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      <text>4.2.2.9 Beyond the fourth quadrant</text>
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      <text><![CDATA[<h4>  Rotation beyond the fourth quadrant <span class="math inline">\((360^\circ&lt;\theta)\)</span>  </h4>

If the vector <span class="math inline">\(OP\)</span> continues to rotate around the circle of unit radius then in the next complete rotation <span class="math inline">\(\theta\)</span> increases from <span class="math inline">\(360^\circ\)</span> to <span class="math inline">\(720^\circ\)</span>. However, a <span class="math inline">\(\theta\)</span> value of, say, <span class="math inline">\(405^\circ\)</span> is indistinguishable from one of <span class="math inline">\(45^\circ\)</span> (just one extra complete revolution is involved).</p>
<p>So <span class="math inline">\(\sin(405^\circ)=\sin(45^\circ)=\dfrac{1}{\sqrt{2}} \text{ and } 
\cos(405^\circ)=\cos(45^\circ)=\dfrac{1}{\sqrt{2}}\)</span></p>
<p>In general <span class="math inline">\(\sin(360^\circ+x^\circ)=\sin(x^\circ),
\cos(360^\circ+x^\circ)=\cos(x^\circ)\)</span></p>

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<h4>Key Point</h4>
<p>If \(n\) is any integer, then</p>
\[\begin{aligned}
\sin (x^\circ+360 n^\circ) &amp;\equiv \sin(x^\circ) \\
\cos (x^\circ+360 n^\circ) &amp;\equiv \cos(x^\circ)
\end{aligned}\]
<p>or, since \(360^\circ\equiv2\pi \text{ radians}\),
\[\begin{aligned}
\sin (x + 2\pi n) &amp;\equiv \sin(x) \\
\cos (x + 2\pi n) &amp;\equiv \cos(x)
\end{aligned}\]
<p>We say that the functions \(\sin(x)\) and \(\cos(x)\) are <strong>periodic</strong> with period (in radian measure) of \(2\pi\).</p>
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