qu.1.topic=Degrees to Radians@

qu.1.1.question=<p>Convert <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$b</mi><mo>&deg;</mo></mrow></math>to radian measure. Round your answer to two decimal places.</p>@
qu.1.1.maple=evalb(abs(($ANSWER)-($RESPONSE))<.005);@
qu.1.1.allow2d=1@
qu.1.1.maple_answer=$ANS@
qu.1.1.type=formula@
qu.1.1.mode=Maple@
qu.1.1.name=Degrees to Radians decimal@
qu.1.1.comment=<p>Multiple the degree measure by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mi>&pi;</mi></mrow><mrow><mn>180</mn></mrow></mfrac></mrow></math>.</p>@
qu.1.1.editing=useHTML@
qu.1.1.hint.1=You know you have to multiple by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mi>&pi;</mi></mrow><mrow><mn>180</mn></mrow></mfrac></mrow></math>or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mn>180</mn></mrow><mrow><mi>&pi;</mi></mrow></mfrac></mrow></math>.@
qu.1.1.hint.2=If you can't remember which to choose, think about converting <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>180</mn></mrow></math> degrees.@
qu.1.1.solution=@
qu.1.1.algorithm=$a=rint(1,360,13);
$n=rint(2);
$b=(-1)^($n)*($a);
$ANS=$b*Pi/180;@
qu.1.1.uid=3341fcbd-5a51-48e1-a402-7c8fb9544853@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Trigonometry Review;
  Difficulty=Easy;
  Sub-Topic=Radian Measure;
@

qu.1.2.question=<p>Convert&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$b</mi><mo>&deg;</mo></mrow></math><sup> </sup>to radian measure. (Your answer should be exact. Enter Pi for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&pi;</mi></mrow></math>.)</p>@
qu.1.2.maple=evalb(($ANSWER)-($RESPONSE) = 0);@
qu.1.2.allow2d=1@
qu.1.2.maple_answer=$ANS;@
qu.1.2.type=formula@
qu.1.2.mode=Maple@
qu.1.2.name=Degrees to Radians PI@
qu.1.2.comment=<p>Multiple the degree measure by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mi>&pi;</mi></mrow><mrow><mn>180</mn></mrow></mfrac></mrow></math>.</p>@
qu.1.2.editing=useHTML@
qu.1.2.hint.1=You know you have to multiple by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mi>&pi;</mi></mrow><mrow><mn>180</mn></mrow></mfrac></mrow></math>or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mn>180</mn></mrow><mrow><mi>&pi;</mi></mrow></mfrac></mrow></math>.@
qu.1.2.hint.2=If you can't remember which to&nbsp;choose, think about converting <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>180</mn></mrow></math> degrees.&nbsp@
qu.1.2.solution=@
qu.1.2.algorithm=$a=switch(rint(17),0,30,45,60,90,120,135,150,180,210,225,240,270,300,315,330,360);
$n=rint(2);
$b=(-1)^$n*$a;
$ANS=maple("$b*Pi/180");@
qu.1.2.uid=28965a99-2bb4-43ae-8444-585fe8ebdb4f@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Topic=Trigonometry Review;
  Course=Introduction to Calculus I;
  Sub-Topic=Radian Measure;
  Difficulty=Easy;
@

qu.2.topic=Radians to Degrees@

qu.2.1.question=<p>Convert the radian measure <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$b</mi></mrow></math> to degrees. Round your answer to two decimal places. (Omit the degree symbol in your answer.)</p>@
qu.2.1.maple=evalb(abs(($ANSWER)-($RESPONSE))<.005);@
qu.2.1.allow2d=1@
qu.2.1.maple_answer=$ANS@
qu.2.1.type=formula@
qu.2.1.mode=Maple@
qu.2.1.name=Radians to Degrees decimal@
qu.2.1.comment=<p>Multiply the radian measure by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mn>180</mn></mrow><mrow><mi>&pi;</mi></mrow></mfrac></mrow></math>.</p>@
qu.2.1.editing=useHTML@
qu.2.1.hint.1=You&nbsp;know you have to multiple by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mi>&pi;</mi></mrow><mrow><mn>180</mn></mrow></mfrac></mrow></math>or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mn>180</mn></mrow><mrow><mi>&pi;</mi></mrow></mfrac></mrow></math>.@
qu.2.1.hint.2=If you can't remember which to choose, think about converting&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&pi;</mi></mrow></math>&nbsp;radians.@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(1,40);
$b=$a/10;
$n=rint(2);
$c=(-1)^($n)*($b);
$c1=mathml("$c");
$ANS=$b*180/Pi;@
qu.2.1.uid=74994830-6338-4365-8f05-29bc9d397a72@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Trigonometry Review;
  Sub-Topic=Radian Measure;
  Difficulty=Easy;
@

qu.2.2.question=<p>Convert the radian measure $b1 to degrees. (Omit the degree symbol in your answer.)</p>@
qu.2.2.maple=evalb(($ANSWER)-($RESPONSE)= 0);@
qu.2.2.allow2d=1@
qu.2.2.maple_answer=$ANS@
qu.2.2.type=formula@
qu.2.2.mode=Maple@
qu.2.2.name=Radians to Degrees PI@
qu.2.2.comment=<p>Multiple the radian measure by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mn>180</mn></mrow><mrow><mi>&pi;</mi></mrow></mfrac></mrow></math>.</p>@
qu.2.2.editing=useHTML@
qu.2.2.hint.1=You know you have to multiple by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mi>&pi;</mi></mrow><mrow><mn>180</mn></mrow></mfrac></mrow></math>or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mn>180</mn></mrow><mrow><mi>&pi;</mi></mrow></mfrac></mrow></math>.@
qu.2.2.hint.2=If you can't remember think about converting <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>radians.@
qu.2.2.solution=@
qu.2.2.algorithm=$a=switch(rint(17),"0","Pi/6","Pi/4","Pi/3","Pi/2","2*Pi/3","3*Pi/4",
"5*Pi/6","Pi","7*Pi/6","5*Pi/4","4*Pi/3","3*Pi/2","5*Pi/3","7*Pi/4",
"11*Pi/6","2*Pi");
$n=rint(2);
$b="(-1)^($n)*($a)";
$b1=maple("printf(MathML[ExportPresentation]($b))");
$ANS=$b*180/Pi;@
qu.2.2.uid=9b719790-be0d-4124-b76a-478c8aecb6ac@
qu.2.2.info=  Course=Introduction to Calculus I;
  Topic=Trigonometry Review;
  Sub-Topic=Radian Measure;
  Difficulty=Easy;
  Author=Jack Weiner, Gord Clement;
@

qu.3.topic=Evaluate Trig(theta)@

qu.3.1.question=<p>Find the exact value of $F. If the answer is not finite, enter U (for undefined!)</p>@
qu.3.1.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.3.1.allow2d=1@
qu.3.1.maple_answer=if ($f=sec and (abs($b)=1/2*Pi or abs($b)=3*Pi/2)) or 
(($f=csc or $f=cot) and (abs($b)=0 or abs($b)=Pi or abs($b)=2*Pi)) then U 
else $f($b) end if@
qu.3.1.type=formula@
qu.3.1.mode=Maple@
qu.3.1.name=ExactCSCSECCOT@
qu.3.1.comment=@
qu.3.1.editing=useHTML@
qu.3.1.hint.1=Remember the special triangles.@
qu.3.1.solution=@
qu.3.1.algorithm=$a=switch(rint(17),"0","Pi/6","Pi/4","Pi/3","Pi/2","2*Pi/3","3*Pi/4",
"5*Pi/6","Pi","7*Pi/6","5*Pi/4","4*Pi/3","3*Pi/2","5*Pi/3","7*Pi/4",
"11*Pi/6","2*Pi");
$n=rint(2);
$b="(-1)^($n)*($a)";
$f=switch(rint(3),"csc","sec","cot");
$F=maple("printf(MathML[ExportPresentation]('$f($b)'))");@
qu.3.1.uid=0a707d85-797e-4b88-8b0a-80bca2f24a1c@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Trigonometry Review;
  Sub-Topic=Evaluation of Trig Function;
  Difficulty=Easy;
@

qu.3.2.question=<p>$F1 equals either $F2 or $F3. Which?</p>@
qu.3.2.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.3.2.allow2d=1@
qu.3.2.maple_answer=$f(-x)@
qu.3.2.type=formula@
qu.3.2.mode=Maple@
qu.3.2.name=trig(-x)@
qu.3.2.comment=@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$f=switch(rint(6),"sin","cos","tan","csc","sec","cot");
$F1=mathml("$f(-x)");
$F2=mathml("-$f(x)");
$F3=mathml("$f(x)");@
qu.3.2.uid=5925c971-1079-49e8-8750-f54d8a8b5333@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Trigonometry Review;
  Sub-Topic=Trig functions: Odd or Even;
  Difficulty=Easy;
@

qu.3.3.question=<p>FInd the exact value of $F. If the answer is not finite, enter U (for undefined!)</p>@
qu.3.3.maple=evalb(($ANSWER)=($RESPONSE));@
qu.3.3.allow2d=1@
qu.3.3.maple_answer=if ($f=tan and (abs($b)=1/2*Pi or abs($b)=3*Pi/2) ) then U else $f($b) end if@
qu.3.3.type=formula@
qu.3.3.mode=Maple@
qu.3.3.name=ExactSINCOSTAN@
qu.3.3.comment=@
qu.3.3.editing=useHTML@
qu.3.3.hint.1=Remember your special triangles.@
qu.3.3.solution=@
qu.3.3.algorithm=$a=switch(rint(17),"0","Pi/6","Pi/4","Pi/3","Pi/2","2*Pi/3","3*Pi/4",
"5*Pi/6","Pi","7*Pi/6","5*Pi/4","4*Pi/3","3*Pi/2","5*Pi/3","7*Pi/4",
"11*Pi/6","2*Pi");
$n=rint(2);
$b="(-1)^($n)*($a)";
$f=switch(rint(3),"sin","cos","tan");
$F=maple("printf(MathML[ExportPresentation]('$f($b)'))");@
qu.3.3.uid=83398eba-ed58-448b-a794-3afc784e11f9@
qu.3.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Trigonometry Review;
  Sub-Topic=Evaluating Trig Functions;
  Difficulty=Easy;
@

qu.4.topic=PeriodTrig@

qu.4.1.question=<p>State, <strong>IN RADIANS, </strong>the exact period of the function $F.</p>@
qu.4.1.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.4.1.allow2d=1@
qu.4.1.maple_answer=$ANS@
qu.4.1.type=formula@
qu.4.1.mode=Maple@
qu.4.1.name=PeriodSINCOS@
qu.4.1.comment=<p>$F is the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> with a horizontal $type by a factor of $size.</p>
<p>Therefore the period is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>2</mn><mi>&pi;</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&times;</mo></mrow></math>$size.</p>@
qu.4.1.editing=useHTML@
qu.4.1.hint.1=Horizontal stretch or compression?@
qu.4.1.solution=@
qu.4.1.algorithm=$a=range(2,5);
$f=switch(rint(4),"sin","cos","csc","sec");
$z=rint(2);
$b=switch($z,"$a*x","(1/$a)*x");
$c=mathml("$b");
$F=mathml("$f($b)");
$ANS=switch($z,"2*Pi/$a", "2*$a*Pi");
$type=switch($z, "compression", "stretch");
$size=switch($z, mathml("1/$a"),mathml("$a"));@
qu.4.1.uid=d9248f82-6b2f-4165-b954-1f422d7e20c4@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Trigonometry Review;
  Sub-Topic=Period of Trig Functions;
  Difficulty=Easy;
@

qu.4.2.question=<p>State, <strong>IN RADIANS</strong>, the exact period of the function $F.</p>@
qu.4.2.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.4.2.allow2d=1@
qu.4.2.maple_answer=$ANS@
qu.4.2.type=formula@
qu.4.2.mode=Maple@
qu.4.2.name=PeriodTANCOT@
qu.4.2.comment=<p>$F is the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> with a horizontal $type by a factor of $size.</p>
<p>Therefore, the period is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&times;</mo></mrow><mrow><mi></mi></mrow></math>$size.</p>@
qu.4.2.editing=useHTML@
qu.4.2.hint.1=Horizontal stretch or compression?@
qu.4.2.solution=@
qu.4.2.algorithm=$a=range(2,5);
$z=rint(2);
$f=switch(rint(2),"tan","cot");
$b=switch($z,"$a*x","(1/$a)*x");
$c=mathml("$b");
$ANS=switch($z,"Pi/($a)","($a)*Pi");
$F=mathml("$f($b)");
$type=switch($z,"compression", "stretch");
$size=switch($z, mathml("1/$a"),"$a");@
qu.4.2.uid=9ac8951f-6c18-4dea-8f3e-77345447e552@
qu.4.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Trigonometry Review;
  Sub-Topic=Period of Trig Functions;
  Difficulty=Easy;
@

qu.5.topic=RangeTrig@

qu.5.1.question=<p>Find the range of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$Q.</p>
<p>Note: Use the letter U for union. For <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>type infinity.</p>
<p>eg: for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mn>5</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math>, you would enter (-infinity,3) U [5,infinity).</p>@
qu.5.1.maple=grade("$RESPONSE","$ANS");@
qu.5.1.allow2d=0@
qu.5.1.maple_answer=show("$ANS");@
qu.5.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.5.1.type=maple@
qu.5.1.mode=Maple@
qu.5.1.name=RangeSinCosIntervalNotation@
qu.5.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>1</mn></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$f</mi><mfenced open='(' close=')' separators=','><mrow><mi>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>1</mn></mrow><mrow></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$sol</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$sol</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$a</mi></mrow></math> (Did the inequality signs change directions?)</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$l</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi mathvariant='normal'></mi></mrow></math>$Q <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$r</mi></mrow></math></p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$A=range(1,5);
$sign = range(0,1);
$a=switch($sign,$A,-$A);
$b=range(2,8);
$c=range(-4,4);
$f=switch(rint(2),'sin','cos');
condition:ne(($a)*($c),0);
$Q=maple("printf(MathML[ExportPresentation](($a)*$f(($b)*x)+($c)))");
$l=-abs($a)+$c;
$r=abs($a)+$c;
$ANS="[$l,$r]";
$sol=switch($sign,"?","?");@
qu.5.1.uid=5508f430-83f6-4fc9-9353-333cc67c70b4@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Trigonometry Review;
  Sub-Topic=Range of Trig Functions;
  Difficulty=Easy;
  Feature=Interval Answer;
@

qu.6.topic=Trig Graphs@

qu.6.1.mode=Multiple Choice@
qu.6.1.name=Identify Trig Graph@
qu.6.1.comment=@
qu.6.1.editing=useHTML@
qu.6.1.solution=@
qu.6.1.algorithm=$f=switch(rint(6),sin(x),cos(x),tan(x),csc(x),sec(x),cot(x));
$m = maple(" 
randomize();
b:=RandomTools[Generate](choose({sin(x),cos(x),tan(x),csc(x),sec(x),cot(x)}minus {$f})):
c:=RandomTools[Generate](choose({sin(x),cos(x),tan(x),csc(x),sec(x),cot(x)}minus {$f,b})):
d:=RandomTools[Generate](choose({sin(x),cos(x),tan(x),csc(x),sec(x),cot(x)}minus {$f,b,c})):
e:=RandomTools[Generate](choose({sin(x),cos(x),tan(x),csc(x),sec(x),cot(x)}minus {$f,b,c,d})):
f:=MathML[ExportPresentation](y=$f):
b,c,d,e,f
");
$B=switch(0,$m);
$C=switch(1,$m);
$D=switch(2,$m);
$E=switch(3,$m);
$F=switch(4,$m);
$pa=plotmaple("plot($f,x=-2*Pi..2*Pi,y=-2..2,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");
$pb=plotmaple("plot($B,x=-2*Pi..2*Pi,y=-2..2,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");
$pc=plotmaple("plot($C,x=-2*Pi..2*Pi,y=-2..2,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");
$pd=plotmaple("plot($D,x=-2*Pi..2*Pi,y=-2..2,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");
$pe=plotmaple("plot($E,x=-2*Pi..2*Pi,y=-2..2,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.6.1.uid=67ed71da-1979-41d3-9fc7-185554220b20@
qu.6.1.info=  Topic=Trigonometry Review;
  Sub-Topic=Graphing Trig Functions;
  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
@
qu.6.1.question=<p>Which of the follwing is the graph of $F?</p>@
qu.6.1.answer=1@
qu.6.1.choice.1=$pa@
qu.6.1.choice.2=$pb@
qu.6.1.choice.3=$pc@
qu.6.1.choice.4=$pd@
qu.6.1.choice.5=$pe@
qu.6.1.fixed=@

qu.7.topic=SinCosLaws@

qu.7.1.question=<p>&nbsp;</p>
<p><img width="196" height="130" src="__BASE_URI__pictures/ABCtriangle.JPG" alt="" /></p>
<p>In the triangle (not drawn to scale!), side <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, side <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>, and in degrees, angle <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$C</mi><mo>&deg;</mo></mrow></math>. Using the Cosine Law, find side <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow></math>. Round your answer to two decimal places.</p>@
qu.7.1.maple=evalb(abs(evalf(($ANSWER)-($RESPONSE)))<0.005);@
qu.7.1.allow2d=1@
qu.7.1.maple_answer=$ANS@
qu.7.1.type=formula@
qu.7.1.mode=Maple@
qu.7.1.name=cos lawSAS@
qu.7.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>a</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>ab</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>C</mi></mrow></mfenced></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$C</mi></mrow></mfenced></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$ANS</mi></mrow></math></p>
<p>Make sure your calculator is in 'Degree' mode.</p>@
qu.7.1.editing=useHTML@
qu.7.1.hint.1=Make&nbsp;sure your calculator is in&nbsp;'Degree'&nbsp;mode.&nbsp@
qu.7.1.solution=@
qu.7.1.algorithm=$a=rint(1,10);
$b=rint(1,10);
$C=rint(5,175,5);
$ANS=sqrt(($a)^2+($b)^2-2*($a)*($b)*cos($C*Pi/180));@
qu.7.1.uid=3f1da51f-1571-4a94-8148-0e512a0967ec@
qu.7.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Trigonometry Review;
  Sub-Topic=Cosine Law;
  Difficulty=Easy;
@

qu.7.2.question=<p><img width="196" height="130" src="__BASE_URI__pictures/ABCtriangle.JPG" alt="" /></p>
<p>In the triangle (not drawn to scale!), angle <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$A</mi><mo>&deg;</mo></mrow></math>, angle <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$B</mi><mo>&deg;</mo></mrow></math>, and side <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>. Find the length of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi></mrow></math> rounded to two decimal places.</p>@
qu.7.2.maple=evalb(abs(evalf(($ANSWER)-($RESPONSE)))<0.005);@
qu.7.2.allow2d=1@
qu.7.2.maple_answer=$ANS@
qu.7.2.type=formula@
qu.7.2.mode=Maple@
qu.7.2.name=sin lawAAS@
qu.7.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mi>b</mi></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow></mfrac></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mi>b</mi></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$B</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$A</mi></mrow></mfenced></mrow></mfrac></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$ANS</mi></mrow></math></p>@
qu.7.2.editing=useHTML@
qu.7.2.solution=@
qu.7.2.algorithm=$a=rint(1,10);
$A=rint(5,85,5);
$B=rint(5,175,5);
condition: lt($A+$B-180,0);
$ANS=($a)*sin($B*Pi/180)/$sin($A*Pi/180);@
qu.7.2.uid=01b251b6-ec2d-4423-947d-7828563ad480@
qu.7.2.info=  Author=Jack Weiner, Gord Clement;
  Topic=Trigonometry Review;
  Sub-Topic=Sine Law;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
@

qu.7.3.question=<p>&nbsp;</p>
<p><img width="196" height="130" src="__BASE_URI__pictures/ABCtriangle.JPG" alt="" /></p>
<p>In the triangle (not drawn to scale!), side <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, side <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>, and side <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>c</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></math>. Using the Cosine Law, find angle <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>C</mi></mrow></math> in degrees rounded to two decimal places.</p>
<p>(Likely, you will first find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>C</mi></mrow></math> in radians. Multiply by <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>180</mn><mrow><mi>&pi;</mi></mrow></mfrac></mrow></math>to convert this radian measure to degrees.)</p>@
qu.7.3.maple=evalb(abs(evalf(($ANSWER)-($RESPONSE)))<0.005);@
qu.7.3.allow2d=1@
qu.7.3.maple_answer=$ANS@
qu.7.3.type=formula@
qu.7.3.mode=Maple@
qu.7.3.name=cos lawSSS@
qu.7.3.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>c</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>a</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>ab</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>C</mi></mrow></mfenced></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$c</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>C</mi></mrow></mfenced></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>C</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$ans</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>C</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$ANS</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>degrees.</p>@
qu.7.3.editing=useHTML@
qu.7.3.solution=@
qu.7.3.algorithm=$a=rint(1,10);
$b=rint(1,10);
$c=rint(3,10);
condition:gt($c,abs(($a)-($b)));
condition:lt($c,($a)+($b));
$ans=(($a)^2+($b)^2-($c)^2)/(2*($a)*($b));
$ANS=arccos($ans)*180/Pi;@
qu.7.3.uid=10a0724b-947d-4ed7-bfb4-66bb6b50ba58@
qu.7.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Trigonometry Review;
  Sub-Topic=Cosine Law;
  Difficulty=Easy;
@

qu.7.4.question=<p><img width="196" height="130" src="__BASE_URI__pictures/ABCtriangle.JPG" alt="" /></p>
<p>In the triangle (not drawn to scale!), side <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>, side <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>, and angle <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$B</mi><mo>&deg;</mo></mrow></math>. Using the Sine Law, find angle <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>A</mi></mrow></math>. If there is no possible triangle, enter N for none. (This happens when <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo><mi>a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></math>.) If there are two solutions for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>A</mi></mrow></math>(which happens when<strong>&nbsp; </strong><strong><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>a</mi></mrow></math></strong>), give the <strong>obtuse angle</strong> (between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>90</mn><mo>&deg;</mo></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>180</mn><mo>&deg;</mo></mrow></math>) solution. Otherwise (when<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo><mi>a</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;rpar;</mo></mrow></math>, give the single <strong>acute angle</strong> (between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn><mo>&deg;</mo></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>90</mn><mo>&deg;</mo></mrow></math>) solution. Give your answer rounded to two decimal places.</p>
<p>&nbsp;</p>
<p><strong>Advance Feedback: </strong>The Sine Law is actually really subtle. Go to the Maple Mathematics Survival Kit page on the Sine Law for a neat geometric investigation showing when you get one solution, two solutions, or none at all.</p>@
qu.7.4.maple=evalb(abs(evalf(($ANSWER)-($RESPONSE)))<0.005);@
qu.7.4.allow2d=1@
qu.7.4.maple_answer=if $c>1 then N
elif(evalf(($a)*sin(($B)*Pi/180))<$b and $b<$a) then evalf(180 - 180*arcsin($c)/Pi) else evalf(180*arcsin($c)/Pi) end if@
qu.7.4.type=formula@
qu.7.4.mode=Maple@
qu.7.4.name=sin law@
qu.7.4.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow><mrow><mi>a</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow><mrow><mi>b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow><mrow><mi>$a</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>$B</mi></mrow></mfenced></mrow><mrow><mi>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$c</mi></mrow></math></p>@
qu.7.4.editing=useHTML@
qu.7.4.solution=@
qu.7.4.algorithm=$a=rint(1,10);
$B=rint(5,85,5);
$b=rint(1,10);
$c=($a)*sin($B*Pi/180)/($b);@
qu.7.4.uid=14c3af44-386d-42ee-9a8f-82fb4532f1a1@
qu.7.4.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus II;
  Topic=Trigonometry Review;
  Sub-Topic=Sine Law;
  Difficulty=Medium;
@

qu.8.topic=Trig Formulas@

qu.8.1.question=<p>We know that <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'> </mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>. Use the appropriate formula to expand $Q in terms of one of these two formulas.</p>
<p>Remember to enter <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;as (sin(x))^2 <strong>NOT </strong>sin^2(x).</p>@
qu.8.1.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.8.1.allow2d=1@
qu.8.1.maple_answer=$ANS@
qu.8.1.type=formula@
qu.8.1.mode=Maple@
qu.8.1.name=trig(2x)@
qu.8.1.comment=<p>
<table width="200" cellspacing="1" cellpadding="1" border="1">
    <tbody>
        <tr>
            <td>
            <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>&nbsp;</p>
            <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced></mrow></math></p>
            <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$A</mi><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced></mrow></math></p>
            </td>
            <td>
            <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math></p>
            <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced></mrow></math></p>
            <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$A</mi><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced></mrow></math></p>
            </td>
        </tr>
    </tbody>
</table>
</p>@
qu.8.1.editing=useHTML@
qu.8.1.solution=@
qu.8.1.algorithm=$a=rint(2,10);
$A = 2*$a;
$z = rint(2);
$f=switch($z,"sin","cos");
$ANS=switch($z, "2*sin(($a)*x)*cos(($a)*x)", "cos(($a)*x)^2-sin(($a)*x)^2");
$sol=switch($z, mathml("2*sin(($a)*x)*cos(($a)*x)"), mathml("cos(($a)*x)^2-sin(($a)*x)^2"));
$Q=mathml("$f(2*$a*x)");@
qu.8.1.uid=d5385a2b-dc93-4fb0-82bf-92e98d373591@
qu.8.1.info=  Author=Jack Weiner;
  Topic=Trigonometry Review;
  Sub-Topic=Using Trig Identities;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
@

qu.8.2.question=<p>We know that <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mi>x</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mi>x</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></math>. Use these formulas to find a formula for $Q in terms of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi><mi>x</mi></mrow></mfenced></mrow></math>. (Enter only the right hand side of the formula.)</p>
<p>Remember that in Maple syntax, <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></math>and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></math>.</p>
<p>&nbsp;</p>@
qu.8.2.maple=evalb((($ANSWER)-($RESPONSE))=0);@
qu.8.2.allow2d=1@
qu.8.2.maple_answer=$ANS@
qu.8.2.type=formula@
qu.8.2.mode=Maple@
qu.8.2.name=(trig(x))^2@
qu.8.2.comment=<p>
<table width="200" cellspacing="1" cellpadding="1" border="1">
    <tbody>
        <tr>
            <td>
            <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mi>x</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></math></p>
            <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></math></p>
            <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi><mi>x</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></math></p>
            </td>
            <td>
            <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mi>x</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></math></p>
            <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></math></p>
            <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi><mi>x</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></math></p>
            </td>
        </tr>
    </tbody>
</table>
</p>@
qu.8.2.editing=useHTML@
qu.8.2.solution=@
qu.8.2.algorithm=$a=rint(2,10);
$b=2*($a);
$z = rint(2);
$f=switch($z,"sin","cos");
$ANS=switch($z, "(1-cos(2*$a*x))/2", "(1+cos(2*$a*x))/2");
$Q=maple("printf(MathML[ExportPresentation]('$f'(($a)*x)^2))");@
qu.8.2.uid=5926dbe3-7186-4990-8b1c-bbe5b0a607d0@
qu.8.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Trigonometry Review;
  Sub-Topic=Using Trig Identities;
  Difficulty=Easy;
@

