The sine function from the unit circle

The graph of the sine function arises from it's definition in terms of the unit circle. An angle $\theta$ subtends an arc length from the point $(1,0)$ to a point shown on the unit circle. That terminal point is shown in red on the unit circle below and the arc is shown in green. By definition, the $y$-coordinate of the red point is $\sin(\theta)$ and its magnitude is represented by the blue line segment.

If we take that green arc and lay it out on the $x$ axis with its initial point at the origin, then its terminal point determines an $x$ value. If we plot the point with that $x$ value and whose $y$ value is given by that same blue segment, we obtain a point on the graph of $y=\sin(x)$. As we vary $\theta$, the graph of the sine function is traced out.