<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>4.3.5.5 ratio of sum of cosines and sines (NOT IN QUIZ)</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Express \(\dfrac{\cos({@b1@}x)-\cos({@a1@}x)}{\sin({@b1@}x)+\sin({@a1@}x)}\) as \(\tan(nx)\) for an integer \(n\).</p>
<p>\( \dfrac{\cos({@b1@}x)-\cos({@a1@}x)}{\sin({@b1@}x)+\sin({@a1@}x)}=\) [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>We first rewrite the numerator and denominator using the sum of sines and cosines formula.</p>
\[\begin{aligned}
\cos({@b1@}x)-\cos({@a1@}x) &amp;\equiv 2 \sin({@A1@}x)\sin({@B1@}x) \\
\sin({@b1@}x)+\sin({@a1@}x) &amp;\equiv 2\sin({@A1@}x)\cos(-{@B1@}x)
\end{aligned}\]
<p>Using these equations we can simplify the original expression into the required form.</p>
\[\dfrac{\cos({@b1@}x)-\cos({@a1@}x)}{\sin({@b1@}x)+\sin({@a1@}x)}\equiv \dfrac{\sin({@B1@}x)}{\cos(-{@B1@}x)}\equiv 
\dfrac{\sin({@B1@}x)}{\cos({@B1@}x)}\equiv \tan({@B1@}x)\]]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2020120600</text>
    </stackversion>
    <questionvariables>
      <text>b1 : 2+rand(10);
a1 : b1 +4 + 2*rand(4);
A1 : (a1+b1)/2;
B1 : (a1-b1)/2;
ta1 : tan(B1*x);
eq : (cos(b1*x)-cos(a1*x))/(sin(b1*x)+sin(a1*x));</text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\(\frac{\cos({@b1@}x)-\cos({@a1@}x)}{\sin({@b1@}x)+\sin({@a1@}x)} = {@ta1@}\)</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>none</multiplicationsign>
    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
    <inversetrig>cos-1</inversetrig>
    <logicsymbol>lang</logicsymbol>
    <matrixparens>[</matrixparens>
    <isbroken>0</isbroken>
    <variantsselectionseed/>
    <input>
      <name>ans1</name>
      <type>algebraic</type>
      <tans>ta1</tans>
      <boxsize>15</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
      <allowwords/>
      <forbidfloat>1</forbidfloat>
      <requirelowestterms>0</requirelowestterms>
      <checkanswertype>0</checkanswertype>
      <mustverify>1</mustverify>
      <showvalidation>3</showvalidation>
      <options/>
    </input>
    <prt>
      <name>prt1</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans1</sans>
        <tans>ta1</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>0.0000000</truescore>
        <truepenalty/>
        <truenextnode>1</truenextnode>
        <trueanswernote>prt1-1-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt1-1-F</falseanswernote>
        <falsefeedback format="html">
          <text><![CDATA[<p dir="ltr" style="text-align: left;">Your answer is not equivalent to&nbsp;\(\dfrac{\cos({@b1@}x)-\cos({@a1@}x)}{\sin({@b1@}x)+\sin({@a1@}x)}\)<br></p>]]></text>
        </falsefeedback>
      </node>
      <node>
        <name>1</name>
        <description/>
        <answertest>EqualComAss</answertest>
        <sans>ans1</sans>
        <tans>ta1</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>prt1-2-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>-</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt1-2-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <deployedseed>58673735</deployedseed>
    <deployedseed>921088209</deployedseed>
    <deployedseed>1574399937</deployedseed>
    <deployedseed>1034499086</deployedseed>
    <deployedseed>1233517893</deployedseed>
    <deployedseed>2072409301</deployedseed>
    <deployedseed>129765847</deployedseed>
    <deployedseed>2003420304</deployedseed>
    <deployedseed>1354640003</deployedseed>
    <deployedseed>2097026109</deployedseed>
    <deployedseed>1754266772</deployedseed>
    <deployedseed>876377695</deployedseed>
    <deployedseed>270514241</deployedseed>
    <deployedseed>455480419</deployedseed>
    <deployedseed>433774061</deployedseed>
    <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>prt1-2-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>eq</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt1-2-F</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
