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    1 ## DESCRIPTION
    2 ## Trigonometric Identities
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('trig')
    6 
    7 ## DBsubject('Precalculus')
    8 ## DBchapter('Trigonometry')
    9 ## DBsection('Trigonometric Identities')
   10 ## Date('01/01/10')
   11 ## Author('Paul Pearson')
   12 ## Institution('Fort Lewis College')
   13 ## TitleText1('Functions Modeling Change')
   14 ## EditionText1('3')
   15 ## AuthorText1('Connally')
   16 ## Section1('7.2')
   17 ## Problem1('36')
   18 
   19 
   20 DOCUMENT();
   21 
   22 loadMacros(
   23 "PGstandard.pl",
   24 "MathObjects.pl",
   25 "AnswerFormatHelp.pl",
   26 "parserPopUp.pl",
   27 "unionTables.pl",
   28 "answerHints.pl",
   29 );
   30 
   31 TEXT(beginproblem());
   32 
   33 $showPartialCorrectAnswers = 1;
   34 
   35 
   36 ###################################
   37 #  Setup
   38 
   39 Context("Numeric")->variables->are(x=>"Real");
   40 
   41 Context()->functions->remove("cos");
   42 package NewFunc;
   43 # this next line makes the function a
   44 # function from reals to reals
   45 our @ISA = qw(Parser::Function::numeric);
   46 sub cos {
   47   shift; my $x = shift;
   48   return CORE::exp(-$x*3.1415926535);
   49 }
   50 package main;
   51 # Make it work on formulas as well as numbers
   52 #sub cos {Parser::Function->call('cos',@_)} # if uncommented, this line will generate error messages
   53 #  Add the new functions to the Context
   54 Context()->functions->add( cos => {class => 'NewFunc', TeX => '\cos'}, );
   55 
   56 
   57 Context()->functions->remove("sin");
   58 package NewFunc;
   59 # this next line makes the function a
   60 # function from reals to reals
   61 our @ISA = qw(Parser::Function::numeric);
   62 sub sin {
   63   shift; my $x = shift;
   64   return CORE::exp($x*3.1415926535);
   65 }
   66 package main;
   67 # Make it work on formulas as well as numbers
   68 #sub cos {Parser::Function->call('cos',@_)} # if uncommented, this line will generate error messages
   69 #  Add the new functions to the Context
   70 Context()->functions->add( sin => {class => 'NewFunc', TeX => '\sin'}, );
   71 
   72 
   73 
   74 $popup = PopUp(["Choose","Identity","Not an identity"],"Identity");
   75 
   76 $ans_eval1 = $popup->cmp();
   77 $ans_eval2 = Formula("sin(x)/cos(x)")->cmp()->withPostFilter(AnswerHints(
   78   Formula("tan(x)") => "Hint: Rewrite tangent in terms of sine and cosine",
   79 ));
   80 $ans_eval3 = Formula("cos(x)/sin(x)")->cmp()->withPostFilter(AnswerHints(
   81   Formula("1/tan(x)") => "Hint: Rewrite tangent in terms of sine and cosine",
   82 ));
   83 $ans_eval4 = Formula("(cos(x))^2+(sin(x))^2")->cmp()->withPostFilter(AnswerHints(
   84   Formula("1") => "Use a trig identity to rewrite 1 in terms of sines and cosines",
   85 ));
   86 
   87 
   88 ###################################
   89 #  Main text
   90 
   91 
   92 ##########
   93 #  PART 1
   94 
   95 BEGIN_TEXT
   96 ${BBOLD}Part 1 of 3:${EBOLD}
   97 $BR
   98 $BR
   99 This is a multi-part problem.  Use a graph to decide whether the equation
  100 $BCENTER
  101 \( \displaystyle \tan(x) + \frac{ 1 }{ \tan(x) } = \frac{ 1 }{ \cos(x) \sin(x) } \)
  102 $ECENTER
  103 $BR
  104 is an identity or not.  \{ $popup->menu() \}
  105 END_TEXT
  106 
  107 ANS($ans_eval1 );
  108 
  109 
  110 
  111 #############
  112 #   PART 2
  113 
  114 $ans_hash1 = $ans_eval1->evaluate($inputs_ref->{ANS_NUM_TO_NAME(1)});
  115 
  116 if (1 == $ans_hash1->{score}) {
  117 
  118 BEGIN_TEXT
  119 $PAR
  120 $HR
  121 ${BBOLD}Part 2 of 3:${EBOLD}
  122 $BR
  123 $BR
  124 Now, let's prove that the equation above is an identity.
  125 Using trigonometric identities, fill in the following answer blanks.
  126 $BR
  127 $BR
  128 \( \displaystyle \tan(x) + \frac{ 1 }{ \tan(x) } = \)
  129 \{ ans_rule(15) \} + \{ ans_rule(15) \}
  130 END_TEXT
  131 
  132 ANS($ans_eval2 );
  133 ANS($ans_eval3 );
  134 
  135 }
  136 
  137 
  138 #############
  139 #   PART 3
  140 
  141 $ans_hash2 = $ans_eval2->evaluate($inputs_ref->{ANS_NUM_TO_NAME(2)});
  142 $ans_hash3 = $ans_eval3->evaluate($inputs_ref->{ANS_NUM_TO_NAME(3)});
  143 
  144 if ( ($ans_hash1->{score}==1) && ($ans_hash2->{score}==1) && ($ans_hash3->{score}==1) ) {
  145 
  146 BEGIN_TEXT
  147 $PAR
  148 $HR
  149 ${BBOLD}Part 3 of 3:${EBOLD}
  150 $BR
  151 $BR
  152 By finding a common denominator, we obtain
  153 $BR
  154 $BR
  155 \{
  156 ColumnTable(
  157 "\( \displaystyle \frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)} = \) ",
  158 ans_rule(20).$HR."\( \cos(x) \sin(x) \)",
  159 indent=>"0", valign=>"MIDDLE", separation=>"10",
  160 );
  161 \}
  162 $BR
  163 Using a trigonometric identity, this equals \( \displaystyle \frac{1}{\cos(x)\sin(x)} \), and therefore we've proved the original identity.
  164 END_TEXT
  165 
  166 ANS($ans_eval4 );
  167 
  168 }
  169 
  170 
  171 
  172 COMMENT("This is a multi-part problem in which the next part is revealed only after the previous part is correct.  Prevents students from entering trivial identities (entering what they were given).");
  173 
  174 
  175 COMMENT('MathObject version');
  176 ENDDOCUMENT();