 0 = 0$, $\sin \frac{\pi}{2} = 1$, and $\sin \pi = 0$. Final understanding: $\sin \theta$ is a fundamental trigonometric function describing vertical position on the unit circle and oscillating between $-1$ and $1$ with a period of $2\pi$.