The Graph of \(y=\tan(t)\)
Exploring the effects of the quotient identity \(\tan(t)=\frac{\sin(t)}{\cos(t)}\) on the behavior of the tangent function will give us a lot of insight into the graph \(y=\tan(t)\text{.}\) Let's make some initial observations.

There is an \(t\)intercept everywhere \(\sin(t)=0\text{.}\) This gives us the following \(t\)intercepts.
\begin{equation*} ...,\,(2\pi,0),\,(\pi,0),\,(0,0),\,(\pi,0),\,(2\pi,0),\,... \end{equation*} 
The is a vertical asymptote at every value of \(t\) where \(\cos(t)=0\text{.}\) This gives us the following vertical asymptotes.
\begin{equation*} ...,\,t=\frac{5\pi}{2},\,t=\frac{3\pi}{2},\,t=\frac{\pi}{2},\,t=\frac{\pi}{2},\,t=\frac{3\pi}{2},\,t=\frac{5\pi}{2},\,... \end{equation*} 
The \(y\)coordinate of the point on the graph of \(y=\tan(t)\) is either \(1\) or \(1\) at every value of \(t\) where the sine and cosine functions have equal or opposite values. This gives us the following points.
\begin{equation*} ...,\,\left(\frac{5\pi}{4},1\right),\,\left(\frac{3\pi}{4},1\right),\,\left(\frac{\pi}{4},1\right),\,\left(\frac{\pi}{4},1\right),\,\left(\frac{3\pi}{4},1\right),\,\left(\frac{5\pi}{4},1\right),\,... \end{equation*}
Let's go ahead and plot what we've discussed to this point. This is done in FigureĀ 14.5.1.
The sign on the tangent value can only change when \(t\) moves from one quadrant to the next. In FigureĀ 14.5.1 this occurs at the asymptotes and the \(t\)intercepts. We can use this fact to infer the behavior of the function as \(t\) approaches the asymptotes from either side, giving us the function graph shown in FigureĀ 14.5.2.
We should make that unlike the sine and cosine functions, the period of \(y=\tan(t)\) is only \(\pi\text{.}\)