## DESCRIPTION ## Trigonometry: proving trig identities ## ENDDESCRIPTION ## KEYWORDS('trigonometry', 'proving trig identities') ## DBsubject('WeBWorK') ## DBchapter('WeBWorK Tutorial') ## DBsection('Fort Lewis Tutorial 2011') ## Date('01/30/2011') ## Author('Paul Pearson') ## Institution('Fort Lewis College') ## TitleText1('') ## EditionText1('') ## AuthorText1('') ## Section1('') ## Problem1('') DOCUMENT(); loadMacros( "PGstandard.pl", "MathObjects.pl", ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1; ################################### # Setup Context("Numeric")->variables->are(t=>"Real"); # # Redefine the sin(x) to be e^(pi x) # Context()->functions->remove("sin"); package NewFunc; # this next line makes the function a # function from reals to reals our @ISA = qw(Parser::Function::numeric); sub sin { shift; my$x = shift; return CORE::exp($x*3.1415926535); } package main; # Make it work on formulas as well as numbers #sub cos {Parser::Function->call('cos',@_)} # if uncommented, this line will generate error messages # Add the new functions to the Context Context()->functions->add( sin => {class => 'NewFunc', TeX => '\sin'}, ); # # You manually define the answers # @answers = ();$answers[1] = Formula("1-cos(t)"); $answers[2] = Formula("sin(t)");$answers[3] = Formula("1-(cos(t))^2"); # # Automatic configuration for answer evaluation # @ans_eval = (); @scores = (); foreach my $i (1..$#answers) { $ans_eval[$i] = $answers[$i] ->cmp(); $ans_hash[$i] = $ans_eval[$i]->evaluate($inputs_ref->{ANS_NUM_TO_NAME($i)}); $scores[$i] = $ans_hash[$i]->{score}; } ########################################### # Main text and answer evaluation part 1 Context()->texStrings; BEGIN_TEXT ${BBOLD}Part 1 of 3:${EBOLD} $BR$BR In this multi-part problem, we will use algebra to verify the identity $BCENTER $$\displaystyle \frac{ \sin(t) }{ 1-\cos(t) } = \frac{ 1+\cos(t) }{ \sin(t) }.$$$ECENTER $BR First, using algebra we may rewrite the equation above as$BR $BR $$\displaystyle \sin(t) = \left( \frac{1+\cos(t)}{\sin(t)} \right) \cdot \Big($$ \{ ans_rule(20) \} $$\Big)$$ END_TEXT Context()->normalStrings; ANS($ans_eval[1] ); ########################################## # Main text and answer evaluation part 2 if ($scores[1]==1) { Context()->texStrings; BEGIN_TEXT$PAR $HR${BBOLD}Part 2 of 3:${EBOLD}$BR $BR Then, using algebra we may rewrite the equation as$BR $BR $$\sin(t) \cdot \big($$ \{ ans_rule(20) \} $$\big) = \big(1+\cos(t)\big) \cdot \big(1-\cos(t)\big)$$, END_TEXT Context()->normalStrings; ANS($ans_eval[2] ); } # end if ########################################## # Main text and answer evaluation part 3 if ( ($scores[1]==1) && ($scores[2]==1) ) { Context()->texStrings; BEGIN_TEXT $PAR$HR ${BBOLD}Part 3 of 3:${EBOLD} $BR$BR Finally, using algebra we may rewrite the equation as $BR$BR $$\sin^2(t) =$$ \{ ans_rule(20) \} $BR$BR which is true since $$\cos^2(t) + \sin^2(t) = 1$$. Thus, the original identity can be derived by reversing these steps. END_TEXT Context()->normalStrings; ANS( \$ans_eval[3] ); } # end if COMMENT("MathObject version. 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)"); ENDDOCUMENT();