The Graph of Tangent Function

Because the graph of $y=\tan x$ is useful for reading off values over its repeated period, and tangent is ODD (so its graph is symmetric about the origin) with the SHORT period $\pi$, it suffices to work out its shape on ONE interval of length $\pi$ — take $\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)$.

Table of values:

Plotting and joining these gives Fig. 4.15: as $x$ approaches $\tfrac{\pi}2$ from below, $\sin x\to1$ while $\cos x\to0^+$, so $\tfrac{\sin x}{\cos x}\to+\infty$ — the vertical line $x=\tfrac{\pi}2$ is a vertical asymptote. Symmetrically, as $x\to-\tfrac{\pi}2$ from above, $\tan x\to-\infty$, so $x=-\tfrac{\pi}2$ is also an asymptote. $\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)$ is called the principal domain of tangent. …