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      <text>4.3.1.2 Explore Pythagoras</text>
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      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Using the exact values, evaluate \(\sin^2 \theta + \cos^2 \theta\) for the following angles:</p>
<ol class="HELM_parts">
<li><p>For \(\theta = \displaystyle{{@a1@}}\),   \(\sin^2 \theta + \cos^2 \theta =\) [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p></li>
<li><p>For \(\theta = \displaystyle{{@a2@}}\),   \(\sin^2 \theta + \cos^2 \theta=\) [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p></li>
<li><p>For \(\theta = \displaystyle{{@a3@}}\),   \(\sin^2 \theta + \cos^2 \theta =\) [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p></li></ol>]]></text>
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      <text><![CDATA[<p>Using a table or otherwise, we can find the exact values of \(\sin \theta\) and \(\cos \theta\) for each of these angles.</p>
<ol class="HELM_parts">
<li><p>For \(\theta = {@a1@}\), we have \(\sin\theta = {@sin(a1)@}\) and \(\cos\theta = {@cos(a1)@}\). Then</p>
\[\sin^2 \theta + \cos^2 \theta = \left({@sin(a1)@}\right)^2 + \left({@cos(a1)@}\right)^2 = 1.\]</li>
<li><p>For \(\theta = {@a2@}\), we have \(\sin\theta = {@sin(a2)@}\) and \(\cos\theta = {@cos(a2)@}\). Then</p>
\[\sin^2 \theta + \cos^2 \theta = \left({@sin(a2)@}\right)^2 + \left({@cos(a2)@}\right)^2 = 1.\]</li>
<li><p>For \(\theta = {@a3@}\), we have \(\sin\theta = {@sin(a3)@}\) and \(\cos\theta = {@cos(a3)@}\). Then</p>
\[\sin^2 \theta + \cos^2 \theta = \left({@sin(a3)@}\right)^2 + \left({@cos(a3)@}\right)^2 = 1.\]</li></ol>]]></text>
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      <text>[a1, a2, a3] : rand_selection([pi/6, pi/4, pi/3, pi/2, 2*pi/3, 3*pi/4, 5*pi/6],3);
ta1 : 1;</text>
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      <text>\(\theta = {@a1@}, {@a2@}, {@a3@}\)</text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
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