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      <text>5.3.11 (Graph of tan)</text>
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      <text><![CDATA[<h3>The graph of the tangent function</h3>

<p>We have previously seen that we can define the tangent function as 
\[
\tan x =\frac{\sin x}{\cos x},
\]
and we have noted that \(\tan\) is positive in the first and third quadrant. From this definition we can also note that the graph of \(y = \tan x\) is zero whenever \(\sin x\) is zero and it will be undefined whenever \(\cos x\) is zero. However, we know precisely the points where the sine and cosine functions are zero:
\[
\sin x = 0 \implies x = 180k^\circ \qquad \mbox{and} \qquad \cos x = 0 \implies x = 90^\circ +180k^\circ \: \mbox{for } k \in \mathbb{N}.
\]
We can use this information to plot the function \(y = \tan x\).
</p>
<br/>
<p>The graph of \(\tan x\) against \(x\) is plotted in the figure below. Note that whereas \(y=\sin x\) and \(y =\cos\theta\) have period \(360^\circ\), \(y=\tan x\) has period \(180^\circ\). Notice the graph repeats every \(180^\circ)\), and has vertical asymptotes at \(90^\circ+180k^\circ\) (which correspond to the point where \(\cos x = 0\)) and \(x\)-intercepts at \(180k^\circ\) for \(k\in \mathbb{N}\).</p>
<p> </p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Graph%20of%20tangent%20function.png?time=1601460391933" alt="graph of tangent function" width="585" height="452" /></p>]]></text>
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<file name="Graph of tangent function.png" path="/" encoding="base64">[encoded data, 115628 characters]</file>
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