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      <text>5.3.7 (Plotting the cosine function)</text>
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      <text><![CDATA[<h3>The general cosine function</h3>
<p> We proceed in a similar way to how we plotted the general sine function.
Since we have defined \(\cos x\) in terms of the coordinates of a point on a circle of unit radius it follows immediately that \(-1\le\sin x \le +1\qquad\text{and}\qquad -1\le \cos x \le +1\) for any value of \(x\). Therefore the amplitude of the cosine graph is \(1\) and the principal axis is the line \(y = 0\).</p>
<p> We can draw the graph of the cosine function, see the figure below. We have labelled the horizontal axis using degrees and have shown two periods. Notice the graph of \(\cos(x)\) also repeats every \(360^\circ\), with maximum values at \(360k^\circ\) and minimum values at \(180^\circ+360k^\circ\) for \(k \in \mathbb{N}\). In addition, notice that \(f(x) = \cos x\) is an even function because \(f(x) = f(-x)\), in other words it is symmetrical about the vertical axis.</p>

<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Labelled%20cosine%20wave.png" alt="labelled diagram of a cos function" width="403" height="259" /></p>

<p><strong>The general cosine function</strong> is the function: \[ y = a \cos b(x-c) + d. \] We will see that \(a\) affects the amplitude of the graph, \(b\) affects the period, \(c\) affects the horizontal translation and \(d\) affects vertical translation. Move the sliders below to see what effect altering each of these values has on the cosine function.</p>
<p> </p>
<p><iframe style="border: 0px;" title="Cosine sliders" src="https://www.geogebra.org/material/iframe/id/sns8hnfw/width/1920/height/964/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" scrolling="no" width="900px" height="450px"> </iframe></p>
<p> </p>
<p>Also notice the graph of \(\cos(x)\) is obtained by shifting the graph of \(\sin(x)\) by \(90^\circ\) to the left. In the plots above, we have extended the graphs to negative values of <span class="math inline">\(x\)</span> using the relations <span class="math inline">\(\sin(-x)=-\sin x,\ \ \cos(-x)=\cos x.\)</span> Both graphs could be extended indefinitely to the left (<span class="math inline">\(x \to -\infty\)</span>) and right (<span class="math inline">\(x \to +\infty)\)</span>.</p>
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<p>We can obtain expressions for \(\cos x\) in terms of the sine function and for \(\sin x\) in terms of the cosine function.</p>
<p>Using the sum identity \(\cos (A-B)\equiv \cos A\cos B+\sin A\sin B\) with \(A=x,\ B=90^\circ\) we obtain 
\[
\cos\left(x-90^\circ \right)\equiv\cos x\ \cos\left(90^\circ \right)+\sin x
\sin \left( 90^\circ \right)\equiv \cos x\ (0)+\sin x
\]
i.e. \(\sin x \equiv\cos\left(x-90^\circ \right)\equiv\cos\left(90^\circ-x\right)\). This result explains why the graph of \(\sin x\) has exactly the same shape as the graph of \(\cos x\) but it is shifted to the left by \(90^\circ\). A similar calculation yields \(\cos x\equiv\sin\left(x+90^\circ\right).\)</p>]]></text>
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