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  <question type="stack">
    <name>
      <text>4.5.1.5  Find a period and frequency</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>We know that the <b>shape</b> of the cosine graph and the sine graph are identical but offset by \(\pi/2\) radians horizontally to the right. Bearing this in mind, attempt the following question.</p>
<p>Write the equation of the wave \(x(t)\), part of which is shown in the following graph. You will need to find the period \(T\), the angular frequency \(\omega\) and the amplitude \(A\) first.</p>
<p>Please write "pi" for the number \( \pi \).</p>
<figure class="HELM_figure">
{@plot(f, [x,0,2*T+1], [y,-A-1,A+1])@}
</figure>
<p>\(T = \) [[input:ans1]] \(\mathrm{s}\) [[validation:ans1]] [[feedback:prt1]]</p>
<p>\(\omega = \) [[input:ans2]] \(\mathrm{rad}/\mathrm{s}\) [[validation:ans2]] [[feedback:prt2]]</p>
<p>\(A =\) [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p>
<p>\( x = \) [[input:ans4]] [[validation:ans4]] [[feedback:prt4]]</p>
<p>The quantity \(x\), a function of \(t\), is referred to as the <b>displacement</b> of the wave.</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>From the shape of the graph we have a <b>sine</b> wave rather than a cosine wave.</p>
<p>The amplitude is \(A = {@A@}\). The period \(T = {@T@}\ \mathrm{s}\), so the angular frequency is</p>
\[\omega = \frac{2\pi}{T} = {@o@}\ \mathrm{rad}/\mathrm{s}.\]
<p>Hence,</p>
\[x = {@ta4@}.\]]]></text>
    </generalfeedback>
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    <idnumber/>
    <stackversion>
      <text>2024072400</text>
    </stackversion>
    <questionvariables>
      <text>T: rand_with_step(2, 5, 1);
o: 2 * %pi / T;
A: rand_with_step(3, 6, 1);
f: A * sin(o * x);

ta1: T;
ta2: o;
ta3: A;
ta4: subst(t, x, f);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\(x = {@ta4@}\)</text>
    </questionnote>
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      <text/>
    </questiondescription>
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      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
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    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
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    <complexno>i</complexno>
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        <tans>ans3*sin(ans2*t)</tans>
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          <text><![CDATA[<p dir="ltr" style="text-align: left;">Your answer is correct based on your earlier values.</p>]]></text>
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    </qtest>
  </question>
</quiz>
