Tangent And Cotangent Graphs | Brilliant Math & Science Wiki

From the definition of the tangent and cotangent functions, we have

\[ \tan( \theta)= \frac{\sin(\theta)}{\cos(\theta)},\quad \cot( \theta)= \frac{\cos(\theta)}{\sin(\theta)}.\]

Thus, \(\tan(\theta)\) is not defined for values of \(\theta\) such that \(\cos(\theta) = 0\). Now, consider the graph of \(\cos (\theta)\):

From this graph, we see that \(\cos(\theta) = 0\) when \(\theta = \frac{\pi}{2} + k\pi\) for any integer \(k\). This implies that the tangent function has vertical asymptotes at these values of \(\theta\).

Does the tangent function approach positive or negative infinity at these asymptotes? As \(\theta\) approaches \(\frac{\pi}{2}\) from below \(\big(\theta\) takes values less than \(\frac{\pi}{2}\) while getting closer and closer to \(\frac{\pi}{2}\big),\) \(\sin (\theta) \) takes positive values that are closer and closer to \(1\), while \(\cos (\theta)\) takes positive values that are closer and closer to \(0\). This shows \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\) is positive and approaches infinity, so \(\tan(\theta)\) has a positive vertical asymptote as \(\theta \rightarrow \frac{\pi}{2} \) from below. By a similar analysis, as \(\theta\) approaches \(\frac{\pi}{2}\) from above \(\big(\theta\) takes values larger than \(\frac{\pi}{2}\) while getting closer and closer to \(\frac{\pi}{2}\big),\) \(\sin (\theta) \) takes positive values that are closer and closer to \(1\), while \(\cos (\theta)\) takes negative values that are closer and closer to \(0\). This shows \(\tan(\theta)\) has a negative vertical asymptote as \(\theta \rightarrow \frac{\pi}{2} \) from above. The following shows the graph of tangent for the domain \(0 \leq \theta \leq 2\pi\):

The graph of tangent over its entire domain is as follows:

Similarly, \(\cot(\theta)\) is not defined for values of \(\theta\) such that \(\sin(\theta) = 0\). From the graph of \(\sin (\theta),\) we see that \(\sin(\theta) = 0\) when \(\theta = 0 + k\pi\) for any integer \(k\), which implies that the cotangent function has vertical asymptotes at these values of \(\theta:\)