The graph of the sine function

The graph of the sine function arises from its definition in terms of the unit circle. An angle \(\theta\) subtends an arc length from the right most point on the unit circle to a terminal point on the same circle. That terminal point is shown in red on the shifted 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 orange 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 orange 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.