##### The Graph of the Function \(y=\sin(t)\)

Although the values of the sine function are dependent upon the graph of the unit circle, the function itself can be graphed. For example, because \(\sin\left(\frac{\pi}{2}\right)=1\) one point that would be on a graph of \(y=\sin(t)\) would be \(\left(\frac{\pi}{2},1\right)\text{.}\)

Several pairs of graphs are about to be presented. It is important that you are mindful of the fact that the variables on the horizontal axes in the graphs of the unit circle are \(x\) whereas the variables on the horizontal axes in the graphs of \(y=\sin(t)\) are \(t\text{.}\)

In FigureĀ 14.4.1 we see that when traveling around the unit circle, as the value of \(t\) increases from \(0\) to \(\frac{\pi}{2}\text{,}\) the \(y\)-coordinate of the point on the unit circle steadily increases from \(0\) to \(1\text{.}\) This means that on the graph of \(y=\sin(t)\text{,}\) as \(t\) increases from \(0\) to \(\frac{\pi}{2}\text{,}\) the function value (\(y\)) will also steadily increase from \(0\) to \(1\text{.}\) This is illustrated in FigureĀ 14.4.2.

In FigureĀ 14.4.3 we see that when traveling around the unit circle, as the value of \(t\) increases from \(\frac{\pi}{2}\) to \(\pi\text{,}\) the \(y\)-coordinate of the point on the unit circle steadily decreases from \(1\) to \(0\text{.}\) This means that on the graph of \(y=\sin(t)\text{,}\) as \(t\) increases from \(\frac{\pi}{2}\) to \(\pi\text{,}\) the function value (\(y\)) will also steadily decrease from \(1\) to \(0\text{.}\) This is illustrated in FigureĀ 14.4.4.

In FigureĀ 14.4.5 we see that when traveling around the unit circle, as the value of \(t\) increases from \(\pi\) to \(\frac{3\pi}{2}\text{,}\) the \(y\)-coordinate of the point on the unit circle steadily decreases from \(0\) to \(-1\text{.}\) This means that on the graph of \(y=\sin(t)\text{,}\) as \(t\) increases from \(\pi\) to \(\frac{3\pi}{2}\text{,}\) the function value (\(y\)) will also steadily decrease from \(0\) to \(-1\text{.}\) This is illustrated in FigureĀ 14.4.6.

In FigureĀ 14.4.7 we see that when traveling around the unit circle, as the value of \(t\) increases from \(\frac{3\pi}{2}\) to \(2\pi\text{,}\) the \(y\)-coordinate of the point on the unit circle steadily increases from \(-1\) to \(0\text{.}\) This means that on the graph of \(y=\sin(t)\text{,}\) as \(t\) increases from \(\frac{3\pi}{2}\) to \(2\pi\text{,}\) the function value (\(y\)) will also steadily increase from \(-1\) to \(0\text{.}\) This is illustrated in FigureĀ 14.4.8.