Consider a point \(A(x,y)\) (the green point on the graph) on the unit circle. The function \[f(\theta) = \sin(\theta)\] is defined as the \(y\)-coordinate of \(A\) and \[f(\theta) = \cos(\theta)\] is defined as the \(x\)-coordinate of \(A\), where \(\theta\) represents the angle measured from the \(+x\)-axis counterclockwise to the terminal arm (the line segment joining the centre to \(A\)). If we plot the graph of either \(f(\theta)\) versus \(\theta\) for \(\theta \in [0,2\pi]\), we will get an oscillating curve (the blue curve) which starts from \((0,0)\) and ends at \((2\pi,0)\). This curve represent one cycle of \(\sin(\theta)\) or \(\cos(\theta)\). You may try to move the slider to change the value of \(\theta\). The red line tells the value of \(f(\theta)\).
Courtesy of Anthony Or, Education Bureau, Hong Kong. firstname.lastname@example.org