qu.1.topic=InverseTheory@

qu.1.1.mode=Non Permuting Multiple Choice@
qu.1.1.name=f has an inverse if@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=@
qu.1.1.uid=9ddb110a-184c-4379-9b69-8be95dc42545@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Functions;
@
qu.1.1.question=<p>For a function to have an inverse, it must be</p>@
qu.1.1.answer=1@
qu.1.1.choice.1=one to one@
qu.1.1.choice.2=continuous@
qu.1.1.choice.3=differentiable@
qu.1.1.choice.4=integrable@
qu.1.1.choice.5=all of the above@
qu.1.1.fixed=@

qu.1.2.mode=Multiple Choice@
qu.1.2.name=derofinverse@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=range(2,10,1);
$b=range(2,10,1);
$c=range(2,10,1);
condition:not(eq($a,$b));
condition:not(eq($a,$c));
condition:not(eq($c,$b));@
qu.1.2.uid=3fc5e996-5eaf-4080-86c0-fd9aed714744@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Functions;
@
qu.1.2.question=<p>If&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math> and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>f</mi><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$c</mi></mrow></mstyle></math> then</p>@
qu.1.2.answer=5@
qu.1.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup ><mi    mathvariant='italic' >f</mi><mrow><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&minus;</mo><mn   mathvariant='normal' >1</mn></mrow></msup><mfenced><mrow><mi   mathvariant='normal' >$b</mi></mrow></mfenced><mo  mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mfrac    ><mn   mathvariant='normal' >1</mn><mi   mathvariant='normal' >$a</mi></mfrac></mrow></math>@
qu.1.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msup><mi>f</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mfenced></mrow><msup><mi></mi><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mi mathvariant='normal'>$c</mi></mfrac></mrow></mstyle></math>@
qu.1.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle   veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msup><mi>f</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mfenced></mrow><msup><mi></mi><mi>&prime;</mi></msup></mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mi mathvariant='normal'>$c</mi></mfrac></mrow></mstyle></math>@
qu.1.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle   veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msup><mi>f</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mfenced></mrow><msup><mi></mi><mi>&prime;</mi></msup></mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mi mathvariant='normal'>$c</mi></mfrac></mrow></mstyle></math>@
qu.1.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle   veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msup><mi>f</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mfenced></mrow><msup><mi></mi><mi>&prime;</mi></msup></mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mi mathvariant='normal'>$c</mi></mfrac></mrow></mstyle></math>@
qu.1.2.fixed=@

qu.2.topic=findinginverse@

qu.2.1.mode=Multiple Choice@
qu.2.1.name=inverse ax^3+b@
qu.2.1.comment=<p>Interchange <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>and solve.</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=range(2,5,1);
$b=range(1,5,1);
condition:eq(gcd($a,$b),1);
$f=$a*x+$b;@
qu.2.1.uid=3c41f29b-414a-481f-9324-d17c0bd3067b@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Functions;
@
qu.2.1.question=<p>The inverse of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&sdot;</mo><msup><mi>x</mi><mn>3</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math> is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>f</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math></p>@
qu.2.1.answer=4@
qu.2.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mn    mathvariant='normal' >1</mn><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'             >&sdot;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >3</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfrac></mrow></math>@
qu.2.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >3</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow></math>@
qu.2.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mfrac    ><mn    mathvariant='normal' >1</mn><mn    mathvariant='normal' >3</mn></mfrac></mrow></msup></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></math>@
qu.2.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup ><mfenced><mrow><mfrac    ><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></mfenced><mrow><mfrac    ><mn    mathvariant='normal' >1</mn><mn    mathvariant='normal' >3</mn></mfrac></mrow></msup></mrow></math>@
qu.2.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'             >&sdot;</mo><msup ><mi    mathvariant='italic' >y</mi><mn    mathvariant='normal' >3</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></math>@
qu.2.1.fixed=@

qu.2.2.mode=Multiple Choice@
qu.2.2.name=inverse arcsin(a*x+b)@
qu.2.2.comment=<p>Interchange <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>and solve.</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=range(2,5,1);
$b=range(1,5,1);
condition:eq(gcd($a,$b),1);
$f=$a*x+$b;@
qu.2.2.uid=e8ea5734-ab84-4e8d-a076-48333fcd3229@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Functions;
@
qu.2.2.question=<p>The inverse of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mstyle></math> is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>f</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math></p>@
qu.2.2.answer=4@
qu.2.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mrow><mi    mathvariant='normal' >arcsin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced></mrow></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></math>@
qu.2.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mrow><mi    mathvariant='normal' >csc</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced></mrow></mrow></mrow></math>@
qu.2.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi    mathvariant='normal' >arcsin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced></mrow></mrow></math>@
qu.2.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mrow><mi    mathvariant='normal' >arcsin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></math>@
qu.2.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi    mathvariant='normal' >arcsin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mfrac    ><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></mfenced></mrow></mrow></math>@
qu.2.2.fixed=@

qu.2.3.mode=Multiple Choice@
qu.2.3.name=inverse ax^3-b@
qu.2.3.comment=<p>Interchange <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>and solve.</p>@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$a=range(2,5,1);
$b=range(1,5,1);
condition:eq(gcd($a,$b),1);
$f=$a*x+$b;@
qu.2.3.uid=efff8f11-7e80-41b8-aac3-c55a39d2eb88@
qu.2.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Functions;
@
qu.2.3.question=<p>The inverse of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math> is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>f</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math></p>@
qu.2.3.answer=4@
qu.2.3.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mn    mathvariant='normal' >1</mn><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'             >&sdot;</mo><msup ><mi    mathvariant='italic' >x</mi><mrow><mn    mathvariant='normal' >3</mn></mrow></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfrac></mrow></math>@
qu.2.3.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='italic' >x</mi><mrow><mn    mathvariant='normal' >3</mn></mrow></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow></math>@
qu.2.3.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mfrac    ><mn    mathvariant='normal' >1</mn><mn    mathvariant='normal' >3</mn></mfrac></mrow></msup></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></math>@
qu.2.3.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup ><mfenced><mrow><mfrac    ><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></mfenced><mrow><mfrac    ><mn    mathvariant='normal' >1</mn><mn    mathvariant='normal' >3</mn></mfrac></mrow></msup></mrow></math>@
qu.2.3.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup ><mfenced><mrow><mfrac    ><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></mfenced><mrow><mfrac    ><mn    mathvariant='normal' >1</mn><mn    mathvariant='normal' >3</mn></mfrac></mrow></msup></mrow></math>@
qu.2.3.fixed=@

qu.2.4.mode=Multiple Choice@
qu.2.4.name=inverse ln(ax+b)@
qu.2.4.comment=<p>Interchange <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>and solve.</p>@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=$a=range(2,5,1);
$b=range(1,5,1);
condition:eq(gcd($a,$b),1);
$f=$a*x+$b;@
qu.2.4.uid=70f310c0-a098-4c15-a905-c526eec4fbc8@
qu.2.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Functions;
@
qu.2.4.question=<p>The inverse of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mstyle></math> is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>f</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math></p>@
qu.2.4.answer=3@
qu.2.4.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mn    mathvariant='normal' >1</mn><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfrac></mrow></math>@
qu.2.4.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><msup ><mi    mathvariant='italic' >e</mi><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></msup></mrow></mrow></math>@
qu.2.4.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><msup ><mi    mathvariant='italic' >e</mi><mi    mathvariant='italic' >x</mi></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></math>@
qu.2.4.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><msup ><mi    mathvariant='italic' >e</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow></msup></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></math>@
qu.2.4.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><msup ><mi    mathvariant='italic' >e</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></msup></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></math>@
qu.2.4.fixed=@

qu.3.topic=exactArcTrig@

qu.3.1.question=<p>State the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$f</mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>. Give your answer in <strong>radian</strong> measure. Use Pi or pi for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&pi;</mi></mrow></mstyle></math>.</p>@
qu.3.1.maple=is(abs(evalf(($RESPONSE))-evalf(($ANSWER)))<.01)@
qu.3.1.allow2d=1@
qu.3.1.maple_answer=$f($a)@
qu.3.1.plot=plot($f,x=-1..1)@
qu.3.1.type=formula@
qu.3.1.mode=Maple@
qu.3.1.name=arccscsec(#)@
qu.3.1.comment=<p>Remember&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$f</mi></mrow></mstyle></math> has range <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lsqb;</mo></mrow></mstyle></math>$r1, $r2 <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$r2, $r3<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1111111em' stretchy='true'>&rsqb;</mo></mrow></mstyle></math>.</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$a=switch(rint(8), "1","sqrt(2)","-1","-sqrt(2)","2/sqrt(3)",
"-2/sqrt(3)","2","-2");
$z=rint(2);
$f=switch($z,"arcsec","arccsc");
$display=mathml("$a");
$r1=switch($z,mathml("0"),mathml("-pi/2"));
$r2=switch($z,mathml("pi/2"),mathml("0"));
$r3=switch($z,mathml("pi"),mathml("pi/2"));@
qu.3.1.uid=18fed9ff-3bd8-49be-aaef-5980934bac5b@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Inverse Trigonometry;
  Sub-Topic=arctrig(#);
@

qu.3.2.question=<p>State the exact value of $display. Give your answer in <strong>radian</strong> measure. Use Pi or pi for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&pi;</mi></mrow></mstyle></math>.</p>@
qu.3.2.maple=is(abs(evalf(($RESPONSE))-evalf(($ANSWER)))<.01)@
qu.3.2.allow2d=1@
qu.3.2.maple_answer=$f($a)@
qu.3.2.plot=plot($f,x=-1..1)@
qu.3.2.type=formula@
qu.3.2.mode=Maple@
qu.3.2.name=arcsincos(#)@
qu.3.2.comment=<p>Remember that&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$f</mi></mrow></mstyle></math> has range $range.</p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$z=rint(2);
$a=switch(rint(9),"1","1/sqrt(2)","-1","-1/sqrt(2)","sqrt(3)/2","-sqrt(3)/2","1/2","-1/2","0");
$f=switch($z,"arcsin","arccos");
$display=mathml("$f($a)");
$r=switch($z,"[-pi/2,pi/2]", "[0,pi]");
$range=maple("printf(MathML[ExportPresentation]($r))");@
qu.3.2.uid=af3cc268-457b-42d8-aaf0-e0d25620793a@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=arctrig(#);
@

qu.3.3.question=<p>State the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$f</mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>. Give your answer in <strong>radian</strong> measure. Use Pi or pi for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&pi;</mi></mrow></mstyle></math>.</p>@
qu.3.3.maple=is(abs(evalf(($RESPONSE))-evalf(($ANSWER)))<.01)@
qu.3.3.allow2d=1@
qu.3.3.maple_answer=$f($a)@
qu.3.3.plot=plot($f,x=-1..1)@
qu.3.3.type=formula@
qu.3.3.mode=Maple@
qu.3.3.name=arctancot(#)@
qu.3.3.comment=<p>Remember&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$f</mi></mrow></mstyle></math> has range $range.</p>@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=$a=switch(rint(7), "1","-1","1/sqrt(3)","-1/sqrt(3)","sqrt(3)","-sqrt(3)","0");
$z=rint(2);
$f=switch($z,"arccot","arctan");
$display=mathml("$a");
$r=switch($z,"(0,pi)","(-pi/2,pi/2)");
$range=mathml("$r");@
qu.3.3.uid=05035eb1-a3e9-4b05-a15e-aaac1f5981cc@
qu.3.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=arctrig(#);
@

qu.4.topic=domainArcTrig@

qu.4.1.mode=Multiple Choice@
qu.4.1.name=domarccsc@
qu.4.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>arccsc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;is undefined for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>1</mn></mrow></mstyle></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>arccsc</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$c<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math> is undefined for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></mstyle></math>$c <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>1</mn></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi></mi></mrow></mstyle></math></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$l</mi><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><mfrac><mi mathvariant='normal'>$r</mi><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mrow></mstyle></math></p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$a=range(2,5);
$b=range(2,7);
$c=mathml("($a*x+$b)");
$r=1-$b;
$l=-1-$b;
condition:eq(gcd($a,$r),1);
condition:eq(gcd($a,$l),1);
condition:eq(gcd($b,$a),1);@
qu.4.1.uid=c097ee89-658c-48c7-ae78-4d56f9359090@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=Domain;
@
qu.4.1.question=<p>Find the domain of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>y</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>arccsc</mi></mrow></mstyle></math>($c).</p>@
qu.4.1.answer=4@
qu.4.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mfrac    ><mrow><mi    mathvariant='normal' >$l</mi></mrow><mi    mathvariant='normal' >$a</mi></mfrac><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mfrac    ><mrow><mi    mathvariant='normal' >$r</mi></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></mfenced></mrow></math>@
qu.4.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mfrac    ><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow></mrow><mn    mathvariant='normal' >2</mn></mfrac></mrow><mrow></mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mn    mathvariant='normal' >0</mn><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&cup;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&lpar;</mo><mn    mathvariant='normal' >0</mn><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mfrac    ><mi    mathvariant='italic' >&pi;</mi><mn    mathvariant='normal' >2</mn></mfrac></mrow></mrow></mfenced></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.4.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='italic' >&infin;</mi></mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mn    mathvariant='normal' >&minus;1</mn><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&cup;</mo><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mn    mathvariant='normal' >1</mn><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mfrac    ><mi    mathvariant='normal' >$l</mi><mi    mathvariant='normal' >$a</mi></mfrac><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&cup;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mrow><mfrac    ><mi    mathvariant='normal' >$r</mi><mi    mathvariant='normal' >$a</mi></mfrac></mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mi    mathvariant='italic' >&infin;</mi></mrow></mrow></mfenced></mrow></math>@
qu.4.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mfrac    ><mrow><mi    mathvariant='normal' >$l</mi></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></mfenced><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&cup;</mo><mfenced><mrow><mrow><mfrac    ><mi    mathvariant='normal' >$r</mi><mi    mathvariant='normal' >$a</mi></mfrac></mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mi    mathvariant='italic' >&infin;</mi></mrow></mrow></mfenced></mrow></math>@
qu.4.1.fixed=@

qu.4.2.mode=Multiple Choice@
qu.4.2.name=domarcsec@
qu.4.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>arcsec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> is undefined for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>1</mn></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>arcsec</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$c<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>is undefined for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></mstyle></math>$c <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>1</mn></mrow></mstyle></math></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$l</mi><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><mfrac><mi mathvariant='normal'>$r</mi><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mrow></mstyle></math></p>@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=$a=range(2,5);
$b=range(2,7);
$c=mathml("($a*x+$b)");
$r=1-$b;
$l=-1-$b;
condition:eq(gcd($a,$r),1);
condition:eq(gcd($a,$l),1);
condition:eq(gcd($b,$a),1);@
qu.4.2.uid=93b47216-43e5-4d33-8fe2-438f68772596@
qu.4.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=Domain;
@
qu.4.2.question=<p>Find the domain of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>arcsec</mi></mrow></mstyle></math>($c).</p>@
qu.4.2.answer=4@
qu.4.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mrow><mfrac><mi mathvariant='normal'>$l</mi><mi mathvariant='normal'>$a</mi></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mi mathvariant='normal'>$r</mi><mi mathvariant='normal'>$a</mi></mfrac></mrow></mfenced></mrow></mstyle></math>@
qu.4.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mrow></mrow><mn    mathvariant='normal' >0</mn><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mfrac    ><mi    mathvariant='italic' >&pi;</mi><mn    mathvariant='normal' >2</mn></mfrac></mrow><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rpar;</mo><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&cup;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&lpar;</mo><mrow><mfrac    ><mi    mathvariant='italic' >&pi;</mi><mn    mathvariant='normal' >2</mn></mfrac></mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='italic' >&pi;</mi><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mrow></mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>@
qu.4.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mi    mathvariant='italic' >&infin;</mi></mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mn    mathvariant='normal' >&minus;1</mn><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&cup;</mo><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mn    mathvariant='normal' >1</mn><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mi    mathvariant='italic' >&infin;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mfrac    ><mi    mathvariant='normal' >$l</mi><mi    mathvariant='normal' >$a</mi></mfrac><mo    mathvariant='normal'  form='postfix' fence='true'  lspace='thinmathspace' rspace='verythinmathspace' stretchy='true'      >&rsqb;</mo><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&cup;</mo></mrow><mo    mathvariant='normal'  form='prefix' fence='true'  lspace='thinmathspace' rspace='thinmathspace' stretchy='true'      >&lsqb;</mo><mrow><mfrac    ><mi    mathvariant='normal' >$r</mi><mi    mathvariant='normal' >$a</mi></mfrac></mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mi    mathvariant='italic' >&infin;</mi></mrow></mrow></mfenced></mrow></math>@
qu.4.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mi    mathvariant='italic' >&infin;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mfrac    ><mrow><mi    mathvariant='normal' >$l</mi></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></mfenced><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&cup;</mo><mfenced><mrow><mrow><mfrac    ><mi    mathvariant='normal' >$r</mi><mi    mathvariant='normal' >$a</mi></mfrac></mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mi    mathvariant='italic' >&infin;</mi></mrow></mrow></mfenced></mrow></math>@
qu.4.2.fixed=@

qu.4.3.mode=Multiple Choice@
qu.4.3.name=domarccos@
qu.4.3.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo></mrow></mstyle></math>$c<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>1</mn></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$l</mi><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mrow><mfrac><mi mathvariant='normal'>$r</mi><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mrow></mstyle></math></p>@
qu.4.3.editing=useHTML@
qu.4.3.solution=@
qu.4.3.algorithm=$a=range(2,5);
$b=range(2,7);
$c=mathml("($a*x+$b)");
$r=1-$b;
$l=-1-$b;
condition:eq(gcd($a,$r),1);
condition:eq(gcd($a,$l),1);
condition:eq(gcd($b,$a),1);@
qu.4.3.uid=cb628524-dbc7-463b-85d4-a91a91efd5a8@
qu.4.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=Domain;
@
qu.4.3.question=<p>Find the domain of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>arccos</mi></mrow></mstyle></math>($c).</p>@
qu.4.3.answer=1@
qu.4.3.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$l</mi></mrow><mi mathvariant='normal'>$a</mi></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mi mathvariant='normal'>$r</mi><mi mathvariant='normal'>$a</mi></mfrac></mrow></mfenced></mrow></mstyle></math>@
qu.4.3.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&pi;</mi></mrow></mfenced></mrow></mstyle></math>@
qu.4.3.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mn    mathvariant='normal' >&minus;1</mn><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mn    mathvariant='normal' >1</mn></mrow></mfenced></mrow></math>@
qu.4.3.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mfrac    ><mrow><mrow><mo    mathvariant='normal'             >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mrow><mi    mathvariant='normal' >$a</mi></mfrac><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mfrac    ><mrow><mrow><mi    mathvariant='italic' >&pi;</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo></mrow><mi    mathvariant='normal' >$b</mi></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.3.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mfrac    ><mrow><mi    mathvariant='normal' >$l</mi></mrow><mi    mathvariant='normal' >$a</mi></mfrac><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mfrac    ><mi    mathvariant='normal' >$r</mi><mi    mathvariant='normal' >$a</mi></mfrac></mrow></mfenced></mrow></math>@
qu.4.3.fixed=@

qu.4.4.mode=Multiple Choice@
qu.4.4.name=domarcsin@
qu.4.4.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo></mrow></mstyle></math>$c<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>1</mn></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$l</mi><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mrow><mfrac><mi mathvariant='normal'>$r</mi><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mrow></mstyle></math></p>@
qu.4.4.editing=useHTML@
qu.4.4.solution=@
qu.4.4.algorithm=$a=range(2,5);
$b=range(2,7);
$c=mathml("($a*x+$b)");
$r=1-$b;
$l=-1-$b;
condition:eq(gcd($a,$r),1);
condition:eq(gcd($a,$l),1);@
qu.4.4.uid=8541bdc4-1a69-4dc3-a22d-517c7ead38f1@
qu.4.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=Domain;
@
qu.4.4.question=<p>Find the domain of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>arcsin</mi></mrow></mstyle></math> ($c).</p>@
qu.4.4.answer=1@
qu.4.4.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mrow><mfrac><mi mathvariant='normal'>$l</mi><mi mathvariant='normal'>$a</mi></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mi mathvariant='normal'>$r</mi><mi mathvariant='normal'>$a</mi></mfrac></mrow></mfenced></mrow></mstyle></math>@
qu.4.4.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi>&pi;</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mrow><mi>&pi;</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>@
qu.4.4.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mn    mathvariant='normal' >&minus;1</mn><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mn    mathvariant='normal' >1</mn></mrow></mfenced></mrow></math>@
qu.4.4.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mfrac    ><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mfrac    ><mi    mathvariant='italic' >&pi;</mi><mn    mathvariant='normal' >2</mn></mfrac><mo    mathvariant='normal'             >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mrow><mi    mathvariant='normal' >$a</mi></mfrac><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mfrac    ><mrow><mfrac    ><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mn    mathvariant='normal' >2</mn></mfrac><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.4.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mfrac    ><mrow><mi    mathvariant='normal' >$l</mi></mrow><mi    mathvariant='normal' >$a</mi></mfrac><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mfrac    ><mi    mathvariant='normal' >$r</mi><mi    mathvariant='normal' >$a</mi></mfrac></mrow></mfenced></mrow></math>@
qu.4.4.fixed=@

qu.5.topic=rangeArcTrig@

qu.5.1.mode=Multiple Choice@
qu.5.1.name=rangearcsin@
qu.5.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi mathvariant='normal'>$a</mi><mrow><mn>2</mn></mrow></mfrac><mi>&pi;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$a</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mrow><mfrac><mi mathvariant='normal'>$a</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mi>&pi;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$l&pi;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b&pi;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$r</mi><mrow><mi>&pi;</mi></mrow></mrow></mstyle></math></p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$a=range(2,5);
$b=range(2,7);
$A=2*$a;
$l=(-$a+2*$b)/2;
$r=($a+2*$b)/2;
$L=-1-2*$b;
$R=1-2*$b;
condition:eq(gcd($L,2*$a),1);
condition:eq(gcd($R,2*$a),1);@
qu.5.1.uid=2e1ebbcd-07fc-4d2f-8adc-e4470402ba50@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=Range;
@
qu.5.1.question=<p>Find the range of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.5.1.answer=2@
qu.5.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mrow><mfrac    ><mi    mathvariant='italic' >&pi;</mi><mn    mathvariant='normal' >2</mn></mfrac></mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mfrac    ><mi    mathvariant='italic' >&pi;</mi><mn    mathvariant='normal' >2</mn></mfrac></mrow></mrow></mfenced></mrow></math>@
qu.5.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mrow><mrow><mrow><mi    mathvariant='normal' >$l</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow></mrow></mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mrow></mrow><mrow><mi    mathvariant='normal' >$r</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow></mrow></mrow></mfenced><mrow></mrow></mrow></math>@
qu.5.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mn    mathvariant='normal' >0</mn><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow></mrow></mfenced></mrow></math>@
qu.5.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mfrac    ><mrow><mi    mathvariant='normal' >$L</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow></mrow><mi    mathvariant='normal' >$A</mi></mfrac><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mfrac    ><mrow><mi    mathvariant='normal' >$R</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow></mrow><mi    mathvariant='normal' >$A</mi></mfrac></mrow></mfenced><mrow></mrow></mrow></math>@
qu.5.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mi    mathvariant='normal' >$l</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$r</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo></mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow></mrow></mfenced></mrow></math>@
qu.5.1.fixed=@

qu.5.2.mode=Multiple Choice@
qu.5.2.name=rangearccos@
qu.5.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>arccos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>&pi;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$a</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>arccos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$a</mi><mi>&pi;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$l</mi><mrow><mi>&pi;</mi></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>arccos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>&pi;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$r</mi></mrow><mrow><mi>&pi;</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.5.2.editing=useHTML@
qu.5.2.solution=@
qu.5.2.algorithm=$a=range(2,10);
$b=range(2,10);
$l=$b;
$r=$a+$b;
$w1=-1-$b;
$w2=1-$b;
condition:eq(gcd($w1,$a),1);
condition:eq(gcd($w2,$a),1);@
qu.5.2.uid=a5a5d15d-e6fd-467d-ad29-3ce4111bd89b@
qu.5.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=Range;
@
qu.5.2.question=<p>Find the range of the function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>arccos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.5.2.answer=2@
qu.5.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mrow><mfrac    ><mi    mathvariant='italic' >&pi;</mi><mn    mathvariant='normal' >2</mn></mfrac></mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mfrac    ><mi    mathvariant='italic' >&pi;</mi><mn    mathvariant='normal' >2</mn></mfrac></mrow></mrow></mfenced></mrow></math>@
qu.5.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mrow><mrow><mrow><mrow><mi    mathvariant='normal' >$l</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow></mrow></mrow></mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mrow></mrow><mrow><mrow><mi    mathvariant='normal' >$r</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow></mrow></mrow></mrow></mfenced><mrow></mrow></mrow></math>@
qu.5.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mn    mathvariant='normal' >0</mn><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow></mrow></mfenced></mrow></math>@
qu.5.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=']'><mrow><mrow><mrow><mrow><mrow><mfrac    ><mrow><mi    mathvariant='normal' >$w1</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></mrow></mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mrow></mrow><mrow><mrow><mfrac    ><mrow><mi    mathvariant='normal' >$w2</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow></mrow><mi    mathvariant='normal' >$a</mi></mfrac></mrow></mrow></mrow></mfenced><mrow></mrow></mrow></math>@
qu.5.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mi    mathvariant='normal' >$l</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo><mi    mathvariant='normal' >$r</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo></mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow></mrow></mfenced></mrow></math>@
qu.5.2.fixed=@

qu.6.topic=arctrig(trigMapleGraded@

qu.6.1.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>arcsec</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$num</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow></mrow><mi mathvariant='normal'>$b</mi></mfrac></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.1.maple=evalb(abs(evalf($ANSWER)-evalf($RESPONSE))<.01);@
qu.6.1.allow2d=1@
qu.6.1.maple_answer=arcsec(sec($num*Pi/$b))@
qu.6.1.type=formula@
qu.6.1.mode=Maple@
qu.6.1.name=arcsec(sec@
qu.6.1.comment=<p>Remember arcsec has range <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mi>U</mi><mfenced open='(' close=']' separators=','><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.1.editing=useHTML@
qu.6.1.solution=@
qu.6.1.algorithm=$a=range(1,9);
$b=range(3,9);
condition:(eq(gcd($a,$b),1));
$n=range(1,2);
$num=(-1)^$n*$a;@
qu.6.1.uid=98e66a01-330c-4877-af35-394457555e15@
qu.6.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=arctrig(trig);
@

qu.6.2.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>arctan</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$num</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow></mrow><mi mathvariant='normal'>$b</mi></mfrac></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.2.maple=evalb(abs(evalf($ANSWER)-evalf($RESPONSE))<.01);@
qu.6.2.allow2d=1@
qu.6.2.maple_answer=arctan(tan($num*Pi/$b))@
qu.6.2.type=formula@
qu.6.2.mode=Maple@
qu.6.2.name=arctan(tan@
qu.6.2.comment=<p>Remember arctan has range <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.2.editing=useHTML@
qu.6.2.solution=@
qu.6.2.algorithm=$a=range(2,9);
$b=range(3,9);
condition:(eq(gcd($a,$b),1));
$n=range(1,2);
$num=(-1)^$n*$a;@
qu.6.2.uid=b514e9a4-ed07-43b1-b5e5-885c3ea1baba@
qu.6.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=arctrig(trig);
@

qu.6.3.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>arccot</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$num</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow></mrow><mi mathvariant='normal'>$b</mi></mfrac></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.3.maple=evalb(abs(evalf($ANSWER)-evalf($RESPONSE))<.01);@
qu.6.3.allow2d=1@
qu.6.3.maple_answer=arccot(cot($num*Pi/$b))@
qu.6.3.type=formula@
qu.6.3.mode=Maple@
qu.6.3.name=arccot(cot@
qu.6.3.comment=<p>Remember arccot has range <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&pi;</mi></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.3.editing=useHTML@
qu.6.3.solution=@
qu.6.3.algorithm=$a=range(1,9);
$b=range(3,9);
condition:(eq(gcd($a,$b),1));
$n=range(1,2);
$num=(-1)^$n*$a;@
qu.6.3.uid=11bfff10-389d-4fb8-945d-cf26be2f804a@
qu.6.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=arctrig(trig);
@

qu.6.4.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$num</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow></mrow><mi mathvariant='normal'>$b</mi></mfrac></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.4.maple=evalb(abs(evalf($ANSWER)-evalf($RESPONSE))<.01);@
qu.6.4.allow2d=1@
qu.6.4.maple_answer=arcsin(sin($num*Pi/$b))@
qu.6.4.type=formula@
qu.6.4.mode=Maple@
qu.6.4.name=arcsin(sin@
qu.6.4.comment=<p>Remember arcsin has range <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.4.editing=useHTML@
qu.6.4.solution=@
qu.6.4.algorithm=$a=range(2,9);
$b=range(2,9);
condition:(eq(gcd($a,$b),1));
$n=range(1,2);
$num=(-1)^$n*$a;@
qu.6.4.uid=d6a821e6-a5c5-470f-ba01-18853d4a0239@
qu.6.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=arctrig(trig);
@

qu.6.5.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>arccos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$num</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow></mrow><mi mathvariant='normal'>$b</mi></mfrac></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.5.maple=evalb(abs(evalf($ANSWER)-evalf($RESPONSE))<.01);@
qu.6.5.allow2d=1@
qu.6.5.maple_answer=arccos(cos($num*Pi/$b))@
qu.6.5.type=formula@
qu.6.5.mode=Maple@
qu.6.5.name=arccos(cos@
qu.6.5.comment=<p>Remember arccos has range <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&pi;</mi></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.5.editing=useHTML@
qu.6.5.solution=@
qu.6.5.algorithm=$a=range(2,9);
$b=range(2,9);
condition:(eq(gcd($a,$b),1));
$n=range(1,2);
$num=(-1)^$n*$a;@
qu.6.5.uid=15cea09e-3029-42b5-8e1a-989f2d1d2e7b@
qu.6.5.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=arctrig(trig);
@

qu.6.6.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>arccsc</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$num</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow></mrow><mi mathvariant='normal'>$b</mi></mfrac></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.6.maple=evalb(abs(evalf($ANSWER)-evalf($RESPONSE))<.01);@
qu.6.6.allow2d=1@
qu.6.6.maple_answer=arccsc(csc($num*Pi/$b))@
qu.6.6.type=formula@
qu.6.6.mode=Maple@
qu.6.6.name=arccsc(csc@
qu.6.6.comment=<p>Remember arccsc has range <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='(' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.6.editing=useHTML@
qu.6.6.solution=@
qu.6.6.algorithm=$a=range(1,9);
$b=range(3,9);
condition:(eq(gcd($a,$b),1));
$n=range(1,2);
$num=(-1)^$n*$a;@
qu.6.6.uid=0aed485e-65f9-459d-9df3-0af750c536ca@
qu.6.6.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=arctrig(trig);
@

qu.7.topic=trig(arctrigXchoice@

qu.7.1.mode=Multiple Choice@
qu.7.1.name=tan(arctan@
qu.7.1.comment=@
qu.7.1.editing=useHTML@
qu.7.1.solution=@
qu.7.1.algorithm=$a=decimal(1,range(-1,1,.1));
condition:ne($a,0);@
qu.7.1.uid=84385436-255b-40cb-9e50-838a89531673@
qu.7.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus II;
  Difficulty=Easy;
  Topic=Inverse Trigonometry;
  Sub-Topic=trig(arctrig);
@
qu.7.1.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>arctan</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.7.1.answer=1@
qu.7.1.choice.1=$a@
qu.7.1.choice.2=Does not exist!@
qu.7.1.choice.3=There are many answers.@
qu.7.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced><mrow><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&minus;</mo><mrow><mfrac    ><mi    mathvariant='italic' >&pi;</mi><mn    mathvariant='normal' >2</mn></mfrac><mo    mathvariant='normal'    separator='true' lspace='0em' rspace='verythickmathspace'       >&comma;</mo></mrow><mfrac    ><mi    mathvariant='italic' >&pi;</mi><mn    mathvariant='normal' >2</mn></mfrac></mrow></mfenced><mrow></mrow></mrow></math>@
qu.7.1.choice.5=${-1*$a}@
qu.7.1.fixed=@

qu.7.2.mode=Multiple Choice@
qu.7.2.name=cot(arccot(@
qu.7.2.comment=@
qu.7.2.editing=useHTML@
qu.7.2.solution=@
qu.7.2.algorithm=$a=decimal(1,range(-15.1,15.1,.2));@
qu.7.2.uid=26503717-cb10-4ec6-b7ba-f5203473687a@
qu.7.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=trig(arctrig);
@
qu.7.2.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>arccot</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.7.2.answer=1@
qu.7.2.choice.1=$a@
qu.7.2.choice.2=0@
qu.7.2.choice.3=Does not exist!@
qu.7.2.choice.4=There are many answers.@
qu.7.2.choice.5=${-1*$a}@
qu.7.2.fixed=@

qu.7.3.mode=Multiple Choice@
qu.7.3.name=sec(arcsec@
qu.7.3.comment=@
qu.7.3.editing=useHTML@
qu.7.3.solution=@
qu.7.3.algorithm=$a=decimal(1,range(1,3,.1));
$b=abs($a);
$c=-1*($a);@
qu.7.3.uid=c14188d6-7429-4f85-be24-9ef609d24d9f@
qu.7.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=trig(arctrig);
@
qu.7.3.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>arcsec</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.7.3.answer=4@
qu.7.3.choice.1=$a@
qu.7.3.choice.2=[-$b,$b]@
qu.7.3.choice.3=There are many answers.@
qu.7.3.choice.4=$c@
qu.7.3.choice.5=Does not exist!@
qu.7.3.fixed=@

qu.7.4.mode=Multiple Choice@
qu.7.4.name=csc(arccsc(@
qu.7.4.comment=@
qu.7.4.editing=useHTML@
qu.7.4.solution=@
qu.7.4.algorithm=$a=decimal(1,range(1,3,.1));
$b=abs($a);
$c=-1*($a);@
qu.7.4.uid=929c1508-1336-40e6-a5ce-28ac65410ab3@
qu.7.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=trig(arctrig);
@
qu.7.4.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>arccsc</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.7.4.answer=4@
qu.7.4.choice.1=$a@
qu.7.4.choice.2=[-$b,$b]@
qu.7.4.choice.3=There are many answers.@
qu.7.4.choice.4=$c@
qu.7.4.choice.5=Does not exist!@
qu.7.4.fixed=@

qu.7.5.mode=Multiple Choice@
qu.7.5.name=cos(arccos(@
qu.7.5.comment=@
qu.7.5.editing=useHTML@
qu.7.5.solution=@
qu.7.5.algorithm=$a=decimal(1,range(-.9,.9,.2));
$b=abs($a);
$c=-1*($a);@
qu.7.5.uid=59c2fb62-d605-4bc3-a668-bbb4c322327d@
qu.7.5.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=trig(arctrig);
@
qu.7.5.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>arccos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.7.5.answer=1@
qu.7.5.choice.1=$a@
qu.7.5.choice.2=There are many answers.@
qu.7.5.choice.3=[-$b,$b]@
qu.7.5.choice.4=$c@
qu.7.5.choice.5=Does not exist!@
qu.7.5.fixed=@

qu.7.6.mode=Multiple Choice@
qu.7.6.name=sin(arsin@
qu.7.6.comment=@
qu.7.6.editing=useHTML@
qu.7.6.solution=@
qu.7.6.algorithm=$a=decimal(1,range(-.9,.9,.2));
$b=abs($a);
$c=-1*($a);@
qu.7.6.uid=09c39dd2-628d-4860-842f-ea667c33b7b0@
qu.7.6.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=trig(arctrig);
@
qu.7.6.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.7.6.answer=1@
qu.7.6.choice.1=$a@
qu.7.6.choice.2=There are many answers.@
qu.7.6.choice.3=[-$b,$b]@
qu.7.6.choice.4=$c@
qu.7.6.choice.5=Does not exist!@
qu.7.6.fixed=@

qu.8.topic=trig(arcDIFFERENTtrig)@

qu.8.1.mode=Multiple Choice@
qu.8.1.name=cos(arcsin)@
qu.8.1.comment=<p>Remember arcsin has range <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>. This means the triangle you draw must be in the first or fourth quadrant, use the CAST rule to determine which.</p>@
qu.8.1.editing=useHTML@
qu.8.1.solution=@
qu.8.1.algorithm=$a=range(-10,10,1);
$b=range(2,10,1);
condition:eq(gcd($a,$b),1);
condition:not(eq($a,0));
condition:not(eq($b,0));
condition:not(le($b,abs($a)));
$c=maple("abs($a/$b)");
$s=($b)^2-($a)^2;@
qu.8.1.uid=5ca2a30d-b10b-4dfb-b57b-54a82e702b5f@
qu.8.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=trig(arctrig);
@
qu.8.1.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mi mathvariant='normal'>$a</mi><mi mathvariant='normal'>$b</mi></mfrac></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.8.1.answer=3@
qu.8.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$c</mi></mrow></math>@
qu.8.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$c</mi></mrow></math>@
qu.8.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msqrt><mrow><mi mathvariant='normal'>$s</mi></mrow></msqrt></mrow><mi mathvariant='normal'>$b</mi></mfrac></mrow></mstyle></math>@
qu.8.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msqrt><mrow><mi mathvariant='normal'>$s</mi></mrow></msqrt></mrow><mi mathvariant='normal'>$b</mi></mfrac></mrow></mstyle></math>@
qu.8.1.choice.5=There are two answers, one +, one -.@
qu.8.1.fixed=@

qu.8.2.mode=Multiple Choice@
qu.8.2.name=sin(arccos)@
qu.8.2.comment=<p>Remember that arccos has range <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&pi;</mi></mrow></mfenced></mrow></mstyle></math>. This&nbsp;means that&nbsp;the triangle you draw should be in the first or second quadrant, use the CAST rule to determine which.</p>@
qu.8.2.editing=useHTML@
qu.8.2.solution=@
qu.8.2.algorithm=$a=range(-10,10,1);
$b=range(2,10,1);
condition:eq(gcd($a,$b),1);
condition:not(eq($a,0));
condition:not(eq($b,0));
condition:not(le($b,abs($a)));
$c=maple("abs($a/$b)");
$s=($b)^2-($a)^2;@
qu.8.2.uid=592778de-4752-44b2-9524-628e5caf5079@
qu.8.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=trig(arctrig);
@
qu.8.2.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>arccos</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mi mathvariant='normal'>$a</mi><mi mathvariant='normal'>$b</mi></mfrac></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.8.2.answer=3@
qu.8.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$c</mi></mrow></math>@
qu.8.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$c</mi></mrow></math>@
qu.8.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msqrt><mrow><mi mathvariant='normal'>$s</mi></mrow></msqrt></mrow><mi mathvariant='normal'>$b</mi></mfrac></mrow></mstyle></math>@
qu.8.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msqrt><mrow><mi mathvariant='normal'>$s</mi></mrow></msqrt></mrow><mi mathvariant='normal'>$b</mi></mfrac></mrow></mstyle></math>@
qu.8.2.choice.5=There are two answers, one +, one -.@
qu.8.2.fixed=@

qu.9.topic=Trig(ArcTrigDNE@

qu.9.1.mode=Multiple Choice@
qu.9.1.name=sec(arcsecDNE@
qu.9.1.comment=<p>$a is not in the domain of arcsec.</p>@
qu.9.1.editing=useHTML@
qu.9.1.solution=@
qu.9.1.algorithm=$a=decimal(2,range(-.9,.9,.1));
$b=abs($a);
$c=-1*($a);@
qu.9.1.uid=2db45c8d-2c15-48d4-bf5a-84a735ad7612@
qu.9.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=trig(arctrig) DNE;
@
qu.9.1.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>arcsec</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.&nbsp;</p>@
qu.9.1.answer=5@
qu.9.1.choice.1=$a@
qu.9.1.choice.2=[-$b,$b]@
qu.9.1.choice.3=There are many answers.@
qu.9.1.choice.4=$c@
qu.9.1.choice.5=Does not exist!@
qu.9.1.fixed=@

qu.9.2.mode=Multiple Choice@
qu.9.2.name=sin(arsinDNE@
qu.9.2.comment=<p>$a is not in the domain of arcsin.</p>@
qu.9.2.editing=useHTML@
qu.9.2.solution=@
qu.9.2.algorithm=$a=decimal(2,range(1.1,3,.01));
$b=abs($a);
$c=-1*($a);@
qu.9.2.uid=2ab804b2-191b-44c8-884b-a9a4e4d87066@
qu.9.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=trig(arctrig) DNE;
@
qu.9.2.question=<p>Find the exact value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.9.2.answer=5@
qu.9.2.choice.1=$a@
qu.9.2.choice.2=[-$b,$b]@
qu.9.2.choice.3=There are many answers.@
qu.9.2.choice.4=$c@
qu.9.2.choice.5=Does not exist!@
qu.9.2.fixed=@

qu.10.topic=Trig(Arctrig(A+B)@

qu.10.1.mode=Multiple Choice@
qu.10.1.name=trig(arcsin(A+B)Xchoice@
qu.10.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>arccos</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$frac<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi mathvariant='normal'>cos</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><mi>&theta;1</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$frac</p>
<p><img width="263" height="189" alt="" src="__BASE_URI__pictures/triangletheta1firstquad.png" /></p>
<p>In this picture <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math> by the Pythagorean Theorem <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mi mathvariant='normal'>$n1</mi></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>From here <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><msqrt><mrow><mi mathvariant='normal'>$n1</mi></mrow></msqrt></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac></mrow></mrow><mrow><mi></mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>arctan</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></mstyle></math>$frac<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$frac</p>
<p><img width="263" height="189" alt="" src="__BASE_URI__pictures/triangletheta2firstquad.png" /></p>
<p>In this picture <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$E</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>, by the Pythagorean Theorem <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>D</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mi mathvariant='normal'>$d1</mi></mrow></msqrt></mrow></mrow></mstyle></math>.</p>
<p>From this we see <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$d1</mi></mrow></msqrt></mrow></mfrac></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$d1</mi></mrow></msqrt></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>Now we just need to use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>&theta;2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;2</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;1</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mo mathvariant='italic' fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><mi>&theta;2</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mi></mi></mrow></mstyle></math></p>@
qu.10.1.editing=useHTML@
qu.10.1.hint.1=Use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><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></mstyle></math>.@
qu.10.1.solution=@
qu.10.1.algorithm=$z=rint(5);
$b=switch($z,2,3,6,7,8);
$a=switch($z,1,range(1,2),range(1,5,4),range(1,5),range(1,5,2));
$display=maple("printf(MathML[ExportPresentation](sin(arccos(-$a/$b)+arctan(-$a/$b))))");
$n1=$b^2-$a^2;
$d1=$b^2+$a^2;
$n2=$a^2;
$frac=mathml("$a/$b");@
qu.10.1.uid=f6c59428-5e79-473a-bfee-ece38972a892@
qu.10.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Inverse Trigonometry;
  Sub-Topic=sin(arccos+arctan);
@
qu.10.1.question=<p>Find the exact value of $display.</p>@
qu.10.1.answer=1@
qu.10.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msqrt><mrow><mi mathvariant='normal'>$n1</mi></mrow></msqrt></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$d1</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mi mathvariant='normal'>$n2</mi><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><mi mathvariant='normal'>$d1</mi></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>@
qu.10.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msqrt><mrow><mi mathvariant='normal'>$n1</mi></mrow></msqrt></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$d1</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi mathvariant='normal'>$n2</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$n1</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><mi mathvariant='normal'>$d1</mi></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>@
qu.10.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><msqrt><mrow><mi mathvariant='normal'>$n1</mi></mrow></msqrt><mrow><msqrt><mrow><mi mathvariant='normal'>$d1</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mi mathvariant='normal'>$a</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$d1</mi></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>@
qu.10.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msqrt><mrow><mi mathvariant='normal'>$n1</mi></mrow></msqrt></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$d1</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mi mathvariant='normal'>$n2</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$n1</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><mi mathvariant='normal'>$d1</mi></mrow></msqrt></mrow></mrow></mfrac></mrow></mstyle></math>@
qu.10.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&sdot;</mo><mrow><msqrt><mrow><mi mathvariant='normal'>$n1</mi></mrow></msqrt></mrow></mrow><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&sdot;</mo><mrow><msqrt><mrow><mi mathvariant='normal'>$d1</mi></mrow></msqrt></mrow></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi mathvariant='normal'>$a</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$d1</mi></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>@
qu.10.1.fixed=@

