ProvingTrigIdentities3

Proving Trig Identities

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This PG code shows how to write a multi-part question in which each new part is revealed only after the previous part is answered correctly. The parts are revealed sequentially on the same html page instead of each part having its own html page. We also cleverly redefine the sine function to require students to simplify their answers when applying well-known trig identities.

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PG problem file	Explanation
Problem tagging data	Problem tagging:
DOCUMENT(); loadMacros( "PGstandard.pl", "MathObjects.pl", "PGML.pl", "scaffold.pl", "PGcourse.pl", ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1;	Initialization: Load the scaffold.pl macro.
Context("Numeric")->variables->are(t=>"Real"); 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; # Add the new functions to the Context Context()->functions->add( sin => {class => 'NewFunc', TeX => '\sin'}, );	Setup: We cleverly redefine the sine function so that when the student enters sin(t), it is interpreted and evaluated internally as exp(pi*t) but displayed to the student as sin(t).
BEGIN_PGML This problem has three parts. A part may be open if it is correct or if it is the first incorrect part. Clicking on the heading for a part toggles whether it is displayed. END_PGML Scaffold::Begin(is_open => "correct_or_first_incorrect"); Section::Begin("Part 1"); BEGIN_PGML In this multi-part problem, we will use algebra to verify the identity >> [` \displaystyle \frac{ \sin(t) }{ 1-\cos(t) } = \frac{ 1+\cos(t) }{ \sin(t) }. `] << First, using algebra we may rewrite the equation above as [` \displaystyle \sin(t) = \left( \frac{1+\cos(t)}{\sin(t)} \right) \cdot \Big( `] [_____________]{"1-cos(t)"} [` \Big) `]. END_PGML Section::End(); Section::Begin("Part 2"); BEGIN_PGML Using algebra we may rewrite the equation as [` \sin(t) \cdot \big( `] [______________]{"sin(t)"} [` \big) = \big(1+\cos(t)\big) \cdot \big(1-\cos(t)\big) `]. END_PGML Section::End(); Section::Begin("Part 3"); BEGIN_PGML Finally, using algebra we may rewrite the equation as [` \sin^2(t) = `] [_______________]{"1-(cos(t))^2"}, which is true since [` \cos^2(t) + \sin^2(t) = 1 .`] Thus, the original identity can be derived by reversing these steps. END_PGML Section::End(); Scaffold::End();	Main Text: This is where we use the scaffold.
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). Uses PGML."); ENDDOCUMENT();	Comment section:

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