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      <text>5.3.2 (Trig functions are waves)</text>
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      <text><![CDATA[<h3>Trigonometric functions as waves</h3>


<p>We have seen that the trigonometric functions \(\cos(\theta)\) and \(\sin(\theta)\) can be defined formally as the \(x\)- and \(y\)-coordinates respectively of the point of intersection of the half-line making an angle \(\theta\) with the positive \(x\)-axis, and the unit circle (radius \(r =1\)) centred at the origin.</p>

<p><iframe scrolling="no" title="Animated unit circle 3" src="https://www.geogebra.org/material/iframe/id/fx9vakmm/width/661/height/544/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" width="661px" height="544px"> </iframe></p>
<p>We can then draw graphs of \(y = \sin \theta\) and \(y = \cos \theta\) by plotting the values of \(\theta\) agains \( \sin \theta\) and \(\cos \theta\). This produces the following graphs for the two functions. \(\sin x\):<br></p>

<p><img src="@@PLUGINFILE@@/sin%20cropped.png" alt="Graph of y=sin(x)" class="img-responsive atto_image_button_text-bottom" width="700" height="382"></p>
<p>\(\cos x\):</p>

<p><img src="@@PLUGINFILE@@/cos%20cropped.png" alt="Graph of y=cos(x)" class="img-responsive atto_image_button_text-bottom" width="700" height="396"><br></p>

<p>From these functions we can then define the rest of the trigonometric functions that we have met. If we zoomed out these graphs, we would see that this wave form repeats itself for both graphs. The wave oscillates about the horizontal axis (labelled the \(x\)-axis in this case) which is often called the <b>principle axis</b> or <b>mean line</b>. This line has the equation
\[
y  = \frac{max + min}{2}.
\]
A <b>maximum</b> point is the point at the top of each crest and a <b>minimum</b> point is the point at the bottom of a dip. The <b>amplitude</b> is the distance between a maximum (or a minimum) and the principal axis, we can find the amplitude by calculating:
\[
Amplitude = \frac{max-min}{2}.
\] </p>]]></text>
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