##DESCRIPTION
## Precalculus, Trigonometric Identities
##ENDDESCRIPTION

## KEYWORDS('precalculus','trigonometry','identities')
## Tagged by skm9b

## DBsubject('Precalculus')
## DBchapter('Analytic Geometry')
## DBsection('Trigonometric Identities')
## Date('')
## Author('')
## Institution('ASU')
## TitleText1('')
## EditionText1('')
## AuthorText1('')
## Section1('')
## Problem1('')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl",
"PGgraphmacros.pl",
"PGasu.pl"
);

TEXT(beginproblem());

$showPartialCorrectAnswers = 1;

$pi = 4*atan2(1,1);
$a=random(3,9,2);
$b=random($a+1,$a+2,1);

$cst=-$a/$b;

#evaluate answers

$bb=$b**2;
$d = $bb-$a**2;
$e = 2*$a**2-$bb;

$snt="-sqrt($d)/$b";
$sn2t="2*$a*sqrt($d)/$bb";
$cs2t="$e/$bb";
$snthalf=sqrt((1-$cst)/2);
$csthalf=-sqrt((1+$cst)/2);

TEXT(EV2(<<EOT));
If \( \cos(t)=-\frac{$a}{$b} \) where \( \pi < t < \frac{3\pi}{2} \), find the
values of the following trigonometric functions. 

$BR
$BBOLD Note: $EBOLD 
Give exact answers, do not use
decimal numbers. The answer should be a fraction or an arithmetic expression. 
If the answer involves a square root it should be enter as sqrt; e.g. the square
root of 2 should be written as sqrt(2).

$BR
$BR

\( \cos(2t) \) =
\{ans_rule(20) \} 
EOT

ANS(exact_no_trig($cs2t));
TEXT(EV2(<<EOT));
$BR
\( \sin(2t) \)= \{ans_rule(20) \} 
EOT
ANS(exact_no_trig($sn2t));
TEXT(EV2(<<EOT));
$BR
\( \cos(\frac{t}{2}) \)= \{ans_rule(20) \}  
EOT
ANS(exact_no_trig($csthalf));

TEXT(EV2(<<EOT));
$BR
\( \sin(\frac{t}{2}) \)= \{ans_rule(20) \} 
EOT

ANS(exact_no_trig($snthalf));
ENDDOCUMENT();        # This should be the last executable line in the problem.