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<quiz>
  <question type="stack">
    <name>
      <text>5.2.5 (Find solutions of cosine equation exact negative)</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<p>Find all of the solutions to the equation {@eqn@} in the range \(0 < x < 360^\circ\) Give your answers as a list, e.g. if you believe the solutions to be \(x= 10^\circ\) and \(x = 20^\circ\) enter <code>[x = 10,x = 20]</code></p>
<p>[[input:ans1]] [[validation:ans1]]</p>]]></text>
    </questiontext>
    <generalfeedback format="moodle_auto_format">
      <text><![CDATA[<p>Since we have {@eqn@} we see that the value of \(\cos\) must be negative so we must have solutions in the second and third quadrants. In other words the solutions are of the form \(180^\circ + x\) and \(180^\circ - x\). We can obtain the angle corresponding to the first quadrant by using our calculator (or remembering) that 
\[
x = \cos^{-1} {@-1*ang@} 
\]
This gives us \(x = {@round(acosd(-ang))@}^\circ\). Hence to find the solutions of {@eqn@} we have
\[
180^\circ - {@round(acosd(-ang))@}^\circ  = {@soln1@}^\circ \qquad \mbox{ and } \qquad 180^\circ + {@round(acosd(-ang))@}^\circ  = {@soln2@}^\circ,
\]
Thus the solutions are
\[
{@ta1@},
\]
and these are the only solutions in the given range.</p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2020081700</text>
    </stackversion>
    <questionvariables>
      <text>cosd(d):=cos(d*%pi/180);
acosd(x):=acos(x)*180/%pi;

ang: rand([-1/2,-1/sqrt(2),-sqrt(3)/2]);
eqn: cos(x) = ang;

soln1: 180-round(acosd(-ang));
soln2: 180+round(acosd(-ang));

ta1: [x=soln1,x=soln2];</text>
    </questionvariables>
    <specificfeedback format="html">
      <text>[[feedback:prt1]]</text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\({@eqn@} \implies {@ta1@}\)</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
    <questionsimplify>1</questionsimplify>
    <assumepositive>0</assumepositive>
    <assumereal>0</assumereal>
    <prtcorrect format="html">
      <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>
    </prtincorrect>
    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
    <multiplicationsign>dot</multiplicationsign>
    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
    <inversetrig>cos-1</inversetrig>
    <logicsymbol>lang</logicsymbol>
    <matrixparens>[</matrixparens>
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    <input>
      <name>ans1</name>
      <type>algebraic</type>
      <tans>ta1</tans>
      <boxsize>15</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint>[x=?,x=?]</syntaxhint>
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    </input>
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      <name>prt1</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
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        <text/>
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      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>setify(ans1)</sans>
        <tans>setify(ta1)</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>prt1-1-T</trueanswernote>
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        <falseanswernote>prt1-1-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
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    </prt>
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    <deployedseed>350835265</deployedseed>
    <deployedseed>1682188139</deployedseed>
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    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ta1</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt1-1-T</expectedanswernote>
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    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>[x=soln2,x=soln1]</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt1-1-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>3</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>[x=soln1,x=soln1]</value>
      </testinput>
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        <name>prt1</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt1-1-F</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
