Trigonometry originated as the study of triangles, but a more general definition of the trigonometric functions uses the unit circle. For a point \(P\) on the unit circle, we define\(\cos\phi\) as the projection of the point \(P\) onto the \(x\)-axis and \(\sin\phi\) as the projection of the point \(P\) onto the \(y\)-axis, where \(\phi\) is the usual angle around the origin, measured counterclockwise from the positive \(x\)-axis. This construction is shown in Figure 2.6.
Figure2.6.The coordinates of a point \(P\) on the unit circle are \((x,y)=(\cos\phi,\sin\phi)\)
Activity2.3.The Relationship between the Unit Circle and Graphs of \(\boldsymbol{\cos\phi}\) and \(\boldsymbol{\sin\phi}\).
To the right of a plot of the unit circle, inspired by Figure 2.6, make a plot of \(\sin\phi\) vs. \(\phi\text{.}\) Align the vertical dimensions and spacing on the two plots so that it’s easy to see the relationship. To see a similar relationship for \(\cos\phi\text{,}\) where should you place the plot of \(\cos\phi\) vs. \(\phi\text{?}\)
Solution.
The graphs of \(\cos\phi\) vs. \(\phi\) should be turned sideways and placed below the unit circle, as illustrated in Figure 2.7. Figure2.7.This combined figure shows the relationship between a point \(P\) moving around the unit circle and the (rectangular) graphs of \(\cos\phi\) and \(\sin\phi\text{,}\) plotted as functions of \(\phi\text{.}\)
