Proving Identities

DOCUMENT();

loadMacros('PGstandard.pl', 'PGML.pl', 'scaffold.pl', 'PGcourse.pl');

Context()->variables->are(t => 'Real');

# Redefine sin(x) to be e^(pi x).
Context()->functions->remove('sin');

package NewFunc;
# The 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' },);

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.

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)}.`]<<

END_PGML

Scaffold::Begin(is_open => 'correct_or_first_incorrect');

Section::Begin('Part 1');
BEGIN_PGML

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)'}{15} [` \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)'}{15}
[`\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'}{15}, 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();

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).  Uses PGML.'
);

ENDDOCUMENT();