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    <name>
      <text>10.3.4.4 Cartesian form to exponential form (NOT IN QUIZ)</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Find the exponential form of the complex number \(z = \displaystyle {@re@} + \displaystyle \left({@im@}\right)i\). Give the argument of \(z\) in terms of \(\pi\), i.e. not in decimal form.</p>
<p>\(z =\quad\)[[input:ans1]][[validation:ans1]][[feedback:prt1]]</p>
<p>Therefore, find the Cartesian form of \(z^{-1}\). Give your answers in decimal form, rounding to 3 decimal places if necessary:</p>
<p>\(z^{-1} =\quad\)[[input:ans2]][[validation:ans2]]\(\quad+\quad i\) [[input:ans3]][[validation:ans3]]</p>
[[feedback:prt2]]]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p> We want to calculate \(z\) in the form \(re^{i\theta}\). We calculate \(\displaystyle{r = \sqrt{\left({@re@}\right)^2 + \left({@im@}\right)^2} = {@arr@}}\) and \(\displaystyle{ \theta = \tan^{-1}\left({@im@}/{@re@}\right) = {@th@} }\). Therefore \[z = {@arr@}e^{{@th@}i}.\] To calculate the inverse, \(z^{-1}\),
    in Cartesian form we use laws of indices and then apply a Key Point from this workbook: \[z^{-1} = {@ta1@}^{-1} = {@arr@}^{-1}e^{-i({@th@})} = \dfrac{1}{{@arr@}}\left(\cos\left(-{@th@}\right) + i\sin\left(-{@th@}\right)\right) = {@ta2@} + {@ta3@}i.\]</p>]]></text>
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    <idnumber/>
    <stackversion>
      <text>2024072400</text>
    </stackversion>
    <questionvariables>
      <text>th: rand([%pi/3,%pi/4,-%pi/3,-%pi/4]);
arr: 10 - rand(9);
re: arr*cos(th);
im: arr*sin(th);
ta2: 0.001*round(1000*cos(-th)/arr);
ta3: 0.001*round(1000*sin(-th)/arr);
simp:false;
ta1: arr*e^(%i*th);</text>
    </questionvariables>
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      <text/>
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    <questionnote format="moodle_auto_format">
      <text>\[ z = {@re@} + ({@im@})i = {@ta1@}  \]
\[ z^{-1} = {@ta2@} + ({@ta3@})i \]</text>
    </questionnote>
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      <text/>
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      <text><![CDATA[<span style="font-size: 1.5em; color: green; --darkreader-inline-color:#7fff7f;" data-darkreader-inline-color=""><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
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      <text><![CDATA[<span style="font-size: 1.5em; color: orange; --darkreader-inline-color:#ffb733;" data-darkreader-inline-color=""><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
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      <text><![CDATA[<span style="font-size: 1.5em; color: red; --darkreader-inline-color:#ff3333;" data-darkreader-inline-color=""><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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    <qtest>
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