The arm of a crane is $20m$ long. The angle of inclination of the boom of the crane has a minimum value of $$ and a maximum value of $.$

$($a$)$ PROVE that the exact value for the vertical displacement, $d,$ of the boom of the crane as it moves from its minimum to its maximum angles is given by

$d=20(sin-sin)$

$($b$)$ Find the exact value and the approximate value $($to one decimal place$)$ of this vertical displacement, if $=60$ and $=45.$