Consider the limacon r $=24$ sin$($$)$ for $0$ $2$$,($a full period$)$

$($Q$1)$locate the $4$ vertical tangent lines.

$($Q$2)$ there is a self intersection at the pole $$ locate the values at which this occurs, and find the slope of the tangent line at respective for a parametrization $($$)=($f$($$)$cos$($$),$ f$($$)$sin$($$))$ with f$($$)=24$sin$($$),0$$2$$.$

$($Q$3)$ Calculate the area of the region bounded between the inner loop and the outer loop