# ProvingTrigIdentities2

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## Proving Trig Identites using a Compound Problem

This PG code shows how to have a multi-part question in which each part is revealed sequentially on its own html page.

• Download file: File:ProvingTrigIdentities2.txt (change the file extension from txt to pg when you save it)
• File location in NPL: FortLewis/Authoring/Templates/Trig/ProvingTrigIdentities2.pg

PG problem file Explanation

Problem tagging:

DOCUMENT();

"PGstandard.pl",
"MathObjects.pl",
"compoundProblem.pl",
"Parser.pl",
"PGunion.pl",
);

TEXT(beginproblem());

BEGIN_PROBLEM();


Initialization:

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

$isProfessor =$studentLogin eq 'professor';

#
#  Set up the compound problem object.
#
$cp = new compoundProblem( parts => 3, totalAnswers => 3, parserValues => 1, allowReset =>$isProfessor,
);
$part =$cp->part;


Setup:

if ($part == 1) { BEGIN_TEXT${BBOLD}Part 1 of 3:${EBOLD}$BR
$BR${BITALIC}Instructions:${EITALIC} You will need to submit your answers twice for each part. The first time you submit your answers they will be checked for correctness. When your answer is correct, check the box for${BITALIC}Go on to next part${EITALIC} and click the submit button. You will not be able to go back to previous parts.$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

ANS( Formula("1-cos(t)")->cmp() );

}


Part 1:

if ($part == 2) { BEGIN_TEXT${BBOLD}Part 2 of 3:${EBOLD}$BR
$BR Step 0: $$\displaystyle \frac{ \sin(t) }{ 1-\cos(t) } = \frac{ 1+\cos(t) }{ \sin(t) }.$$$BR
$BR Step 1: $$\displaystyle \sin(t) = \left( \frac{1+\cos(t)}{\sin(t)} \right) \cdot ( 1 - \cos(t) ).$$$BR
$HR$BR
We may use algebra to rewrite the equation from Step 1 as
$BR$BR
$$\sin(t) \cdot \big($$
\{ ans_rule(20) \}
$$\big) = \big(1+\cos(t)\big) \cdot \big(1-\cos(t)\big)$$.
END_TEXT

ANS( Formula("sin(t)")->cmp() );

}


Part 2:

if ($part == 3) { BEGIN_TEXT${BBOLD}Part 3 of 3:${EBOLD}$BR
$BR Step 0: $$\displaystyle \frac{ \sin(t) }{ 1-\cos(t) } = \frac{ 1+\cos(t) }{ \sin(t) }.$$$BR
$BR Step 1: $$\displaystyle \sin(t) = \left( \frac{1+\cos(t)}{\sin(t)} \right) \cdot ( 1 - \cos(t) ).$$$BR
$BR Step 2: $$\displaystyle \sin(t) \sin(t) = (1+\cos(t))(1-\cos(t))$$$BR
$HR$BR
Finally, using algebra we may rewrite the equation from step 2 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

ANS( Formula("1-(cos(t))^2")->cmp() );

}

END_PROBLEM();

ENDDOCUMENT();


Part 3: