What you show will work fine, except that it needs the angle to be expressed in radians. That's not what most people think of first, so a conversion from degrees to radians is in order,

my $pi = 4 * atan2 1, 1;
while (1) {
    print "Angle? ";
    chomp ( my $angle = <STDIN>);
    last if $angle eq '';
    print sin( $angle * $pi / 180 ), $/;
}
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That is for fractional degrees expressed as decimal or float notation. Your users may think of dms notation, which I leave as an exercise ;-)

As you've been advised, Math::Trig may be very helpful.

After Compline,
Zaxo